1

A Gauss-Seidel Iteration Scheme forReference-Free 3-D Histological Image

ReconstructionSimone Gaffling?, Volker Daum, Stefan Steidl, Andreas Maier, Harald Köstler and

Joachim Hornegger, Member, IEEE

Abstract—Three-dimensional (3-D) reconstruction of histo-logical slice sequences offers great benefits in the investigationof different morphologies. It features very high-resolutionwhich is still unmatched by in-vivo 3-D imaging modalities,and tissue staining further enhances visibility and contrast.One important step during reconstruction is the reversal ofslice deformations introduced during histological slice prepa-ration, a process also called image unwarping. Most methodsuse an external reference, or rely on conservative stoppingcriteria during the unwarping optimization to prevent straight-ening of naturally curved morphology. Our approach showsthat the problem of unwarping is based on the superposition oflow-frequency anatomy and high-frequency errors. We presentan iterative scheme that transfers the ideas of the Gauss-Seidelmethod to image stacks to separate the anatomy from thedeformation. In particular, the scheme is universally applicablewithout restriction to a specific unwarping method, and usesno external reference. The deformation artifacts are effectivelyreduced in the resulting histology volumes, while the naturalcurvature of the anatomy is preserved. The validity of ourmethod is shown on synthetic data, simulated histology datausing a CT data set and real histology data. In the case of thesimulated histology where the ground truth was known, themean Target Registration Error (TRE) between the unwarpedand original volume could be reduced to less than 1 pixel onaverage after 6 iterations of our proposed method. 1

Index Terms—3-D Histology, Reconstruction, Registration,Reference-free, Gauss-Seidel

I. INTRODUCTION

A. Biomedical Motivation

Imaging modalities that are able to directly visualize 3-Dobjects, e.g., µ-CTs/MRs, only recently achieve resolutionvalues of about 0.7 µm. However, they still do not match

Asterisk indicates corresponding author.S. Gaffling (email: simone.gaffling@cs.fau.de), S. Steidl, A. Maier and J.

Hornegger are with the Pattern Recognition Lab, Department of Com-puter Science, Friedrich-Alexander-University of Erlangen Nuremberg(FAU), Martensstr. 3, 91058 Erlangen, Germany

V. Daum is with Chimaera GmbH, Am Weichselgarten 7, 91058 Erlan-gen, Germany

H. Köstler is with the Chair of System Simulation, Department ofComputer Science, Friedrich-Alexander-University of Erlangen Nurem-berg (FAU), Cauerstr. 11, 91058 Erlangen, Germany

S. Gaffling, A. Maier , H. Köstler and J. Hornegger are members of theGraduate School in Advanced Optical Technologies (SAOT), Paul-Gordan-Str. 6, 91052 Erlangen, Germany

1Copyright (c) 2010 IEEE. Personal use of this material is permit-ted. However, permission to use this material for any other purposesmust be obtained from the IEEE by sending a request to pubs-permissions@ieee.org.

the high resolution of conventional light microscopes,and lack the possibility to stain structures of interest toenhance the contrast and thus their visibility. Therefore,the investigation of histological image sequences to gaininsight into 3-D morphological structures is still a regulartask in biomedical laboratories.

Especially when the spatial relationship betweenanatomical structures and their progression has to beinvestigated, a significant disadvantage is that the spatialconnectivity of structures is lost during histological slicepreparation. Envisioning a 3-D structure from an imagesequence is challenging, especially for large data sets. Incontrast to that, looking at a 3-D object from differentangles helps to infer spatial relationships very fast. Inaddition, the extraction of quantitative values is ofteneasier and more reliably done using volumetric data.

A possibility to combine the advantages of 3-D imagingwith the staining and high-resolution properties of lightmicroscopy is to create a 3-D reconstruction of the originaltissue from the histological image sequence. Restoring theanatomy such that it closely recovers the original in-vivotissue sample can tremendously facilitate the perceptionof the morphology and spatial relations. As both theslice preparation and digitizing process introduce artifacts,however, merely stacking the 2-D images is not an option.The reconstruction routine – consisting of all processingsteps that are necessary to achieve the reconstruction – istherefore closely related to the histological slice prepara-tion process.

B. Histological Slice Preparation and Preprocessing

To generate a sequence of digital histology images,an extensive procedure is necessary. After extraction, thetissue is fixed and embedded into a synthetic material.Depending on the specific synthetic that is used, thisblock is then cut into slices, typically with a thickness ofabout 0.5− 20µm. The slice thickness is usually chosensuch that the structure of interest varies slowly withthe number of sections, to be able to investigate thestructure morphology. After the slices were placed on anobject plate, the embedding synthetic is removed. Then,the histological staining is applied, typically a series ofchemical treatments with histological dyes, alcohol anddistilled water at different temperatures. This procedure

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Figure 1: Left: Smooth, undisturbed volume rendering of the hippocampal region showing the pyramidal layer anddentate gyrus of a T2*-weighted MRI of a mouse brain [1]. Middle: Structures after stacking the rigidly aligned, butstill deformed histological slices [2]. Deformations (red/dashed) within the slice plane show low to moderate curvature,while the deformation of the anatomy along the stack corresponds to high-frequency errors. Right: Reconstructionresult using our approach. For references regarding color be referred to the web version of our article.

greatly enhances the visibility of the anatomical structuresof interest. Finally the stained slices are viewed under amicroscope, and digitized using a microscope-mountedcamera. Alternatively semi-automatic systems capable offocusing, capturing and stitching several partial images tothe final image covering the object of interest are used.

Unfortunately, each of the aforementioned steps intro-duces artifacts which degrade the quality of the resultingdigital images representing the original tissue slice. While,e.g., tissue shrinking during fixation and embedding is as-sumed to be small and therefore negligible, other artifactshave to be addressed to ensure that the 3-D reconstructionresult is as similar to the original in-vivo tissue sample aspossible. Therefore, most reconstruction methods includea preprocessing stage, linear alignment as well as a strategyto reverse the slice deformations.

Consequently, the first step is to normalize the slice in-tensities. Inconsistent lighting conditions during digitizinglead to an intensity bias on every image. They are usuallybrighter in the center region than at the borders of the im-age, which has to be corrected [3], [4]. Varying conditionsduring the staining procedure lead to differently distinctcolorings of the tissue slices, such that the same tissueclass shows a different color in different images. Differentprocedures were suggested to normalize these intensities[5], [6], [7].

The most severe artifacts, however, stem from the cut-ting procedure, as it introduces tissue deformations - orwarps - within the slice plane, and generally destroys thespatial relationship between slices and thus the connec-tivity of the original anatomy.

Therefore, one crucial step in reconstruction is theglobal alignment of the slices. Its task is to rigidly trans-form the sections such that translational offset and rota-tion differences between slice images are compensated.This will restore the global true shape of the original

tissue sample, which is a prerequisite for an accuratereconstruction. Global shape here means the large-scaleoverall form of the structure of interest, not accounting forthe smaller-scale distortions still contained in the images.In the absence of ground-truth reference information,there is unfortunately no guarantee that the calculatedglobal shape will represent the actual shape.

Still, there are different approaches how to achieve goodresults. First, the actual structure of interest is quite oftenembedded in surrounding tissue, which can regularizethe rigid registration of slices such that the global shapecorresponds to the real tissue shape without the need offurther correction. A second possibility is the use of anatlas image as external reference, which might be helpfulto recover a realistic global shape. Sometimes, fiducialmarkers were used for alignment. Finally, due to the lowernumber of degrees of freedom in rigid registration ascompared to non-rigid registration, manual refinementof an automatically created rigidly registered volume isalso an option. You can find more information about thiscrucial step in the references of section I-D

Our article focuses on a strategy for reversal of slicedeformations, also called image unwarping. We thereforeassume for the sake of this article that all previous steps,i.e., preprocessing and linear alignment, have already beentaken. For more detailed information about these steps,also refer to [8], [9], [10], [11] and references therein.

C. Problem Definition

To justify the approach we are taking for slice unwarp-ing, one has to take a closer look at the problem at hand.When cutting the tissue into slices, section thickness ischosen such that the structure of interest varies slowly withthe number of sections. If done otherwise, the shape of thestructure might be undersampled, and any investigationof the morphology obsolete. The natural morphological

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structure along the stack is therefore comparable to afunction with low or moderate spatial frequency, sampledat the slice locations.

In addition, depending on the robustness of the tissueand diligence during the preparation process, the slicepreparation leads to artifacts such as tears, foldings andgaps corresponding to sharp local deformations. While weincluded an elaboration of the impact and treatment ofthese defects later in this article, the focus of our work arethe more global, smooth in-plane deformations that areinevitably present in the final tissue slices.

As these smooth deformations are different for eachslice, merely stacking the rigidly registered slice im-ages leads to high-frequency disturbances of the originalanatomy along the stack. They present the main reasonlinearly aligned reconstructions appear jagged and discon-tinuous.

Along the stack, the slowly oscillating “anatomic func-tion” is basically superimposed with the highly oscillat-ing disturbances due to the slice deformations. Figure1 visually illustrates this fact. The actual problem onehas to solve in histological image unwarping is then toeliminate the high-frequency components from the defor-mation while preserving the low frequency componentsrepresenting the natural anatomy. This has to be achievedby unwarping the individual histology images, emulatingthe smooth in-plane deformation imposed during cutting.

D. Related Work

3-D reconstruction of slice images has a long tradi-tion, going back as far as 1883 [12]. The invention andsteady advancement of computer technology enabled thedevelopment of more and more sophisticated methodsfor spatial restoration of disassembled anatomical struc-tures. For an extensive review of the history and differentmethodologies developed for 3-D histological image re-construction see [8] and [13].

Many techniques rely on prior knowledge about theundeformed, original structure to guide the reconstruc-tions. How this knowledge is incorporated differs, and in-cludes mostly fiducial markers (e.g., in [14], [15]), blockfacephotographs (e.g., [16], [17]), or volume images acquiredusing 3-D imaging modalities (e.g., [18], [19]). While ex-ternal knowledge undeniably helps to restore the originalshape and connectivity, it also has to be stated that thisinformation is often not available. The reasons for this aremanifold.

Blockface photographs require the acquisition of an im-age of the unstained surface of the embedded tissue block.On the one hand this is tedious, as this has to be donefor each individual slice, interrupting the regular workflow.On the other hand, this is only possible for certain typesof staining, as it has to be applied on-the-fly on eachindividual slice after one slice is cut from the block. Prioracquisition of a volume image, e.g., from a µ-CT/µ-MR, isoften not possible as the devices are not available, andthe resolution is mostly still too small to visualize the

relevant structures. Apart from that, histological stainingmight be necessary to make the structures of interestvisible in the first place, which of course is not possible inthis case. As stated above, while the usage of an atlas asexternal reference may be sufficient to recover a realisticglobal shape, it certainly would not be appropriate forperforming a non-rigid correction. Last, there is a largepool of highly valuable historical slice sequences, wherenaturally a reference is not available. Any reconstructionmethod that aims to reconstruct one of these sequenceshas to be able to forgo such a reference.

There have been some attempts to achieve an anatom-ically sound reconstruction without the use of an ex-ternal reference. Braumann et al. [20] use a series ofthree registration steps for reconstruction. After initial rigidalignment, the slices are registered using a polynomialnon-linear registration, and later the alignment is refinedusing a curvature-based registration approach. Ju et al.[2] propose to calculate pairwise warps between adjacent,bilaterally filtered slices, and the final warp for an imageis then given by a binomially weighted sum of the warpsfrom that image to images in a certain neighborhood ofsize d . Wirtz et al. [21] optimize an objective function mea-suring the elastic potential of the deformation field, whichis extended for image stacks. Similarly, Schmitt et al. [8]achieve global alignment using a principal axes transformand optimizing rotation and shearing. This is followedby an elastic non-rigid registration routine. Chakravartyet al. developed two different approaches. In [22] theyused slice-to-slice non-linear registration, minimizing themean distance between segmented contours. Their secondapproach [23] extends their previous work, and similarly toJu et al. [2], uses an average deformation field calculatedfrom the deformations to the four predecessing and foursuccessing slices. Pitiot et al. [24], [25] use block-matchingto estimate a displacement field between neighboringimages. They then use regularization with adaptable rigid-ity, incorporating also the geometry and topology of theimages. The image stack is processed starting with amanually chosen reference slice. Bagci et al. [26] proposeto map the histological images into a feature space callededgeness space, and use elastic registration of the slices inthis feature space, starting with an automatically chosenbest reference slice. Scheibe et al. [27] use non-rigidregistration based on an intensity similarity measure andregularization of the deformation based on the opticalflow. Slices are registered to their respective predecessorin a forward-backward iteration scheme. Finally, Cifor etal. [13] propose a smoothness-driven method, calculating amin-max curvature flow restricted to 2-D planes. From theflow, arbitrarily flexible transformations can be calculated.

E. Contribution

First, we define the goal of histological image unwarpingto separate superimposed functions of different frequen-cies, as stated in section I-C. We introduce a non-rigidregistration scheme that transfers the iterative Gauss-Seidel method to images. That way we can reverse the

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artificially introduced slice deformations and restore thesmooth spatial connectivity of the tissue morphology,while preserving the natural curvature of structures. Inparticular, we achieve this without the use of an externalreference. While most approaches dealing with histologicalimage unwarping present a method for the unwarpingmethod itself, i.e., emulating the deformation, we insteadfocus on how this method has to be applied on an imagestack to achieve the desired result. The user is thereforenot bound to a specific type of non-rigid registration, butinstead is able to use whatever method works best for thedata at hand.

F. Outline

The article is organized as follows. In section II, wedescribe the employed methods we use for reference-freehistological image reconstruction. First, we explain thenon-rigid, non-parametric image registration method weuse for image unwarping in Section II-A. We then give ashort explanation of the iteration scheme and convergencebehavior of the Gauss-Seidel method, which our recon-struction scheme is based on, in Section II-B1. In SectionII-C, we transfer the previously described mathematicalconcepts into the domain of images and image registra-tion, and finalize the section with an algorithmic overviewof our approach. Section III describes the data and ex-periments that were used to evaluate our method, andshows qualitative and quantitative results on simulatedand real data. The article is concluded with a summaryand discussion in section IV.

II. METHODS

The unwarping strategy of an entire histological imagestack requires the reversal of the artificial deformation ofeach individual section. This process is guided by severalassumptions and requirements.

As stated before, one prerequisite for a truthful recon-struction is that the global shape of the original tissue wascorrectly recovered in the initial linear alignment step. Afailed linear alignment of the slices, e.g., a global rotationor tilt, corresponds to a low frequency error. Since ourmethod is specifically targeted at high frequency artifacts,it will not be able to restore errors of the global shape.

Assumptions regarding the slice deformations itself arethat they are smooth in accordance with the elasticityof organic material, are restricted to deformations withinthe plane, and deformations of one slice are independentfrom deformations of neighboring slices. An additionalrequirement is that the connectivity and run of anatomicalstructures along the stack is assumed to be smooth afterreversing the deformation of each individual slice. Andlast, the natural curvature of the anatomy along the stackhas to be preserved.

While the assumptions about the nature of the deforma-tions are mostly relevant for unwarping individual slices,the requirements of smooth progression of structures andpreservation of the natural curvature demand to take

into account the neighborhood of the sections that arecurrently processed, or even the entire stack of images, andtherefore require global optimization strategies. In fact, acommon and well-known problem in histological imagereconstruction is known as aperture or banana problem[8], [28], [2], [26]. It stems from the fact that individualtreatment of the slices according to the first and secondassumption – i.e., reversing the deformation within theslice plane such that the connectivity of structures alongthe stack is restored and smooth – often lead to resultsthat violate the third criterion, basically straightening thenatural curvature. Note that this effect can also occurduring the linear alignment of the slices, which is whythis step has to be performed with great care. Therefore,it is important to ensure that all assumptions and require-ments are simultaneously fulfilled, as this is the only wayto achieve an anatomically correct reconstruction of theentire histological image sequence.

A. In-plane Deformations

The most challenging part in histological image recon-struction is the reversal of the tissue slice deformations.Beside the cutting direction during slice preparation –and therefore a possible bias of the deformation in onedirection – the individual deformations of slices are con-sidered to be independent of each other. A commonly usedstrategy to reverse these deformations is 2-D non-rigidregistration. It can be used to compensate motion betweenimages such that corresponding content is mapped ontoeach other. The types of transforms, and the optimizationstrategies that are used, however, differ significantly, andare often tailored to the data at hand. Similar to [8],we will employ a non-rigid, non-parametric registrationformulation in a custom implementation[29].

Mathematically, the aim of a 2-D image registration is tofind a mapping u :R2→R2 between a reference image Rand a template image T , such that the deformed templateimage Tu(x) = T (x−u(x)) is similar to R . The similarity ofthe images is measured by a distance measure D, whichis minimal if the similarity is maximal. Additionally werequire the deformation u to be constrained in somesense, which in our case means we want it to be smoothaccording to the elastic deformation of the tissue.

The smoothness is measured by the regularizer R . Allin all, we thus want to solve

u∗ = argminu

D(R , Tu) +αR(u) :=NPREG (R , T ) , (1)

where α is a weighting parameter that decides whetherwe prefer a better match or a smoother deformation.Throughout this article, NPREG is used as an abbreviationfor non-rigid, non-parametric image registration betweenimages R and T .

As we only have to deal with mono-modal registrationproblems in this work, we employ the well known sum-of-squared differences (SSD) as distance measure. It isdefined as

DSSD(R , Tu) =1

|Ω |ˆΩ

(R (x)−Tu(x))2dx ,

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where Ω is the computational domain of the registrationand |Ω| its area. As regularizer we employ the curvatureregularization, popularized in [30]:

RCURV(u) =1

|Ω |ˆΩ

‖∆u(x) ‖22 dx .

The curvature regularizer minimizes the norm of thesecond order derivatives of the deformation field u, givenby the Laplace operator ∆. We choose the curvature regu-larizer, because it does not penalize affine transformations.This can be an advantage in the context of histologicalimage unwarping, as the non-linear transform betweenreference and template images might additionally containan affine component. For a more in-depth discussionplease refer to [31], [29] and [32].

This non-rigid registration method gives us a means togenerate a deformation field that is in general able toemulate (or therefore reverse) the smooth global defor-mations that were imposed on the slices during cutting.Although the individual slice deformations are deemedindependent from each other, as we do not have anexternal reference, we need to take the neighborhood intoaccount. The specific deformations can therefore only becalculated when the anatomical smoothness along thestack is considered.

B. Elimination of High-Frequency Errors

As was shown in Figure 1, the deformations within theslice plane lead to high-frequency disturbances of the orig-inally smooth structures along the stack. Mathematically,high frequencies correspond to high curvature, i.e., largevalues for the second derivative of a function. As stated insection I-C, the problem to solve is therefore to eliminatehigh-curvature components from a function where lowand high curvature components are superimposed. Toreverse this, our reconstruction scheme is based on aGauss-Seidel iteration scheme. We take advantage of thesmoothing property of Gauss-Seidel to eliminate the high-frequency deformations of our digital histology volumeto restore the original, smooth progression and curvatureof the in-vivo anatomical structures. We will first explainthe idea and properties we exploit on a fictional 1-Dmodel problem, introducing the iterative Jacobi, weightedJacobi and Gauss-Seidel methods, and later reformulatethe equations in terms of images and deformation fields.

1) Model Problem: In the following section we willdemonstrate the effect of the employed iterative solu-tion schemes on different frequency components of agiven function f when these methods are applied. Forthis purpose, we assume for the moment that the idealfunction f ? : [0, 1]→ R we would like to restore given aninitial function f has a minimal absolute curvature. In theideal case, this corresponds to a second-order differentialequation

f ?′′ (x ) = 0 0< x < 1 (2)

on a domain Ω ∈ (0, 1). For simplicity we assume Dirichletboundary conditions, i.e., boundary values that are knownand therefore fixed,

f (0) = f (1) = 0 .

The function domain Ω is first discretized into m subin-tervals of uniform width h = 1/m , resulting in m + 1 gridpoints xi = i ·h , i := i ∈N |0≤ i ≤m. At each of the m−1interior grid points xi , the second derivative of f can beexpressed by the central-difference approximation

f ′′ (xi ) =f (xi−1)−2 f (xi )+ f (xi+1)

h 2

!= 0 . (3)

Eliminating h 2 by multiplication, we solve each i -th equa-tion for the i -th unknown. Every discretized function valuecan therefore be expressed as linear combination of itsneighbors

f (xi ) =1

2

f (xi−1) + f (xi+1)

, (4)

and we can generate a vector of discrete values approxi-mating our function,

f =f (x1), f (x2), . . . , f (xm−1)

T=f1, f2, . . . , fm−1

T.

Starting with the discretized values f (0)i of given functionf , we can iteratively update these values using Eq. (4),which after t iterations are given by

f (t )i =1

2

f (t−1)

i−1 + f (t−1)i+1

. (5)

This can be interpreted such that for the current iterationneighboring values are assumed to be correct, and areused to calculate a better approximation for the grid pointin between. All values of f are either simultaneously orsubsequently updated, and the next iteration is started.This iterative scheme is a special form of the well-knownJacobi method [33].

A popular modification of this scheme is to use the up-dated value of Eq. (5) just as intermediate value f (t )i , i n t e r m .

The final update value f (t )i is then a weighted combinationof the old function value f (t−1)

i and this intermediate value,i.e.,

f (t )i = (1−ω) f (t−1)i +ω f (t )i , i n t e r m . (6)

Adjusting the weight ω can lead to faster or slower conver-gence, and can be tailored to the specific type of problemat hand.

To investigate the convergence and smoothing prop-erties, it is useful to look into the methods in matrixnotation. Multiplied with −1 for optimization reasons, Eq.(3) is equivalent to a system of linear equations

2 −1 0 · · ·−1 2 −1

0...

...−1 2

f1

f2...

fm−1

=

00...0

Af= 0.

The symmetric, positive definite Matrix A is decomposedinto its components - diagonal matrix D, lower and upper

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triangular matrices L and U - and the iteration scheme ofthe Jacobi method can be rewritten as

f(t ) =D−1 (L+U)︸ ︷︷ ︸R

f(t−1)

with iteration matrix R. For the weighted Jacobi method(Eq. 6), the iteration matrix changes to

Rω = (1−ω)I+ωR .

2) Smoothing and convergence: As the iterative Jacobiand Gauss-Seidel methods were invented in the early 19thcentury, their convergence and smoothing properties arewell understood, e.g., compare [34] and references therein.Therefore we will only provide a brief overview about themain parts of the theory related to our problem.

In general, the remaining error of our original modelproblem (2) after t iterations is defined as the differencebetween the ground-truth solution f? and the currentapproximation f(t )

e= f?− f(t ).

The iteration scheme says that a new approximation isgiven by

f(1) =Rf(0).

The real solution f? is a fixed point of the iteration process,i.e., f? = Rf?, which means that after convergence, furtheriterations do not change the solution anymore. The errorof the iteration scheme after the first iteration is thereforegiven by

e(1) =Re(0),

and after t iterations

e(t ) =Rt e(0). (7)

Note that the superscript t without brackets denotes anexponent rather than just indicating the iteration number,which is given in brackets.

Using, e.g., the L2 vector and matrix norms, one canshow that the error is bound by

‖e(t )‖2 ≤ ‖R‖t2 ‖e(0)‖2.

The method will converge for limt→∞Rt = 0, which is the case

if and only if the spectral radius of the matrix

ρ (R) := max1≤k≤m−1

|λk (R) |< 1,

that is, if the largest eigenvalue of the iteration matrixis smaller than 1. The eigenvalues of the Jacobi iterationmatrix R and weighted Jacobi method Rω are given by [34]

λk (R) = 1−2 sin2

kπ

2m

,

λk (Rω) = 1−2ωsin2

kπ

2m

,

for 1 ≤ k ≤ m − 1, and the k corresponding eigenvectorscan be given by their j -th components [35]

wk , j = sin

j kπ

m

, 1≤ k ≤m −1, 0≤ j ≤m .

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Af= 0.

The symmetric, positive definite Matrix A is decomposedinto its components - diagonal matrix D, lower and uppertriangular matrices L and U - and the iteration scheme ofthe Jacobi method can be rewritten as

f(t ) =D−1 (L+U)︸ ︷︷ ︸R

f(t−1)

with iteration matrix R. For the weighted Jacobi method(Eq. 6), the iteration matrix changes to

Rω = (1−ω)I+ωR .

2) Smoothing and convergence: As the iterative Jacobiand Gauss-Seidel methods were invented in the early 19thcentury, their convergence and smoothing properties arewell understood, e.g., compare [34] and references therein.Therefore we will only provide a brief overview about themain parts of the theory related to our problem.

In general, the remaining error of our original modelproblem (2) after t iterations is defined as the differencebetween the ground-truth solution f? and the currentapproximation f(t )

e= f?− f(t ).

The iteration scheme says that a new approximation isgiven by

f(1) =Rf(0).

The real solution f? is a fixed point of the iteration process,i.e., f? = Rf?, which means that after convergence, furtheriterations do not change the solution anymore. The errorof the iteration scheme after the first iteration is thereforegiven by

e(1) =Re(0),

and after t iterations

e(t ) =Rt e(0). (7)

Note that the superscript t without brackets denotes anexponent rather than just indicating the iteration number,which is given in brackets.

Using, e.g., the L2 vector and matrix norms, one canshow that the error is bound by

‖e(t )‖2 ≤ ‖R‖t2 ‖e(0)‖2.

The method will converge for limt→∞Rt = 0, which is the case

if and only if the spectral radius of the matrix

ρ (R) := max1≤k≤m−1

|λk (R) |< 1,

that is, if the largest eigenvalue of the iteration matrixis smaller than 1. The eigenvalues of the Jacobi iterationmatrix R and weighted Jacobi method Rω are given by [34]

λk (R) = 1−2 sin2

kπ

2m

,

λk (Rω) = 1−2ωsin2

kπ

2m

,

0

20

40

60

80

100

0 10 20 30 40 50 60

Weighted Jacobi, w = 23

Jacobi

Gauss-Seidel

Error reduction for different frequencies

Iter

atio

ns

Wavenumber k

Figure 2: Initial guesses, corresponding to the initial er-rors, are functions of different frequencies (given by wavenumber k ). Plotted is the number of iterations requiredto reduce the error by a factor of at least 100 for a modelproblem with m = 64. For the weighted Jacobi and Gauss-Seidel method, high frequencies (k ≥ m

2 ) are reducedrapidly, while low frequencies need a considerably largernumber of iterations to be reduced by the same factor.Adapted from [34].

for 1 ≤ k ≤ m − 1, and the k corresponding eigenvectorscan be given by their j -th components [35]

wk , j = sin

j kπ

m

, 1≤ k ≤m −1, 0≤ j ≤m .

The eigenvectors therefore simply correspond to differentfrequencies (or modes), depending on the so-called wavenumber k , which indicates the number of half-sine wavesthe eigenvectors consist of over the m +1 subintervals.

The error of our model problem can be expressed asweighted linear combination of these eigenvectors,

e(0) =m−1∑k=1

ck wk ,

with weights ck ∈R, and using Eq. 7, it can be rearrangedto see that after t iterations, the error is given by

e(t ) =m−1∑k=1

ck (λk )t wk .

The frequency component in the error given by wk istherefore reduced by a factor of (λk )

t after t iterations,and the smaller the eigenvalue, the faster this process is.Of interest here is especially the weighted Jacobi method.The proper choice of weight ω guarantees optimal conver-gence, and it can be shown that the ideal weight for thatpurpose is ω= 2

3 [34]. Then, about half of the eigenvalues –those corresponding to the high-frequency wave numbersk ≥ m

2 – are of low magnitude. These high frequenciesare therefore reduced rapidly, while lower frequencies aredamped slowly, cmp. also Fig. 2.

Figure 2: Initial guesses, corresponding to the initial er-rors, are functions of different frequencies (given by wavenumber k ). Plotted is the number of iterations requiredto reduce the error by a factor of at least 100 for a modelproblem with m = 64. For the weighted Jacobi and Gauss-Seidel method, high frequencies (k ≥ m

2 ) are reducedrapidly, while low frequencies need a considerably largernumber of iterations to be reduced by the same factor.Adapted from [34].

The eigenvectors therefore simply correspond to differentfrequencies (or modes), depending on the so-called wavenumber k , which indicates the number of half-sine wavesthe eigenvectors consist of over the m +1 subintervals.

The error of our model problem can be expressed asweighted linear combination of these eigenvectors,

e(0) =m−1∑k=1

ck wk ,

with weights ck ∈R, and using Eq. 7, it can be rearrangedto see that after t iterations, the error is given by

e(t ) =m−1∑k=1

ck (λk )t wk .

The frequency component in the error given by wk istherefore reduced by a factor of (λk )

t after t iterations,and the smaller the eigenvalue, the faster this process is.Of interest here is especially the weighted Jacobi method.The proper choice of weight ω guarantees optimal conver-gence, and it can be shown that the ideal weight for thatpurpose is ω= 2

3 [34]. Then, about half of the eigenvalues –those corresponding to the high-frequency wave numbersk ≥ m

2 – are of low magnitude. These high frequenciesare therefore reduced rapidly, while lower frequencies aredamped slowly, cmp. also Fig. 2.

3) 1-D Gauss-Seidel Method: The Gauss-Seidel methoddiffers from the Jacobi scheme outlined above in Eq. 5 suchthat it uses the already updated function value f (t )i−1 insteadof f (t−1)

i−1 . The resulting method is called Gauss-Seidel

77

-2

-1

0

1

2

0 20 40 60 80 100

-1

0

1

0 20 40 60 80 100

Initial 30Iterations: 5 10

Evolution of functions for Gauss-Seidel method

Figure 3: Evolution of functions when applying the Gauss-Seidel method on the 1-D model problem. Initial guessesconsisted of functions of different frequencies correspond-ing to wave number k = 10 (top row) and superimposedfrequencies with wave number k1 = 4, k2 = 20 (bottomrow). The evolution of the functions is shown for iterations5, 10 and 30.

3) 1-D Gauss-Seidel Method: The Gauss-Seidel methoddiffers from the Jacobi scheme outlined above in Eq. 5 suchthat it uses the already updated function value f (t )i−1 insteadof f (t−1)

i−1 . The resulting method is called Gauss-Seidelmethod, and the update scheme is modified accordinglyto

f (t )i =1

2

f (t )i−1+ f (t−1)

i+1

, (8)

and in matrix notation

f(t ) = (D−L)−1 U︸ ︷︷ ︸RG

f(t−1).

As outlined in [35], the eigenvalues and eigenvectorschange to

λk (RG ) = cos2

kπ

m

, 1≤ k ≤m −1

and

wk , j = [λk (RG )]j2 sin

j kπ

m

=cos

kπ

m

j

sin

j kπ

m

.

Although different frequencies are mixed here, the generalproperty that smooth modes are damped slowly, whilethe high-frequency modes are eliminated rapidly remainsunchanged for the Gauss-Seidel method [34]. This is alsodemonstrated in Fig. 2, where the Jacobi, weighted Jacobi(for the optimal weight w = 2

3 ) and the Gauss-Seidelmethod were applied to functions of different frequencies,defined by wave number k . Shown is the number ofiterations that are needed to reduce the error by a factorof at least 100.

C. Reformulation of Gauss-Seidel for Image Stacks

First, it is important to note again that – unlike assumedfor the model problem 2 – it is not our goal to minimizethe curvature such that f ” (x ) = 0. This would corre-spond to eliminating any curvature of structures in ouranatomical volume. Instead, we merely use the property ofGauss-Seidel to degrade high-frequency variances very fastwhile mostly preserving low frequencies to our advantage.Furthermore, applying the Gauss-Seidel method on imagesequences requires some adjustments of the operationsand the notation. A side-by-side view comparing the op-erations used for the 1-D model problem and as used inthe context of image sequences is depicted in Fig. 4.

1) Transfer of properties and operations to image se-quences: The discrete function values fi are from nowon replaced by the histological images Ii . As statedabove, minimizing the curvature now means that thehigh-frequency disturbances perpendicular to the sliceplanes should be eliminated, while preserving the lowerfrequency progression of the anatomical structures alongthe stack. More important, however, is that the differenceor offset di , j between function values - which in the 1-Dcase could easily be calculated by subtraction of neighbor-ing function values - has to be defined for images. Thisrequires more in-depth discussion, as the application ofthe subtraction operator to digital images imposes certainconstraints.

The offset di , j modifies a function value f j such thatit is most similar (or equal) to another function valuefi . For our histology images, this offset is defined as thedeformation field u(x , y ) = u (x) as defined in section II-A,relocating the pixels of one image I j such that it is mostsimilar to another image Ii , Ii = I j (x−u (x)) =: I j u. Incontrast to the 1-D real case, however, where correcting afunction value by the offset lets it assume an exactly cal-culated value, a real image transformed by a deformationfield will never exactly look like the image it was registeredto. This is because the deformation is restricted such thatit mimics the deformations that are imposed on the tissueslices during cutting.

2) Iteration scheme: To express the Gauss-Seidel itera-tive update scheme in Eq. 8 such that it can be transferredto images and deformation fields, we expand and reformu-late it,

f (t )i =1

2

f (t )i−1+ f (t−1)

i+1

=1

2

f (t )i−1+ f (t )i−1− f (t )i−1+ f (t−1)

i+1

=1

2

f (t )i−1+ f (t )i−1+

f (t−1)

i+1 − f (t )i−1

︸ ︷︷ ︸

:=di−1,i+1

=1

2

2 · f (t )i−1+di−1,i+1

= f (t )i−1+1

2di−1,i+1, (9)

Figure 3: Evolution of functions when applying the Gauss-Seidel method on the 1-D model problem. Initial guessesconsisted of functions of different frequencies correspond-ing to wave number k = 10 (top row) and superimposedfrequencies with wave number k1 = 4, k2 = 20 (bottomrow). The evolution of the functions is shown for iterations5, 10 and 30.

method, and the update scheme is modified accordinglyto

f (t )i =1

2

f (t )i−1+ f (t−1)

i+1

, (8)

and in matrix notation

f(t ) = (D−L)−1 U︸ ︷︷ ︸RG

f(t−1).

As outlined in [35], the eigenvalues and eigenvectorschange to

λk (RG ) = cos2

kπ

m

, 1≤ k ≤m −1

and

wk , j = [λk (RG )]j2 sin

j kπ

m

=cos

kπ

m

j

sin

j kπ

m

.

Although different frequencies are mixed here, the generalproperty that smooth modes are damped slowly, whilethe high-frequency modes are eliminated rapidly remainsunchanged for the Gauss-Seidel method [34]. This is alsodemonstrated in Fig. 2, where the Jacobi, weighted Jacobi(for the optimal weight w = 2

3 ) and the Gauss-Seidelmethod were applied to functions of different frequencies,defined by wave number k . Shown is the number ofiterations that are needed to reduce the error by a factorof at least 100.

C. Reformulation of Gauss-Seidel for Image Stacks

First, it is important to note again that – unlike assumedfor the model problem 2 – it is not our goal to minimize

the curvature such that f ” (x ) = 0. This would corre-spond to eliminating any curvature of structures in ouranatomical volume. Instead, we merely use the property ofGauss-Seidel to degrade high-frequency variances very fastwhile mostly preserving low frequencies to our advantage.Furthermore, applying the Gauss-Seidel method on imagesequences requires some adjustments of the operationsand the notation. A side-by-side view comparing the op-erations used for the 1-D model problem and as used inthe context of image sequences is depicted in Fig. 4.

1) Transfer of properties and operations to imagesequences: The discrete function values fi are from nowon replaced by the histological images Ii . As statedabove, minimizing the curvature now means that thehigh-frequency disturbances perpendicular to the sliceplanes should be eliminated, while preserving the lowerfrequency progression of the anatomical structures alongthe stack. More important, however, is that the differenceor offset di , j between function values - which in the1-D case could easily be calculated by subtraction ofneighboring function values - has to be defined forimages. This requires more in-depth discussion, as theapplication of the subtraction operator to digital imagesimposes certain constraints.

The offset di , j modifies a function value f j such thatit is most similar (or equal) to another function valuefi . For our histology images, this offset is defined as thedeformation field u(x , y ) = u (x) as defined in section II-A,relocating the pixels of one image I j such that it is mostsimilar to another image Ii , Ii = I j (x−u (x)) =: I j u. Incontrast to the 1-D real case, however, where correcting afunction value by the offset lets it assume an exactly cal-culated value, a real image transformed by a deformationfield will never exactly look like the image it was registeredto. This is because the deformation is restricted such thatit mimics the deformations that are imposed on the tissueslices during cutting.

2) Iteration scheme: To express the Gauss-Seidel itera-tive update scheme in Eq. 8 such that it can be transferredto images and deformation fields, we expand and reformu-late it,

f (t )i =1

2

f (t )i−1+ f (t−1)

i+1

=1

2

f (t )i−1+ f (t )i−1− f (t )i−1+ f (t−1)

i+1

=1

2

f (t )i−1+ f (t )i−1+

f (t−1)

i+1 − f (t )i−1

︸ ︷︷ ︸

:=di−1,i+1

=1

2

2 · f (t )i−1+di−1,i+1

= f (t )i−1+1

2di−1,i+1, (9)

where di , j denotes the difference between the current bestapproximations of the function values fi and f j , whichin R1 corresponds to a simple subtraction di , j = f j − fi .

8

8

(11)

I j (x) = Ii (x) ui j

= Ii

x−ui j (x)

=

di j := f j − fi

y

x

Image Sequence

di j ∈R1, e.g., di j = 5

SubtractionDefinition of Difference

Application

Gauss-Seidel Update

Non-rigid Registration

x

y

y

xf5

f4

f3

f2

f1

f (x ) ∈R1

1-D Real Function

I1 I2 I3 I4 I5

I (x) ∈Ru×v

ui j (x) :=NPREGIi , I j

ui j (x) ∈Ru×v×2, e.g.,

Ii I jui j

f j = fi +di j

di jf j

fi

f (t−1)i

f (t )i = f (t )i−1+12 di−1,i+1

12 di−1,i+1f (t )i−1

f (t−1)i+1

Iz = I (t )z−1

12 ·uz−1,z+1

I (t )zI (t−1)zIz

I (t )z = I (t−1)z uz ,z

IzI (t )z−1 I (t−1)z I (t−1)

z+1

uz−1,z+1

uz ,z

(9)

(12)

Figure 4: Comparison of corresponding components andoperations for 1-D model problem (left column) and imagesequence (right column) when the Gauss-Seidel methodis applied. Row 1, Input: in case of the model problem asequence of discretized function values fi ∈R1, and in ourcase an image sequence Ii ∈ Ru×v . Row 2, difference di j

between two input elements: calculated by subtraction inthe model problem. For images, the element transformingone image into the other image – corresponding to thedifference – is given by the deformation field u i j cal-culated using non-rigid registration. Row 3, applicationof difference di j : by simple addition in fi ∈ R1, while adeformation field to transform image Ii to image I j isapplied via resampling. Row 4, Gauss-Seidel update: forthe model problem achieved by relocating the currentvalue to the midpoint between its neighbors (by addinghalf the difference between these neighbors). In our case,an image is updated by transforming it into the meanshape of its neighbors, which first requires the creationof an interpolated intermediate image Iz .

where di , j denotes the difference between the current bestapproximations of the function values fi and f j , whichin R1 corresponds to a simple subtraction di , j = f j − fi .The current function value f (t )i is given by the previousfunction value, plus half the distance from that to thefollowing function value, i.e., simple linear interpolation.

Equation (9) basically says that the new updated valueis given by the preceding value, plus half the differencebetween the preceding and succeeding value. In terms ofimages and deformation fields, this update is equivalent tochanging an image by half the deformation transformingthe preceding image to the succeeding image, please referalso to Fig. 4.

Therefore, the Gauss-Seidel update formula as appliedto images is given by

I (t )z = I (t )z−1

1

2·uz−1,z+1

(10)

Here, Iz (x , y ) =: Iz is the digital image at position z ofthe image sequence. As detailed above, the differenceuz−1,z+1

x , y

=: uz−1,z+1 – effectively the corresponding

concept to the difference di , j of the model problem – isdefined as the deformation field that transforms imageI (t−1)

z+1 into image I (t )z−1, and is obtained using non-rigidregistration between these two images. Unfortunately thiswould discard any tissue information that was originallycontained in slice Iz which is currently updated. Thereforeanother adjustment has to be made to ensure that allanatomical information that is contained in the sliceimages is used for reconstruction. The t th approximationof image Iz is not given by merely deforming the previousimage, as indicated in Eq. (10), but instead this approx-imation is used as interpolated intermediate image Iz

(t ),

as described in [36]. This intermediate image is then usedas reference, to which the original histology image is non-rigidly registered. In this way, the anatomy contained inthe original slice image Iz is preserved, but deformed tomatch the interpolated image calculated by Gauss-Seideliteration in Eq. (10).

The iteration scheme is therefore modified to

Iz = I (t )z−1

1

2·uz−1,z+1

(11)

with Iz(t )

indicating the artificial, interpolated image usingthe neighboring images of the t -th iteration, and the actualnew image at position z is given by

I (t )z = I (t−1)z uz ,z (12)

where uz ,z is the deformation field between the originalimage at position z of iteration t −1 and the interpolatedimage Iz .

3) Boundary conditions: In general, boundary condi-tions specify the values or the behavior of the solutionto a (partial) differential equation at the boundaries ofthe domain. Two of the most commonly used types ofboundary conditions are Dirichlet boundary conditions,which define specific function values of the solution at

Figure 4: Comparison of corresponding components andoperations for 1-D model problem (left column) and imagesequence (right column) when the Gauss-Seidel methodis applied. Row 1, Input: in case of the model problem asequence of discretized function values fi ∈R1, and in ourcase an image sequence Ii ∈ Ru×v . Row 2, difference di j

between two input elements: calculated by subtraction inthe model problem. For images, the element transformingone image into the other image – corresponding to thedifference – is given by the deformation field u i j cal-culated using non-rigid registration. Row 3, applicationof difference di j : by simple addition in fi ∈ R1, while adeformation field to transform image Ii to image I j isapplied via resampling. Row 4, Gauss-Seidel update: forthe model problem achieved by relocating the currentvalue to the midpoint between its neighbors (by addinghalf the difference between these neighbors). In our case,an image is updated by transforming it into the meanshape of its neighbors, which first requires the creationof an interpolated intermediate image Iz .

The current function value f (t )i is given by the previousfunction value, plus half the distance from that to thefollowing function value, i.e., simple linear interpolation.

Equation (9) basically says that the new updated valueis given by the preceding value, plus half the difference

between the preceding and succeeding value. In terms ofimages and deformation fields, this update is equivalent tochanging an image by half the deformation transformingthe preceding image to the succeeding image, please referalso to Fig. 4.

Therefore, the Gauss-Seidel update formula as appliedto images is given by

I (t )z = I (t )z−1

1

2·uz−1,z+1

(10)

Here, Iz (x , y ) =: Iz is the digital image at position z ofthe image sequence. As detailed above, the differenceuz−1,z+1

x , y

=: uz−1,z+1 – effectively the corresponding

concept to the difference di , j of the model problem – isdefined as the deformation field that transforms imageI (t−1)

z+1 into image I (t )z−1, and is obtained using non-rigidregistration between these two images. Unfortunately thiswould discard any tissue information that was originallycontained in slice Iz which is currently updated. Thereforeanother adjustment has to be made to ensure that allanatomical information that is contained in the sliceimages is used for reconstruction. The t -th approximationof image Iz is not given by merely deforming the previousimage, as indicated in Eq. (10), but instead this approx-imation is used as interpolated intermediate image Iz

(t ),

as described in [36]. This intermediate image is then usedas reference, to which the original histology image is non-rigidly registered. In this way, the anatomy contained inthe original slice image Iz is preserved, but deformed tomatch the interpolated image calculated by Gauss-Seideliteration in Eq. (10).

The iteration scheme is therefore modified to

Iz = I (t )z−1

1

2·uz−1,z+1

(11)

with Iz(t )

indicating the artificial, interpolated image usingthe neighboring images of the t -th iteration, and the actualnew image at position z is given by

I (t )z = I (t−1)z uz ,z (12)

where uz ,z is the deformation field between the originalimage at position z of iteration t −1 and the interpolatedimage Iz .

3) Boundary conditions: In general, boundary condi-tions specify the values or the behavior of the solutionto a (partial) differential equation at the boundaries ofthe domain. Two of the most commonly used types ofboundary conditions are Dirichlet boundary conditions,which define specific function values of the solution atthe boundary, or Neumann boundary conditions, impos-ing constraints on the derivative of the function at theboundary. In both cases, these values or derivatives areknown in advance.

In our case, the function values at the boundary of ourdomain, i.e., the corrected and unwarped images at thebeginning and the end of the image stack, are unknown. Aswe do not use any prior information via reference volumes,we also cannot reliably assume any specific behavior of

9

the “anatomic function” at these positions. In addition,the images at the boundaries of the image stack also haveto be unwarped during the optimization routine.

To achieve this, our domain is enlarged at both endsby one image respectively. Which kind of image is useddepends on the boundary condition one wants to assume.

Dirichlet boundary conditions correspond to copyingone image - e.g., the original histological image prior tounwarping - as fixed boundary value. This works wellif it is known that the images at the borders do notshow strong deformations. After t iterations, the prob-lem domain then consists of the updated images andthe original boundary images as boundary conditions,Ω(t ) :=

I (0)1 =: I (t )0 , I (t )1 , I (t )2 , . . . , I (t )Z , I (t )Z+1 := I (0)Z

. However, a

mostly artifact-free image can generally not be guaranteed,and in extreme cases such an outlier could adversely affecta significant portion at the end of the stack.

To at least dampen the effect of such a possible outlierslice, we choose to use Neumann boundary conditions.Here, the derivative of the function is set to zero. Thisis achieved by copying the last updated version of theboundary images of the image stack as new boundarycondition images after each iteration of the optimizationroutine. After t iterations, the problem domain is thengiven by Ω(t ) :=

I (t )1 =: I (t )0 , I (t )1 , I (t )2 , . . . , I (t )Z , I (t )Z+1 := I (t )Z

.

4) Convergence: Regarding the convergence, note thatthe theoretical convergence behavior in practice is affectedby the regularization of the deformation within the sliceplane x-y and the severity of the deformations. Conver-gence will therefore be slower than what theory predicts.In addition, the assumption of uniform subintervals -equivalent to equally spaced histology slices - might notbe fulfilled, since tissue slices often get lost or are severelydisrupted such that they are not suitable to be used forreconstruction. Just removing such an image from thestack is not an option, since this would compress theanatomy at this position. Instead, we use interpolation ofslices as outlined in [36] to fill the gap. This restores theuniformity of the subintervals, and therefore the conver-gence requirements outlined above.

5) Algorithm: To prevent repeated propagation of pos-sible defects in the slices along the stack, the Gauss-Seidel scheme is iteratively applied to the stack, with alter-nating directions (also known as symmetric Gauss-Seidelmethod). This also effectively reduces any bias effect thatnon-rigid registration methods often show when they areapplied, depending on the direction along the stack. Aftera first update of the entire stack, including the boundarycondition images, the scheme is repeated in reverse order,starting with the images that were updated last in theprevious iteration. The algorithm for our proposed imageunwarping strategy is depicted in Algorithm 1. To preventartifacts from repeated interpolation during the updatestep, we accumulate all calculated update deformationfields for a specific image I j into one deformation fieldu(t )j , a c c u = u(t−1)

j ... u(0)j , which is finally applied to the

original image I (0)j to generate the image I (t )j after t

Algorithm 1 Proposed method for image unwarping usinga Gauss-Seidel iteration scheme and Neumann boundaryconditions.

Input

I (0)1 , ..., I (0)Z histological image sequenceN number of iterations

Expand image domain by Neumann b.c.I (0)0 ← I (0)1 , I (0)Z+1← I (0)Z

for iteration t = 1, ..., N

for image j = 1, ..., Z

u j−1, j+1 =NPREGI (t )j−1, I (t−1)

j+1

I j = I (t−1)j+1

12 ·u j−1, j+1

u? =NPREGI j , I (t−1)

j

u(t )j , a c c u = u(t−1)j , a c c u u?

I (t )j = I (0)j u(t )j , a c c u updateend

I (t )0 ← I (t )1 , I (t )Z+1← I (t )Z

reverse sequenceI (t )0 , I (t )1 , ..., I (t )Z , I (t )Z+1

end

Output

I (N )1 , ..., I (N )Z unwarped image sequence

iterations. As was already mentioned before, this algorithmcan be used with other registration methods tailoredto the reconstruction problem at hand. Note howeverthat our algorithm requires the scaling and compositionof calculated slice transforms. If the chosen registrationmethod does not directly offer this possibility, we proposeto create a dense deformation field from the calculatedslice transformation, which should be possible regardlessof the choice of deformation model.

III. EVALUATION AND RESULTS

To evaluate the performance of our reconstructionmethod, it is tested on synthetic and real data sets. Eachof the experiments was designed to evaluate a differentcharacteristic of our method.

The applicability of the Gauss-Seidel method in thecontext of images and deformation fields, and the conver-gence and smoothing properties for different frequenciesis demonstrated using synthetic slice data sets. The sec-ond experiment uses a CT slice sequence that was non-rigidly transformed to quantify the error reduction overthe number of iterations for more complex data for whichthe ground truth is known. Finally, we show the qualityof our reconstructions using a real histology data set. Wecompared our results to atlas and MR images, and recon-struction results by [2], who provide their reconstructionsonline.

10

(a) Wave number k = 10.

(b) Superimposed wave numbers k1 = 4 and k2 = 20.

Figure 5: Volume renderings of data sets with differentfrequencies, defined by the wave number k , evolving overtime. Shown are the original data sets, and after 5, 10, 15,20, 25 and 30 iterations. Note the similarity of the resultsto the results shown for the 1-D model problem (fig. 3).

A. Synthetic Data

Every data set consists of n = 100 images of size 121×121. A white circle with a diameter of 21 pixels is placed onblack background. The circle center changes its positionalong the image sequence according to a sinusoidal func-tion. Starting at the first image with center

xc , yc

, the

coordinates of the circle center with coordinatesxc c , yc c

on slice z is given by

xc c , yc c

z=xc , yc

−a · sin

z kπ

n

.

Here, a denotes the amplitude of the sinusoid, andwas chosen as a = 10. k denotes the wave numberof the desired frequency, which corresponds to k halfsine waves along the entire slice sequence. This emu-lates a smoothly varying anatomical structure. Severaldata sets were prepared in this manner, with frequenciesk = 4, 10, 20, 40, 60, 80, 90. In addition, data sets with twosuperimposed frequencies - one low frequency of k1 = 4and a high frequency with k2 = 20 resp. k2 = 40 weregenerated using

xc c , yc c

z=xc , yc

−a ·sin

z k1π

n

+ sin

z k2π

n

.

To measure the effect of the Gauss-Seidel registrationscheme on the different frequency components in theslice volumes, the mean squared error (MSE) between therespective volumes and a reference volume consisting ofa centered cylinder is calculated. As the largest part of theconvergence happens during the first iterations, the MSEwas calculated for each of the first 10 iterations, and every

10

is demonstrated using synthetic slice data sets. The sec-ond experiment uses a CT slice sequence that was non-rigidly transformed to quantify the error reduction overthe number of iterations for more complex data for whichthe ground truth is known. Finally, we show the qualityof our reconstructions using a real histology data set. Wecompared our results to atlas and MR images, and recon-struction results by [2], who provide their reconstructionsonline.

(a) Wave number k = 10.

(b) Superimposed wave numbers k1 = 4 and k2 = 20.

Figure 5: Volume renderings of data sets with differentfrequencies, defined by the wave number k , evolving overtime. Shown are the original data sets, and after 5, 10, 15,20, 25 and 30 iterations. Note the similarity of the resultsto the results shown for the 1-D model problem (fig. 3).

A. Synthetic Data

Every data set consists of n = 100 images of size 121×121. A white circle with a diameter of 21 pixels is placed onblack background. The circle center changes its positionalong the image sequence according to a sinusoidal func-tion. Starting at the first image with center

xc , yc

, the

coordinates of the circle center with coordinatesxc c , yc c

on slice z is given by

xc c , yc c

z=xc , yc

−a · sin

z kπ

n

.

Here, a denotes the amplitude of the sinusoid, andwas chosen as a = 10. k denotes the wave numberof the desired frequency, which corresponds to k halfsine waves along the entire slice sequence. This emu-lates a smoothly varying anatomical structure. Severaldata sets were prepared in this manner, with frequenciesk = 4, 10, 20, 40, 60, 80, 90. In addition, data sets with twosuperimposed frequencies - one low frequency of k1 = 4

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

0 5 10 15 20 25 30

Cylinder volume= 4/20= 4/40= 4

= 10= 20= 40= 60= 80

Convergence of synthetic sinusoidal data sets

Iterations

Mea

nSq

uar

edE

rro

r

for different wave numbers k

kk

k

kkkkk

Figure 6: Reduction of MSE for volumes of different fre-quencies. The horizontal dashed line represents the MSEbetween the cylindrical volume and a rigidly registeredversion of itself that was shifted by 2.5 pixels in x and y .

and a high frequency with k2 = 20 resp. k2 = 40 weregenerated using

xc c , yc c

z=xc , yc

−a ·sin

z k1π

n

+ sin

z k2π

n

.

To measure the effect of the Gauss-Seidel registrationscheme on the different frequency components in theslice volumes, the mean squared error (MSE) between therespective volumes and a reference volume consisting ofa centered cylinder is calculated. As the largest part of theconvergence happens during the first iterations, the MSEwas calculated for each of the first 10 iterations, and every5th after that, for a total of 30 iterations. You can see thedifferences in convergence for volumes with different wavenumbers in fig. 6. To help interpret the magnitude of theMSE values, the MSE between the original volume and arigidly registered shifted version of the original volume (by2.5 pixels in x and y ) is plotted as horizontal line.

Figure 5 visually shows volume renderings after severaliterations for two different data sets. The first one wascreated with a moderate wave number k = 10. The fre-quency is gradually decreased, and still slightly visible after30 iterations. The second data shows two superimposedfrequencies with wave numbers k1 = 4 and k2 = 20. Here,the high frequency component is eliminated after about 15iterations, while the low-frequency component is mostlypreserved even after 30 iterations. The slightly flattenedappearance at the upper and lower ends of this dataset is an effect of the boundary condition, which kindof propagates the zero derivative condition into the dataset. However, this experiment quantitatively as well asvisually confirms the theory that the low frequencies aremostly preserved, while the high frequencies are efficientlyeliminated, even when frequencies are superimposed. Inaddition, we could show that the Gauss-Seidel method isapplicable in the domain of images and deformation fields,

Figure 6: Reduction of MSE for volumes of different fre-quencies. The horizontal dashed line represents the MSEbetween the cylindrical volume and a rigidly registeredversion of itself that was shifted by 2.5 pixels in x and y .

5th after that, for a total of 30 iterations. You can see thedifferences in convergence for volumes with different wavenumbers in fig. 6. To help interpret the magnitude of theMSE values, the MSE between the original volume and arigidly registered shifted version of the original volume (by2.5 pixels in x and y ) is plotted as horizontal line.

Figure 5 visually shows volume renderings after severaliterations for two different data sets. The first one wascreated with a moderate wave number k = 10. The fre-quency is gradually decreased, and still slightly visible after30 iterations. The second data shows two superimposedfrequencies with wave numbers k1 = 4 and k2 = 20. Here,the high frequency component is eliminated after about 15iterations, while the low-frequency component is mostlypreserved even after 30 iterations. The slightly flattenedappearance at the upper and lower ends of this dataset is an effect of the boundary condition, which kindof propagates the zero derivative condition into the dataset. However, this experiment quantitatively as well asvisually confirms the theory that the low frequencies aremostly preserved, while the high frequencies are efficientlyeliminated, even when frequencies are superimposed. Inaddition, we could show that the Gauss-Seidel method isapplicable in the domain of images and deformation fields,while featuring the same smoothing properties as for realfunctions.

To justify our decision to use the Gauss-Seidel method,we repeated the experiment on three data sets represent-ing low (k = 4/20), medium (k = 10) and high frequency(k = 80) functions using the Jacobi method instead ofGauss-Seidel. Here, the calculation of the interpolatedintermediate slice Iz in Eq. 11 depends not on the alreadyupdated slice I (t )z−1, but on the previous iteration t−1, I (t−1)

z−1 .Beside this difference, all other parts of the algorithmwere identical to our experiment using the Gauss-Seidel

1111

0

1000

2000

3000

4000

5000

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k = 4/20 Jac

k = 10 GSk = 10 Jac

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Figure 7: Reduction of MSE for selected volumes, re-constructed using the Gauss-Seidel method (GS) and theJacobi method (Jac).

while featuring the same smoothing properties as for realfunctions.

To justify our decision to use the Gauss-Seidel method,we repeated the experiment on three data sets represent-ing low (k = 4/20), medium (k = 10) and high frequency(k = 80) functions using the Jacobi method instead ofGauss-Seidel. Here, the calculation of the interpolatedintermediate slice Iz in Eq. 11 depends not on the alreadyupdated slice I (t )z−1, but on the previous iteration t−1, I (t−1)

z−1 .Beside this difference, all other parts of the algorithmwere identical to our experiment using the Gauss-Seidelmethod.

Figure 7 shows a comparison of the convergence usingboth methods. Especially for the high-frequency data setk = 80, the Gauss-Seidel method converges significantlyfaster than the Jacobi method in the first iterations, justas theory predicts, cmp. Fig. 2. For all experiments, theMSE values using the Gauss-Seidel method are lower thanthe MSE values using the Jacobi method. However, thedifference between both methods in the later iterationsis not very substantial, being about 2%. Since our goal isto eliminate the high-frequency errors as fast as possible,using the Gauss-Seidel method is a natural choice, andwas therefore exclusively used for the other experiments.If for some reasons the Jacobi method is used, e.g., dueto simpler parallelization of the method, which will bediscussed in section III-E – the resulting reconstructionswill show a similar quality as those created using theGauss-Seidel method.

B. Synthetic Histology Data

To show the efficacy of our approach on more complexdata, we generated a synthetic histological slice sequenceusing a CT data set of an armadillo [37]. A subset of100 CT images, cropped and resized to 464× 387 pixels,were individually transformed with non-rigid transforms.As other artifacts typical for digital histology images are

Figure 8: Synthetic CT data set with biased deformations.Rows 1-4: Deformed data set, and after 5, 10 and 15iterations. Bottom row: Original CT data.

not present, the reconstruction quality independently ofthe preprocessing methods - including the quality of thelinear alignment prior to unwarping - can be evaluated.

The deformations for each image were synthesized usingB-Splines with random offsets at the B-Spline grid points.The maximum dislocation at a grid point was restrictedto ±10 pixels, which was empirically chosen to yielddeformations that were visually similar to real histologicalimages. Two experiments were implemented. First, tendifferent versions of the deformed CT data set were createdusing the mentioned method (these volumes will be calledunbiased volumes).

Second, the process generating the random deforma-tions was adapted to emphasize offsets in one direction.This corresponds more to real histology data, which ismore likely deformed along the cutting direction. Again,ten different volumes were generated, which will be de-noted biased volumes. The first experiment thus enablesto quantify the error stemming from the reconstructionalgorithm independently from a deformation bias. Thesecond experiment then demonstrates the effect of thebias on the reconstruction result.

Qualitatively, as can be seen in Figure 8, the deforma-tions are immensely reduced after 5 iterations, and after15 iterations there are barely differences to the original CTdata set visible.

Since the ground truth deformation is known in thisexperiment, the Target Registration Error (TRE) as definedby Fitzpatrick et al. [38] was calculated after each iteration.The TRE denotes the difference between the true locationof a pixel before it was dislocated using the generateddeformations, and the location of that pixel after ourunwarping strategy was applied, which ideally should bezero. In case of the datasets that were deformed by abiased random deformation, the bias leads to the anatomybeing shifted in a certain direction on average. As this

Figure 7: Reduction of MSE for selected volumes, re-constructed using the Gauss-Seidel method (GS) and theJacobi method (Jac).

method.Figure 7 shows a comparison of the convergence using

both methods. Especially for the high-frequency data setk = 80, the Gauss-Seidel method converges significantlyfaster than the Jacobi method in the first iterations, justas theory predicts, cmp. Fig. 2. For all experiments, theMSE values using the Gauss-Seidel method are lower thanthe MSE values using the Jacobi method. However, thedifference between both methods in the later iterationsis not very substantial, being about 2%. Since our goal isto eliminate the high-frequency errors as fast as possible,using the Gauss-Seidel method is a natural choice, andwas therefore exclusively used for the other experiments.If for some reasons the Jacobi method is used, e.g., dueto simpler parallelization of the method, which will bediscussed in section III-E – the resulting reconstructionswill show a similar quality as those created using theGauss-Seidel method.

B. Synthetic Histology Data

To show the efficacy of our approach on more complexdata, we generated a synthetic histological slice sequenceusing a CT data set of an armadillo [37]. A subset of100 CT images, cropped and resized to 464× 387 pixels,were individually transformed with non-rigid transforms.As other artifacts typical for digital histology images arenot present, the reconstruction quality independently ofthe preprocessing methods - including the quality of thelinear alignment prior to unwarping - can be evaluated.

The deformations for each image were synthesized usingB-Splines with random offsets at the B-Spline grid points.The maximum dislocation at a grid point was restrictedto ±10 pixels, which was empirically chosen to yielddeformations that were visually similar to real histologicalimages. Two experiments were implemented. First, tendifferent versions of the deformed CT data set were created

Figure 8: Synthetic CT data set with biased deformations.Rows 1-4: Deformed data set, and after 5, 10 and 15iterations. Bottom row: Original CT data.

using the mentioned method (these volumes will be calledunbiased volumes).

Second, the process generating the random deforma-tions was adapted to emphasize offsets in one direction.This corresponds more to real histology data, which ismore likely deformed along the cutting direction. Again,ten different volumes were generated, which will be de-noted biased volumes. The first experiment thus enablesto quantify the error stemming from the reconstructionalgorithm independently from a deformation bias. Thesecond experiment then demonstrates the effect of thebias on the reconstruction result.

Qualitatively, as can be seen in Figure 8, the deforma-tions are immensely reduced after 5 iterations, and after15 iterations there are barely differences to the original CTdata set visible.

Since the ground truth deformation is known in thisexperiment, the Target Registration Error (TRE) as definedby Fitzpatrick et al. [38] was calculated after each iteration.The TRE denotes the difference between the true locationof a pixel before it was dislocated using the generateddeformations, and the location of that pixel after ourunwarping strategy was applied, which ideally should bezero. In case of the datasets that were deformed by abiased random deformation, the bias leads to the anatomybeing shifted in a certain direction on average. As thisglobal shift is not relevant for the reconstruction, we haveto subtract the global mean shift µutrue

from the groundtruth deformation. In addition we ignore pixels belongingto the background (air, pixels outside the volume and fieldof view). The background values are determined by simplethresholding. The resulting mask M(x,y) which is 0 for allbackground pixels and 1 otherwise is used together withthe mean shift to calculate the TRE as

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Figure 9: Mean and standard deviation of the Target Regis-tration Error TRE, calculated for the two sets of deformedCT data sets. For more details, please refer to the text.

global shift is not relevant for the reconstruction, we haveto subtract the global mean shift µutrue

from the groundtruth deformation. In addition we ignore pixels belongingto the background (air, pixels outside the volume and fieldof view). The background values are determined by simplethresholding. The resulting mask M(x,y) which is 0 for allbackground pixels and 1 otherwise is used together withthe mean shift to calculate the TRE as

TRE=1

n

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Mx , y

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where n denotes the number of pixels in the volume,uz ,true and uz ,calc denote the ground truth deformation andcalculated deformation of slice image z respectively, andMx , y

is a binary mask image with an entry 1 if there is

anatomy at the corresponding location and 0 otherwise.In this way, we get the mean offset from the ground truthlocation per pixel for a given volume.

Figure 9 shows the mean and standard deviation of theTarget Registration Error, calculated for the two experi-ments. The TRE converges very fast to an average offsetof about 1 pixel. The reduction in the TRE shows that atleast for the deformed CT data sets our method is able toreverse a high amount of the actual deformations in thedata sets.

Since the ground truth deformation of histology datasets is usually not known, we additionally quantify theachieved smoothness of the reconstructed volume. To dothis, one inevitably has to compare neighboring pixelsand evaluate their similarity. Therefore in theory, everysimilarity measure used in image registration can be usedfor this purpose (examples of measures used for thispurpose are cross-correlation [39] and Sum of SquaredDistance [21]). Due to their similarity to the optimizationcriterion in our non-rigid registration objective function,these measures are inappropriate for evaluation purposes.

Convergence of Mean GLCM Contrast for CT data

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Figure 10: Mean and standard deviation of the GLCMcontrast feature, calculated for the two sets of deformedCT data sets.

Another popular measure is the Correspondence Align-ment Measure (CAM,[40], [26]), which assumes that in anideal reconstruction, a specific point lies exactly at themidpoint between its two neighbors. However, this is alsobasically what we optimize via the Gauss-Seidel Measureand additionally requires the correct identification of cor-responding points. Last, the calculation of overlap wasused before, e.g., in [39], which requires the segmentationof the structures of interest

Instead, we choose the contrast feature calculated fromthe gray-level co-occurrence matrices (GLCMs) as intro-duced by Haralick et al. [41] to quantify the achievedsmoothness of the reconstructions, similar to [39] and [13].Thereby we are independent from correspondence searchand segmentation issues, and avoid a measure too similarto our optimization criterion.

The GLCM contrast feature measures the amount ofcontrast in an image by investigating the frequency ofcertain intensity pairings of neighboring pixels. Beforereconstruction, neighboring pixels in axial direction (alongthe stack) more likely show high intensity differences – thatis, higher contrast – as opposed to after reconstruction(cmp. top row and bottom row of Figure 8). Decreasingcontrast therefore corresponds to a better match of neigh-boring slices, and consequently to smoother morphologi-cal structures along the stack.

For a given volume, we add up the values calculated foreach volume slice extracted in axial direction. Figure 10shows the decrease of the contrast GLCM measure over theiterations. It shows that our unwarping strategy effectivelyrestores anatomical smoothness for the volume, mostlyover the first few iterations.

As we have already stated in the introduction, a smoothreconstruction does not guarantee an anatomically cor-rect reconstruction. Curved structures might be effectivelystraightened by the reconstruction method, leading toextremely good smoothness values, while at the same

Figure 9: Mean and standard deviation of the Target Regis-tration Error TRE, calculated for the two sets of deformedCT data sets. For more details, please refer to the text. 12

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Figure 9: Mean and standard deviation of the Target Regis-tration Error TRE, calculated for the two sets of deformedCT data sets. For more details, please refer to the text.

global shift is not relevant for the reconstruction, we haveto subtract the global mean shift µutrue

from the groundtruth deformation. In addition we ignore pixels belongingto the background (air, pixels outside the volume and fieldof view). The background values are determined by simplethresholding. The resulting mask M(x,y) which is 0 for allbackground pixels and 1 otherwise is used together withthe mean shift to calculate the TRE as

TRE=1

n

∑x ,y ,z

Mx , y

· uz ,true

x , y

−µutrue−uz ,calc

x , y

2

where n denotes the number of pixels in the volume,uz ,true and uz ,calc denote the ground truth deformation andcalculated deformation of slice image z respectively, andMx , y

is a binary mask image with an entry 1 if there is

anatomy at the corresponding location and 0 otherwise.In this way, we get the mean offset from the ground truthlocation per pixel for a given volume.

Figure 9 shows the mean and standard deviation of theTarget Registration Error, calculated for the two experi-ments. The TRE converges very fast to an average offsetof about 1 pixel. The reduction in the TRE shows that atleast for the deformed CT data sets our method is able toreverse a high amount of the actual deformations in thedata sets.

Since the ground truth deformation of histology datasets is usually not known, we additionally quantify theachieved smoothness of the reconstructed volume. To dothis, one inevitably has to compare neighboring pixelsand evaluate their similarity. Therefore in theory, everysimilarity measure used in image registration can be usedfor this purpose (examples of measures used for thispurpose are cross-correlation [39] and Sum of SquaredDistance [21]). Due to their similarity to the optimizationcriterion in our non-rigid registration objective function,these measures are inappropriate for evaluation purposes.

Convergence of Mean GLCM Contrast for CT data

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Figure 10: Mean and standard deviation of the GLCMcontrast feature, calculated for the two sets of deformedCT data sets.

Another popular measure is the Correspondence Align-ment Measure (CAM,[40], [26]), which assumes that in anideal reconstruction, a specific point lies exactly at themidpoint between its two neighbors. However, this is alsobasically what we optimize via the Gauss-Seidel Measureand additionally requires the correct identification of cor-responding points. Last, the calculation of overlap wasused before, e.g., in [39], which requires the segmentationof the structures of interest

Instead, we choose the contrast feature calculated fromthe gray-level co-occurrence matrices (GLCMs) as intro-duced by Haralick et al. [41] to quantify the achievedsmoothness of the reconstructions, similar to [39] and [13].Thereby we are independent from correspondence searchand segmentation issues, and avoid a measure too similarto our optimization criterion.

The GLCM contrast feature measures the amount ofcontrast in an image by investigating the frequency ofcertain intensity pairings of neighboring pixels. Beforereconstruction, neighboring pixels in axial direction (alongthe stack) more likely show high intensity differences – thatis, higher contrast – as opposed to after reconstruction(cmp. top row and bottom row of Figure 8). Decreasingcontrast therefore corresponds to a better match of neigh-boring slices, and consequently to smoother morphologi-cal structures along the stack.

For a given volume, we add up the values calculated foreach volume slice extracted in axial direction. Figure 10shows the decrease of the contrast GLCM measure over theiterations. It shows that our unwarping strategy effectivelyrestores anatomical smoothness for the volume, mostlyover the first few iterations.

As we have already stated in the introduction, a smoothreconstruction does not guarantee an anatomically cor-rect reconstruction. Curved structures might be effectivelystraightened by the reconstruction method, leading toextremely good smoothness values, while at the same

Figure 10: Mean and standard deviation of the GLCMcontrast feature, calculated for the two sets of deformedCT data sets.

where n denotes the number of pixels in the volume,uz ,true and uz ,calc denote the ground truth deformation andcalculated deformation of slice image z respectively, andMx , y

is a binary mask image with an entry 1 if there is

anatomy at the corresponding location and 0 otherwise.In this way, we get the mean offset from the ground truthlocation per pixel for a given volume.

Figure 9 shows the mean and standard deviation of theTarget Registration Error, calculated for the two experi-ments. The TRE converges very fast to an average offsetof about 1 pixel. The reduction in the TRE shows that atleast for the deformed CT data sets our method is able toreverse a high amount of the actual deformations in thedata sets.

Since the ground truth deformation of histology datasets is usually not known, we additionally quantify the

achieved smoothness of the reconstructed volume. To dothis, one inevitably has to compare neighboring pixelsand evaluate their similarity. Therefore in theory, everysimilarity measure used in image registration can be usedfor this purpose (examples of measures used for thispurpose are cross-correlation [39] and Sum of SquaredDistance [21]). Due to their similarity to the optimizationcriterion in our non-rigid registration objective function,these measures are inappropriate for evaluation purposes.Another popular measure is the Correspondence Align-ment Measure (CAM,[40], [26]), which assumes that in anideal reconstruction, a specific point lies exactly at themidpoint between its two neighbors. However, this is alsobasically what we optimize via the Gauss-Seidel Measureand additionally requires the correct identification of cor-responding points. Last, the calculation of overlap wasused before, e.g., in [39], which requires the segmentationof the structures of interest

Instead, we choose the contrast feature calculated fromthe gray-level co-occurrence matrices (GLCMs) as intro-duced by Haralick et al. [41] to quantify the achievedsmoothness of the reconstructions, similar to [39] and [13].Thereby we are independent from correspondence searchand segmentation issues, and avoid a measure too similarto our optimization criterion.

The GLCM contrast feature measures the amount ofcontrast in an image by investigating the frequency ofcertain intensity pairings of neighboring pixels. Beforereconstruction, neighboring pixels in axial direction (alongthe stack) more likely show high intensity differences – thatis, higher contrast – as opposed to after reconstruction(cmp. top row and bottom row of Figure 8). Decreasingcontrast therefore corresponds to a better match of neigh-boring slices, and consequently to smoother morphologi-cal structures along the stack.

For a given volume, we add up the values calculated foreach volume slice extracted in axial direction. Figure 10shows the decrease of the contrast GLCM measure over theiterations. It shows that our unwarping strategy effectivelyrestores anatomical smoothness for the volume, mostlyover the first few iterations.

As we have already stated in the introduction, a smoothreconstruction does not guarantee an anatomically cor-rect reconstruction. Curved structures might be effectivelystraightened by the reconstruction method, leading toextremely good smoothness values, while at the sametime introducing strong unnatural deformations to theanatomy. However, our experiments show a strong cor-relation of the GLCM measure with the TRE. The fact thatour method is able to approximate the true deformationsof the slices confirms the intuition that using our method,smoother looking data sets are indeed closer to the realanatomy.

C. Histology Data

Finally, we applied our method to a Nissl-stained dataset of a mouse brain, available online from Ju et al. [2]. The

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Figure 11: Convergence of the GLCM contrast featurecalculated for the successive versions of the image stackover the iterations, for two different deformation models.

time introducing strong unnatural deformations to theanatomy. However, our experiments show a strong cor-relation of the GLCM measure with the TRE. The fact thatour method is able to approximate the true deformationsof the slices confirms the intuition that using our method,smoother looking data sets are indeed closer to the realanatomy.

C. Histology Data

Finally, we applied our method to a Nissl-stained dataset of a mouse brain, available online from Ju et al. [2]. Thedata set consists of 350 coronal, cryo-sectioned and Nissl-stained histological images with a resolution of 850× 670pixels, 25 µm slice thickness and 15 µm per pixel.

Ju et al. provide an unaligned, original data set withoutintensity normalization but homogeneous illumination, aswell as a rigidly registered and normalized version of thedata, and their reconstruction result.

As this article deals with the reversal of the local defor-mations of the slices, we applied our method to the alignedand intensity-normalized data set. The normalization ofthe slice intensities also enabled us to employ the sum-of-squared differences (SSD) measure in our experiment.It is faster to calculate than, e.g., the multimodal measureMutual Information, which would normally be appropriatein cases with inter-slice intensity differences, and experi-ments using the MI delivered similar results, while takingmore computation time. Optimal values for the weightingfactor of the regularizer were empirically chosen, as wasthe number of 10 iterations for our method.

To show that our initial statement – that our methodworks independently from the chosen non-rigid registra-tion method – is valid, we performed a second reconstruc-tion of the mouse brain data set. Instead of our presentednon-parametric, non-rigid registration method, we useda registration method using a parametric B-Spline based

deformation model as proposed by Rueckert et al. [42] andimplemented in the Insight Toolkit [43].

Figure 11 shows the Convergence of the GLCM contrastfeature values, calculated for successive versions of theimage stack over the iterations for both experiments.

In both cases, the contrast values decrease significantlyin the first iterations, indicating an increase in contoursmoothness. The slightly slower convergence of the B-Spline-based registration can be attributed to the lessflexible deformation in this specific setup as compared toour non-parametric approach.

To compare our result with other state-of-the-art imag-ing modalities, we created virtual slices through the re-constructed volume. Figure 12 shows a direct comparisonof virtual sections to an atlas image from Paxino’s Atlas[44], and Fig. 13 to a T1-weighted MR volume from theWaxholm space [1].

Furthermore, we compared our reconstruction result toJu’s reconstruction, which is also provided online. As youcan see in Figure 14, the brain structures in general appearmore clearly, and boundaries are significantly smootherin our reconstruction. In addition, the fiber-like structuresthat are barely visible on the left are clearly visible withour method, but are also visible in the MR volume 13. Thisis most likely due to the calculation of the deformation onthe bilaterally filtered images, which in this case preventsmatching of the connected structures. For comparison,the corresponding slices extracted from the rigidly alignedinput stack are shown in the top row. Even more profoundis the comparison of the original and warped coronalsections, cmp. Figure 15. The warp filtering approachclearly introduces unnatural deformations to the slice,whereas the unwarped slice using our approach appearsunaffected by artificial deformations.

1) Effect of slice defects on reconstruction result: Asmentioned already, due to the severe mechanical stressthat is imposed on the tissue during slice preparation andstaining – e.g., cutting, relocation, heating, etc. – and de-pending on the robustness of the tissue that is processed,the final slice images can show different artifacts, whichinevitably have an influence on the reconstruction result.

How severe the effect of the defects on the reconstruc-tion result is depends on the size of the defect, and theregularization of the allowed deformation in the non-rigidregistration method. Smaller defects, e.g., small tears orthe bubble-like defects visible in the brain data set, donot show significant effects in our experiments. The reasonis usually that the change in the calculated deformationfield due to the defect is so local that the regularizationterm – which emphasizes smooth non-local deformations– prevents said deformation.

If the defect is large (e.g., large parts missing from theslice, or larger folds), the corresponding image parts haveenough weight in the similarity measure part of the non-rigid registration objective function so that unwanted de-formations are introduced as far as the regularizer allowsthem. Such a defect should therefore always be treated be-fore reconstruction. Strategies for this include eliminating

Figure 11: Convergence of the GLCM contrast featurecalculated for the successive versions of the image stackover the iterations, for two different deformation models.

data set consists of 350 coronal, cryo-sectioned and Nissl-stained histological images with a resolution of 850× 670pixels, 25 µm slice thickness and 15 µm per pixel.

Ju et al. provide an unaligned, original data set withoutintensity normalization but homogeneous illumination, aswell as a rigidly registered and normalized version of thedata, and their reconstruction result.

As this article deals with the reversal of the local defor-mations of the slices, we applied our method to the alignedand intensity-normalized data set. The normalization ofthe slice intensities also enabled us to employ the sum-of-squared differences (SSD) measure in our experiment.It is faster to calculate than, e.g., the multimodal measureMutual Information (MI), which would normally be appro-priate in cases with inter-slice intensity differences. Exper-iments using the MI delivered similar results, while takingmore computation time. Optimal values for the weightingfactor of the regularizer were empirically chosen, as wasthe number of 10 iterations for our method.

To show that our initial statement – that our methodworks independently from the chosen non-rigid registra-tion method – is valid, we performed a second reconstruc-tion of the mouse brain data set. Instead of our presentednon-parametric, non-rigid registration method, we useda registration method using a parametric B-Spline baseddeformation model as proposed by Rueckert et al. [42] andimplemented in the Insight Toolkit [43].

Figure 11 shows the convergence of the GLCM contrastfeature values, calculated for successive versions of theimage stack over the iterations for both experimentalsetups.

In both cases, the contrast values decrease significantlyin the first iterations, indicating an increase in contoursmoothness. The slightly slower convergence of the B-Spline-based registration can be attributed to the lessflexible deformation in this specific setup as compared toour non-parametric approach.

To compare our result with other state-of-the-art imag-ing modalities, we created virtual slices through the re-constructed volume. Figure 12 shows a direct comparisonof virtual sections to an atlas image from Paxino’s Atlas[44], and Fig. 13 to a T1-weighted MR volume from theWaxholm space [1].

Furthermore, we compared our reconstruction result toJu’s reconstruction, which is also provided online. As youcan see in Figure 14, the brain structures in general appearmore clearly, and boundaries are significantly smootherin our reconstruction. In addition, the fiber-like structuresthat are barely visible on the left are clearly visible withour method, but are also visible in the MR volume 13. Thisis most likely due to the calculation of the deformation onthe bilaterally filtered images, which in this case preventsmatching of the connected structures. For comparison,the corresponding slices extracted from the rigidly alignedinput stack are shown in the top row. Even more profoundis the comparison of the original and warped coronalsections, cmp. Figure 15. The warp filtering approachclearly introduces unnatural deformations to the slice,whereas the unwarped slice using our approach appearsunaffected by artificial deformations.

1) Effect of slice defects on reconstruction result: Asmentioned already, due to the severe mechanical stressthat is imposed on the tissue during slice preparation andstaining – e.g., cutting, relocation, heating, etc. – and de-pending on the robustness of the tissue that is processed,the final slice images can show different artifacts, whichinevitably have an influence on the reconstruction result.

How severe the effect of the defects on the reconstruc-tion result is depends on the size of the defect, and theregularization of the allowed deformation in the non-rigidregistration method. Smaller defects, e.g., small tears orthe bubble-like defects visible in the brain data set, donot show significant effects in our experiments. The reasonis usually that the change in the calculated deformationfield due to the defect is so local that the regularizationterm – which emphasizes smooth non-local deformations– prevents said deformation.

If the defect is large (e.g., large parts missing from theslice, or larger folds), the corresponding image parts haveenough weight in the similarity measure part of the non-rigid registration objective function so that unwanted de-formations are introduced as far as the regularizer allowsthem. Such a defect should therefore always be treated be-fore reconstruction. Strategies for this include eliminatingthe slice if the loss of anatomical information is tolerable,repairing by complete or partial interpolation of the defectslice, cmp. [45],[36],[27], or by using artifact-tolerant non-rigid registration methods [46], [47]. For medium-sizeddefects, masking the defect image parts in the registrationmight also be sufficient to achieve a satisfactory result.

If such a large defect is overlooked, however, the result-ing unwanted deformations will be propagated along thestack, spreading to the left and right of the original defectslice, an effect demonstrated in Figure 16.

We performed an experiment illustrating this effect.

14

Figure 12: Left: Virtual section through the reconstructed volume. Right: Real histology slice from Paxino’s Atlas [44]

(a) (b)

Figure 13: a) Virtual section through the reconstructed volume. b) Sagittal section of a T1-weighted MR volume fromthe Waxholm space [1].

Here, we manually erased a significant amount of thetissue of slice 30, simulating a larger defect like it mightoccur in histological slice sequences. When unwarpingthe volumes once with the corrupt slice, the effect ofpropagated distortions of the tissue is clearly visible ascompared to the original reconstruction, cmp. Figure 16.After 6 iterations, the effect was visible from slices 21 to39. When replacing the defective slice with an interpolatedone [36] before applying our method, the morphology inthe reconstructed image stack again seems to be intact,with only small differences to the original reconstruction.

D. Complexity and Convergence

Usually histological image reconstruction is not a time-critical process. The time that is needed for image unwarp-ing mostly depends on the non-rigid registration methodthat is used, and the image size. The Gauss-Seidel schemewe employ is generally independent from the type of non-rigid registration method that is used, but determines thenumber of registration operations that are needed. For adata set consisting of K images (excluding the boundarycondition images) and using N iterations, the number ofnon-rigid registrations is given by

#(NPREG) = 2 ·K ·N .

15

Figure 14: Virtual axial (left) and sagittal (right) sections. Top: After rigid alignment. Middle: Reconstruction from Ju etal. [2]. Bottom: Our approach. Inset: Detail image from Paxino’s Atlas (plate 110) [44] showing the fiber bundles in thecaudoputamen region.

16

Figure 15: Visualization of the effect of in-plane deformations used for unwarping. Note the middle section of theimages. Left: From Ju et al. [2]. Middle: Original histology slice. Right: Our proposed method.

Figure 16: Visualization of the effect larger defects have on the reconstruction result after 6 iterations. Left: Ground truthreconstruction with intact slice 30. Middle: Distorted reconstruction for artificially destroyed slice 30. Lines indicateboundaries of the distorted region. Right: Reconstruction result using an interpolated replacement for slice 30 (topright).

Depending on the data set at hand, this number can getquite large. The reconstruction of the mouse brain dataset with its 350 slice images and 10 iterations required intotal 7000 registrations. For the image size of 850 × 670pixels, one non-rigid registration needed in average 19.7seconds to finish on a 3.2 GHz workstation with 6 GBRAM. The total time to reverse the deformations of theentire image stack therefore was 38:24:11 hh:m:s. Anotheradvantage of our method is that due to the sequential typeof processing, the memory requirements are low, even forvery highly resolved histology images.As with any fixed-point iteration method, the decisionof when to stop the iteration process also arises in ourmethod. Here, we propose to investigate the convergenceof the method as stopping criterion. A fixed-point iterationmethod is converged, when the change of the calculatedsolution from one to the next iteration is smaller than

a – usually empirically chosen – constant. In our case,one could calculate the MSD between successive versionsof the image stack over the iterations, cmp., e.g., Figure17, and falls below a certain limit. Another possibility is,of course, to let an anatomical expert decide when thereconstruction result is satisfactory.

E. Parallelization

Due to the potentially high number of images andtherefore registration operations, the parallelization of ourmethod would potentially lead to a significant accelerationof the reconstruction process. Since the update of a valueusing the Jacobi method depends only on the predecessorand successor of the same iteration number, paralleliza-tion is more straight-forward here. One strategy is, e.g., toassign equal parts of the data set to different processors.After updating the images within the blocks, only the

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Figure 17: MSD between successive versions of the im-age stack over the iterations. The first value denotes theMSD between the rigidly registered volume, and the firstiteration.

E. Parallelization

Due to the potentially high number of images andtherefore registration operations, the parallelization of ourmethod would potentially lead to a significant accelerationof the reconstruction process. Since the update of a valueusing the Jacobi method depends only on the predecessorand successor of the same iteration number, paralleliza-tion is more straight-forward here. One strategy is, e.g., toassign equal parts of the data set to different processors.After updating the images within the blocks, only theimages at the boundaries of the different blocks have tobe updated, which can also be done in parallel. Since theupdate of a value in the classical Gauss-Seidel methoduses an already updated predecessor, parallelization isnot as obvious. Still, it is possible using a method calledRed-Black Gauss-Seidel, in which the data is alternatelydistributed into two disjoint subsets, red and black. Withineach of the subsets, the individual update operationsare independent of each other, and can therefore beperformed by several processors [34].

IV. SUMMARY

We could show that the iterative Gauss-Seidel methodis applicable for image unwarping of histological slicesequences. The methods’ property of quickly eliminatinghigh-frequency disturbances and degrading low-frequencycomponents only very slowly is well suited to reversethe deformations due to slice cutting, while preservingthe naturally smooth curvature of anatomical structures.The final quality of the reconstruction is comparable tostate-of-the-art technologies like atlases and MR devices.Furthermore, as we showed in our experiments, our ap-proach can be applied independently from the non-rigidregistration method that is used, and is therefore highlyversatile.

In principle, both the Jacobi and Gauss-Seidel methodscan be used. While the dampening of high-frequencyartifacts is faster for the Gauss-Seidel method during thefirst iterations, the Jacobi method might offer benefits asit is easier to parallelize than Gauss-Seidel.

However, the reconstruction result strongly depends onthe quality of the global alignment. The overall shape ofthe tissue stack will not be altered much during unwarp-ing, and therefore has to be verified in advance. This isusually achieved using rigid registration. If corrections ofan automatically achieved alignment are required, manualrefinement is still an option, which in contrast to that isprohibitive for image unwarping due to the high numberof degrees of freedom that would have to be definedmanually in this case.

Furthermore, our method assumes that the low-frequency components of the unwanted artificial defor-mations still have a higher frequency than the high-frequency components of the anatomical deformations.Especially extremely curved structures will be subject tosome straightening, although in most cases this effectshould not be of high relevance.

There are always at least partly defective images con-tained in a histological slice sequence. These defects neg-atively affect the deformation fields calculated for imageunwarping, e.g., by “pulling” tissue into regions where thetissue is missing naturally or due to cutting artifacts. Thiscan then be propagated to neighboring images, leadingto unnaturally deformed parts in a certain neighborhoodaround the defect. This might be prevented using repairingor replacement strategies before unwarping.

To conclude, our approach is able to effectively reversethe deformations imposed on histological tissue sectionsduring cutting. Its versatility enables easy adaption to thespecific task at hand. Due to the sequential processingand the possibilities for parallelization, it is also suitablefor very highly-resolved histology images and image se-quences consisting of a large number of slices. Thereforeit offers a valuable contribution for the anatomically soundreconstruction of histological image sequences.

ACKNOWLEDGMENT

We thank Ju et al. for providing the histology data setand their reconstructions online. T1 and T2* data setcourtesy of the Duke Center for In Vivo Microscopy, anNIH/NIBIB Biomedical Technology Resource Center (P41EB015897). We also thank Chimaera GmbH (Erlangen) forproviding the registration libraries. The authors gratefullyacknowledge funding of the Erlangen Graduate School inAdvanced Optical Technologies (SAOT) by the GermanNational Science Foundation (DFG) in the framework ofthe excellence initiative.

REFERENCES

[1] G. A. Johnson, A. Badea, J. Brandenburg, G. Cofer, B. Fubara, S. Liu,and J. Nissanov, “Waxholm space: An image-based reference forcoordinating mouse brain research,” NeuroImage, vol. 53, no. 2, pp.365–372, Nov. 2010.

Figure 17: MSD between successive versions of the im-age stack over the iterations. The first value denotes theMSD between the rigidly registered volume, and the firstiteration.

images at the boundaries of the different blocks have tobe updated, which can also be done in parallel. Since theupdate of a value in the classical Gauss-Seidel methoduses an already updated predecessor, parallelization isnot as obvious. Still, it is possible using a method calledRed-Black Gauss-Seidel, in which the data is alternatelydistributed into two disjoint subsets, red and black. Withineach of the subsets, the individual update operationsare independent of each other, and can therefore beperformed by several processors [34].

IV. SUMMARY

We could show that the iterative Gauss-Seidel methodis applicable for image unwarping of histological slicesequences. The methods’ property of quickly eliminatinghigh-frequency disturbances and degrading low-frequencycomponents only very slowly is well suited to reversethe deformations due to slice cutting, while preservingthe naturally smooth curvature of anatomical structures.The final quality of the reconstruction is comparable tostate-of-the-art technologies like atlases and MR devices.Furthermore, as we showed in our experiments, our ap-proach can be applied independently from the non-rigidregistration method that is used, and is therefore highlyversatile.

In principle, both the Jacobi and Gauss-Seidel methodscan be used. While the dampening of high-frequencyartifacts is faster for the Gauss-Seidel method during thefirst iterations, the Jacobi method might offer benefits asit is easier to parallelize than Gauss-Seidel.

However, the reconstruction result strongly depends onthe quality of the global alignment. The overall shape ofthe tissue stack will not be altered much during unwarp-ing, and therefore has to be verified in advance. This isusually achieved using rigid registration. If corrections of

an automatically achieved alignment are required, manualrefinement is still an option, which in contrast to that isprohibitive for image unwarping due to the high numberof degrees of freedom that would have to be definedmanually in this case.

Furthermore, our method assumes that the low-frequency components of the unwanted artificial defor-mations still have a higher frequency than the high-frequency components of the anatomical deformations.Especially extremely curved structures will be subject tosome straightening, although in most cases this effectshould not be of high relevance.

There are always at least partly defective images con-tained in a histological slice sequence. These defects neg-atively affect the deformation fields calculated for imageunwarping, e.g., by “pulling” tissue into regions where thetissue is missing naturally or due to cutting artifacts. Thiscan then be propagated to neighboring images, leadingto unnaturally deformed parts in a certain neighborhoodaround the defect. This might be prevented using repairingor replacement strategies before unwarping.

To conclude, our approach is able to effectively reversethe deformations imposed on histological tissue sectionsduring cutting. Its versatility enables easy adaption to thespecific task at hand. Due to the sequential processingand the possibilities for parallelization, it is also suitablefor very highly-resolved histology images and image se-quences consisting of a large number of slices. Thereforeit offers a valuable contribution for the anatomically soundreconstruction of histological image sequences.

ACKNOWLEDGMENT

We thank Ju et al. for providing the histology data setand their reconstructions online. T1 and T2* data setcourtesy of the Duke Center for In Vivo Microscopy, anNIH/NIBIB Biomedical Technology Resource Center (P41EB015897). We also thank Chimaera GmbH (Erlangen) forproviding the registration libraries. The authors gratefullyacknowledge funding of the Erlangen Graduate School inAdvanced Optical Technologies (SAOT) by the GermanNational Science Foundation (DFG) in the framework ofthe excellence initiative.

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