Direct Surface Extraction from 3D Freehand Ultrasound Imagesdpai/papers/ZhRoPa02.pdfmodels were...

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Direct Surface Extraction from 3D Freehand Ultrasound Images Youwei Zhang * Robert Rohling Dinesh K. Pai The University of British Columbia ABSTRACT This paper presents a new technique for the extraction of surfaces from 3D ultrasound data. Surface extraction from ultrasound data is challenging for a number of reasons including noise and artifacts in the images and non-uniform data sampling. A method is proposed to fit an approximating radial basis function to the group of data samples. An explicit surface is then obtained by iso-surfacing the function. In most previous 3D ultrasound research, a pre-processing step is taken to interpolate the data into a regular voxel array and a corresponding loss of resolution. We are the first to represent the set of semi-structured ultrasound pixel data as a single function. From this we were able to extract surfaces without first reconstructing the irregularly spaced pixels into a regular 3D voxel array. CR Categories: 1.3.3 [Computer Graphics]: Computational Ge- ometry and Object Modelling—surface, object representation; Keywords: Radial Basis Functions, Ultrasound, Isosurface, 3D Freehand Ultrasound, Direct Surface Extraction, Unstructured data 1 I NTRODUCTION Research into 3D medical imaging has been ongoing for several decades. Considerable success has been found in visualizing 3D magnetic resonance (MR) and computed tomography (CT) images. Algorithms to extract surfaces of the skin, bone and many organs are well developed for the relatively low noise, structured MR and CT data [25]. Surface extraction from 3D ultrasound (3DUS) has been a more challenging problem. 3D US has a number of peculiar properties that inhibit sur- face extraction. First, the ultrasound images contain a significant amount of speckle and artifacts [12]. The images also show a strong directionality - the distinctiveness of boundaries depends on the viewing direction and therefore placement of the hand-held probe [19]. 3D freehand ultrasound - the most common technique - is based on acquisition of a set of 2D cross-sectional ultrasound im- ages [15, 26]. What is also required is a sensor on the hand-held probe that measures that probe’s spatial position and orientation. With such a system, a sonographer moves the sensorized probe over a region of interest. Typically a single organ is scanned from tip to tail. The set of acquired images, together with the sensor readings, comprise a 3D US data set. An example of a 3D freehand US data set is shown in Fig. 1. Although each cross-sectional image is composed of regularly spaced pixels, the spacing between images is irregular because it is determined by the sonographer’s hand mo- tion. The last issue is a main reason why many standard surface * Department of Computer Science, 2366 Main Mall, Vancouver BC, Canada V6T1Z4, [email protected] Department of Electrical and Computer Engineering, 2356 Main Mall, Vancouver BC, Canada V6T1Z4, [email protected] Department of Computer Science, 2366 Main Mall, Vancouver BC, Canada V6T1Z4, [email protected] extraction techniques (developed for regularly-spaced MR and CT) cannot be applied directly to 3D freehand US data. Figure 1: Freehand ultrasound image slices, note that images are irregularly spaced. Most previous work on surface extraction in MR and CT as- sumes that the data lies on a regular 3D grid of volume elements - hereafter called a voxel array, for example [21]. Although freehand US data can be interpolated into a voxel array, a loss of resolution normally results [18]. Nevertheless a short summary of voxel-based surface extraction methods is presented here. The previous work can be classified into several main categories: marching algorithms, level set methods, and contour connecting methods. There are several existing methods for surface extraction. The Marching Cubes technique [13] and its variations [16, 8, 11, 23]have been the most popular approach to extract isodensity sur- face from volume data; it produces triangles within voxels that con- tain the surface. The main difficulty associated with this approach is that no variety of the algorithm exists for scattered data. An alternative type of approach is to use parameterized curves and surfaces. This approach has been most widely used in 3D im- ages, where they are known as snakes or active contours. These models were first introduced by Terzopoulos et al [22, 14], and have been widely used, for instance, see the balloon models of Cohen and Cohen [5], and the active contour models of Blake and Isard [3]. By interpolating contours from ultrasound image slices, a surface could be reconstructed. An important difficulty with these models is good initialization; once initialized they work well for tracking. Initialization is often performed manually and accurate tracking de- pends on optimizing a number of parameters to the characteristics of the image. There are some available methods for extracting implicit sur- faces from unorganized point clouds, for instance, from laser scan- ners [4]. Implicit surfaces methods like Level Set method [27, 20] and Radial Basis Function [4, 7] extract the zero isocontour from a signed distance function. The surface is embedded as a (zero) level set of a higher dimensional function. Our approach is closest to these methods, but differs in that we work directly with intensity data rather constructing a density based on signed distance func- tions. We propose in this paper to construct a surface directly from the

Transcript of Direct Surface Extraction from 3D Freehand Ultrasound Imagesdpai/papers/ZhRoPa02.pdfmodels were...

  • Direct Surface Extraction from 3D Freehand Ultrasound Images

    Youwei Zhang ∗ Robert Rohling † Dinesh K. Pai ‡

    The University of British Columbia

    ABSTRACT

    This paper presents a new technique for the extraction of surfacesfrom 3D ultrasound data. Surface extraction from ultrasound data ischallenging for a number of reasons including noise and artifacts inthe images and non-uniform data sampling. A method is proposedto fit an approximating radial basis function to the group of datasamples. An explicit surface is then obtained by iso-surfacing thefunction. In most previous 3D ultrasound research, a pre-processingstep is taken to interpolate the data into a regular voxel array and acorresponding loss of resolution. We are the first to represent the setof semi-structured ultrasound pixel data as a single function. Fromthis we were able to extract surfaces without first reconstructing theirregularly spaced pixels into a regular 3D voxel array.

    CR Categories: 1.3.3 [Computer Graphics]: Computational Ge-ometry and Object Modelling—surface, object representation;

    Keywords: Radial Basis Functions, Ultrasound, Isosurface, 3DFreehand Ultrasound, Direct Surface Extraction, Unstructured data

    1 INTRODUCTION

    Research into 3D medical imaging has been ongoing for severaldecades. Considerable success has been found in visualizing 3Dmagnetic resonance (MR) and computed tomography (CT) images.Algorithms to extract surfaces of the skin, bone and many organsare well developed for the relatively low noise, structured MR andCT data [25]. Surface extraction from 3D ultrasound (3DUS) hasbeen a more challenging problem.

    3D US has a number of peculiar properties that inhibit sur-face extraction. First, the ultrasound images contain a significantamount of speckle and artifacts [12]. The images also show a strongdirectionality - the distinctiveness of boundaries depends on theviewing direction and therefore placement of the hand-held probe[19]. 3D freehandultrasound - the most common technique - isbased on acquisition of a set of 2D cross-sectional ultrasound im-ages [15, 26]. What is also required is a sensor on the hand-heldprobe that measures that probe’s spatial position and orientation.With such a system, a sonographer moves the sensorized probe overa region of interest. Typically a single organ is scanned from tip totail. The set of acquired images, together with the sensor readings,comprise a 3D US data set. An example of a 3D freehand US dataset is shown in Fig. 1. Although each cross-sectional image iscomposed of regularly spaced pixels, the spacing between imagesis irregular because it is determined by the sonographer’s hand mo-tion. The last issue is a main reason why many standard surface

    ∗Department of Computer Science, 2366 Main Mall, Vancouver BC,Canada V6T1Z4, [email protected]

    †Department of Electrical and Computer Engineering, 2356 Main Mall,Vancouver BC, Canada V6T1Z4, [email protected]

    ‡Department of Computer Science, 2366 Main Mall, Vancouver BC,Canada V6T1Z4, [email protected]

    extraction techniques (developed for regularly-spaced MR and CT)cannot be applied directly to 3D freehand US data.

    Figure 1: Freehand ultrasound image slices, note that images areirregularly spaced.

    Most previous work on surface extraction in MR and CT as-sumes that the data lies on a regular 3D grid of volume elements -hereafter called a voxel array, for example [21]. Although freehandUS data can be interpolated into a voxel array, a loss of resolutionnormally results [18]. Nevertheless a short summary of voxel-basedsurface extraction methods is presented here. The previous workcan be classified into several main categories: marching algorithms,level set methods, and contour connecting methods.

    There are several existing methods for surface extraction. TheMarching Cubes technique [13] and its variations [16, 8, 11,23]have been the most popular approach to extract isodensity sur-face from volume data; it produces triangles within voxels that con-tain the surface. The main difficulty associated with this approachis that no variety of the algorithm exists for scattered data.

    An alternative type of approach is to use parameterized curvesand surfaces. This approach has been most widely used in 3D im-ages, where they are known as snakes or active contours. Thesemodels were first introduced by Terzopoulos et al [22, 14], and havebeen widely used, for instance, see the balloon models of Cohenand Cohen [5], and the active contour models of Blake and Isard [3].By interpolating contours from ultrasound image slices, a surfacecould be reconstructed. An important difficulty with these modelsis good initialization; once initialized they work well for tracking.Initialization is often performed manually and accurate tracking de-pends on optimizing a number of parameters to the characteristicsof the image.

    There are some available methods for extracting implicit sur-faces from unorganized point clouds, for instance, from laser scan-ners [4]. Implicit surfaces methods like Level Set method [27, 20]and Radial Basis Function [4, 7] extract the zero isocontour froma signed distance function. The surface is embedded as a (zero)level set of a higher dimensional function. Our approach is closestto these methods, but differs in that we work directly with intensitydata rather constructing a density based on signed distance func-tions.

    We propose in this paper to construct a surface directly from the

    Melanie K Tory

    Administrator45

  • freehand US data without first interpolating into a voxel array. Inthis paper we will describe:

    • a new 3D freehand acquisition system

    • a method for fitting a function to noisy ultrasound pixel data

    • a method to calculate an explicit surface from the function

    • the application of such methods to anatomy imaged in vivo

    2 3D FREEHAND ULTRASOUND

    Figure 2: System setup of freehand ultrasound imaging.

    Fig. 2 shows our system setup. For measurement, we use a newultrasound machine recently developed by Ultrasonix Medical Cor-poration1. One reason for this choice is its open architecture. In-stead of conventional, closed-architecture, stand-alone ultrasoundmachines, a machine with an open architecture allows researchersto overcome current restrictions to data and parameters internal tothe machine, as well as closely couple new software applicationsand hardware to the ultrasound system. A second reason is thatdirect access to high quality digital images is available without anintermediate analog conversion stage and corresponding loss of im-age quality.

    The Ultrasonix machine consists of a standard ultrasound probethat is connected to a generic PC through programmable hardware.The PC contains software for ultrasound signal processing and ahigh-quality monitor to display the resulting images. With an openarchitecture, the various stages of image formation can be directlyaccessed. The visualization work described in this paper is the firstproject to be undertaken on the Ultasonix machine.

    We use the OPTOTRAK 30202 as the tracking device. The OP-TOTRAK 3020 is an trinocular system, which consists of threecalibrated CCD cameras arranged linearly. It measures 3D mo-tion and position by tracking markers attached to a probe. Asthe probe moves, OPTOTRAK detects the positions of the infraredLED markers and simultaneously calculates the precise 3D positionfor each marker. The maximum inaccuracy of 3D markers’ posi-tion is less than 0.1mm, but overall system accuracy is determinedmainly by calibration.

    1www.ultrasonix.com2Northern Digital Ltd. www.ndigital.com

    Figure 3: Z-wire calibration phantom

    We calibrate our probe with the Z-wire phantom [17, 6]. See Fig.3. The goal of calibration is to determine the transformation matrixfrom 2D ultrasound image coordinates to 3D probe coordinates.With a wisely designed Z-wire phantom, corresponding points canbe easily found from the ratio property of similar triangles, whichthen define the transform matrix. The overall system accuracy isapproximately1mm.

    Several objects, including a real human hand, were used in oursurface extraction experiments. In order to avoid the deformationcaused by contact of probe with human hand, we scan the hand in awater bath. Because the difference between water and human tissueis relatively small, ultrasound is able to penetrate through the skininto internal tissues.

    3 SURFACE MODELLING

    Figure 4: Ultrasound image of a finger cross section. 1 the noiseand speckle; 2 the skin surface, which is a bright boundary; 3 thebone surface; 4 the vein; 5 interior of bone, since ultrasound cannot penetrate inside the bone, this is a dark region in the image

    In ultrasound images, organ boundaries produce higher intensityechoes than surrounding areas. These echoes result in pixels withhigher grey-level values in the image. See Fig. 4 for an example ofintensity distribution.

    Another challenge for surface extraction from ultrasound imageslies where there is a steep vertical slope. The vertical section willcause a break on the ultrasound images, as shown in Fig. 5. Aneffective surface extraction method should be able to smoothly in-terpolate across these missing parts by using neighboring data.

    The 3D data can be represented with a single functionf(x, y, z).The surface is implicitly described as the set of points(x, y, z)where the function attains a constant level equal to the echo inten-

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  • Figure 5: Ultrasound images of toy duck, note the gaps at two in-dicated positions. The gap exists because the vertical section of theboundary cannot reflect ultrasound.

    sity at the organ boundary. An explicit representation - in the formof a mesh of polygons - can also be extracted by an iso-surfacingalgorithm (such as Marching Cubes).

    More formally: given function valuesfi at N distinct pointsxi(xi, yi, zi) find the functionf(x, y, z) such thatf(xi, yi, zi) =fi for i = 1, .., N. This is a classic interpolation problem.

    Then a surface is desired wheref(x, y, z) attains a level ofk -the echo intensity at the organ boundary. The surface is obtainedby an iso-surfacing algorithm that fits a mesh of polygons to thesurface wheref(x, y, z) = k.

    4 RADIAL BASIS FUNCTION INTERPOLA -TION

    Radial basis functions (RBF’s) are proposed forf(x, y, z). Thereare several reasons for this choice:

    • Compact description of a single function;

    • Can interpolate sparse, non-uniformly spaced data;

    • Can both interpolate and approximate data;

    • Can be evaluated anywhere to generate meshes of desired res-olution;

    • Gradients and higher derivatives can be calculated analyti-cally. These are also continuous and smooth depending onthe choice of basis functions;

    • Surface normals are easily computed;

    • Iso-surfaces extracted from RBF’s are manifold (i.e. non self-intersecting).

    Traditional RBF’s requireO(N3) arithmetic operations for fitting,so the computational complexity rapidly increases as the data setsize increases. Also the system of equations is not always wellconditioned. For these two reasons, direct solution of this equationis impractical for any data sets larger than a few thousand points.We propose that new fast methods for RBF’s are feasible for thisproblem [4].

    Fast RBF methods were developed recently for surface recon-struction from laser range finger data. In that problem, surfacepoints(xi, yi, zi) were given and a functionf(x, y, z) was desiredsuch thatf(x, y, z) = 0 on the surface. To solve that problem(for non trivial solutions), off surface points were introduced ar-tificially along with a signed-distance function. This resulted in

    the expanded problem:f(xi, yi, zi) = 0 for on-surface points, andf(xi, yi, zi) = di (not equals 0) for off surface points, wheredi isthe estimate of the signed distance to the surface. We adopt this for-mulation to our problem of fitting RBF’s to pixel data. But insteadof usingdi as ”off-surface distance”, we substitute the intensity ofthe pixel. Then, instead of iso-surfacing at zero, we iso-surface atthe mean intensity of the boundary.

    Consider a set of 3D pointsxi, i = 1, .., N have the intensityvaluespi. The basic idea of Radial Basis Function interpolation isto find a splineS(x) that fits the data points as smoothly as possible.With this constraint, we find theS(x) that minimizes

    N∑i=1

    |pi − S(xi)|2 + ωI(s) (1)

    The first term is the deviation of the spline from the data points,which represents the interpolation error. The second term,I(s), isa smoothness function of surfaces, normally chosen to minimizeone or more partial derivatives of the function.ω is the weightwhich determines the relative cost of this two components.ω = 0means that the RBF solution fit the given centers perfectly, withoutany smoothing.

    The general solution can be expressed as

    S(x) = t(x) +

    N∑i=1

    λiR(|x,xi|) (2)

    where|x,xi| is the Euclidean distance;t(x) is a polynomial oflow degree called the trend function,R(x, xi) is a radial basis func-tion (RBF), andxi are the centers of the RBF. There are severalchoices for the basis function. The format depends on choice ofI(s). The biharmonic splineR(r) = r, (r = |x,xi|) is well suitedto the representation of 3D objects. It can be characterized as thesmoothestinterpolation of data, in the sense that it is the interpolantwhich minimizes the integral of the 2nd derivative.

    The solution can be written in matrix form as(A TT T 0

    )(λc

    )=

    (p0

    )(3)

    where

    Ai,j = R(|xi,xj |), i, j = 1, ..., N

    Ti,j = tj(xi), i = 1, ..., N, j = 1, ..., l

    p is the vector of intensity of pixelspi, {t1, . . . , tl} is a basis forpolynomials, andλ is a vector of the coefficientsλi in equation 2.

    In 3D, the biharmonic spline case gives

    c = (c1, c2, c3, c4)

    T is the matrix withith row (1, xi, yi, zi), andl - the degree of thispolynomial - equals 4 here.

    When we take weightω into consideration in Equation 1. Equa-tion 3 can be modified as(

    A + ωI TT T 0

    )(λc

    )=

    (p0

    )(4)

    For our experiments,ω is adjusted to smooth the speckle compo-nents of the ultrasound iamges.

    Beatson et al. [2] propose a fast fitting and evaluation algorithmbased on the Fast Multipole Method (FMM) [10]. For an evaluationpoint, FMM computes an approximate evaluation for those pointsfar from it, while using direct evaluation for those near it. A pre-defined evaluation accuracy can be selected to divide all points into

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  • Figure Number of Peak Fitting SurfacingInterpolation points RAM(MB) time time

    Duck I 9,907 475 58s 70sDuck II 99,087 545 1047s 391sHand I 9,920 156 50s 73sHand II 99,479 209 1340s 765s

    Table 1: Comparison of RBF fitting times on a 1GHz Intel P3 with 1GB RAM

    the two categories. Fitting accuracy is another parameter we canadjust in this formulation. It is specified as the maximum alloweddeviation of RBF values at interpolation points. Both storage andcomputation can be significantly decreased with suitable values forfitting accuracy and evaluation accuracy.

    As an additional way to speed computation, the number of datapoints is reduced selectively. In most cases, a small subset of cen-ters is able to produce the same approximation as the original inputdata points. Though it adds additional computation, center reduc-tion brings the benefit of smaller memory requirements and fasterevaluation time without losing accuracy. In our experimental ap-proach, we also perform an automatic pre-processing step to furthertrim the data and improve computational performance.

    5 AUTOMATIC SEGMENTATION

    The goal of the automatic segmentation algorithm is to trim pixelsfar from the estimated boundary of the surface. There is no needto have a high level of accuracy at this stage. The only need is toavoid removing pixels that depict the boundary. As described inprevious sections, we use the intensity value of the pixels as a quasidistance function in the calculation of the radial basis function co-efficients. It is therefore important to also extract pixels that arenearby the boundary. For both of these reasons, the desired seg-mentation consists of a band of pixels surrounding the boundary ineach ultrasound image.

    The perform this, an initial segmentation is drawn by the useron the first image and a simple tracker is used to automaticallypropagate the segmentation to the following frames. The simpletracker is guided by the presence of boundary features that followthe boundary as it changes from image to image. The features thatare used here are edgels (edge elements) detected along each col-umn of pixels. Edges are detected along columns because eachimage is created from ultrasound echoes that travel in the verticaldirection. Thus the physics of ultrasound image creation suggeststhat edges are best detected along vertical lines.

    Edgels are identified by large gradients. The largestM suchgradients are selected in each image, whereM is smaller than thenumber of columns. This reduces the number of occasional falseedgels around speckle or other artifacts.

    The segmentation region is propatated by the following algo-rithm:

    For each image i:

    1. Assign prior probabilities P(u,v) =C1 to pixelspu,v (whereu,v are the row,column indices) inside the segmented areaBi(whereB0 is created manually). Assign P(u,v) =C0 to pixelsoutside (C1 = 1.0 andC0 = 0.2)

    2. Locate the edgelsEi,k (k=1 . . . M )

    3. For all pixelspu,v in the segmented areaBi:

    (a) Find the nearest edgelEi,k

    (b) Calculate Likelihood function L(u,v) as the proximityof the nearest edgel, weighted using a Gaussian cen-tered atpu,v (Gaussian with standard deviation of 15pixels and truncated to zero at± 32 pixels)

    4. Convolve Likelihood with the Prior:P (u, v)new = L(u, v)⊗P (u, v) + C0 over all u,v

    5. Threshold the new Likelihood at value T (T=0.35) to obtainBi+1

    In this way, the segmentation region propagates smoothly fromimage to image without deviations into regions far from the bound-ary. All pixels that fall within the segmented areas are then used asthe inputs to the RBF interpolation.

    6 SURFACE EXTRACTION

    Once the RBF weights are calculated, the next step is to create anexplicit surface along iso-values. Conventional surface extractionmethods such as Marching Cubes [13] are optimized for visualizinga complete volume data, which creates a large number of poor as-pect ratio triangles. Some mesh simplification algorithms like Ver-tex Clustering can significantly reduce the number of triangles butnot in general maintain the topology. Treece [23] proposes a Regu-larized Marching Tetrahedra which combines Marching Tetrahedraand Vertex Clustering to generate an iso-surfaces with smaller num-ber of triangles of improved aspect ratio, and still keep the topologyconsistency. Treece’s method was adopted by J.Carr et al[4], and isalso used here.

    7 RESULTS

    Two examinations were performed. The first was an examinationof an artificial phantom based on a toy duck. The second wasan examination of the human hand scannedin vivo. In the phan-tom examination, the duck was scanned in two sweeps because thecross-section of the duck was larger than the field of view of theultrasound probe. The first sweep of the duck imaged the head,starting from the nose. The second sweep imaged the torso, fromchest to tail. The hand examination was done in three sweeps.

    The automatic segmentation algorithm was then applied to thedata sets to get a reasonable computation time. In fact, additionalpruning was also performed to see the effect of large data reductionon surface quality. To do this, pixels from the set of images wererandomly deleted. A small amount of deletion resulted in the “highresolution” data set, and larger amount of deletion resulted in the“low resolution” data set. A similar process was undertaken on thehand examination. Some statistics of the computational demandsare listed in Table 1.

    Retaining more than 100k points only serves to highlight thespeckle (noise) on surface, although retaining less than 10k pointsstarts to remove surface features. So in practice, this must be donecarefully; this could cause the surface to be artificially trimmedmore than what would be acceptable in clinical practice. We seethat fitting time follows the expectedO(n log n) complexity.

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  • (a) toy duck phantom (b) point cloud (c) no smoothing with 9,907 points

    (d) no smoothing with 99,087points

    (e) medium smoothing (ω = 3)with 99,087 points

    (f) large smoothing (ω = 9) with98,087 points

    Figure 6: Reconstructed surface meshes of the artificial phantom.

    The results of the duck phantom examination are shown in Fig.6. Fig. 6(a) and 6(b) show the phantom and the point cloud of theinput data. Fig. 6(c) and 6(d) show the surface extracted from thelow resolution and high resolution data respectively. In both cases,the iso-surfacing was performed at a resolution of0.5mm, and aisovalue of 50 (between the background intensity level of approxi-mately 15 and the boundary intensity of approximately 130).

    These results represent the first time that RBF functions havebeen fit to non-regular 3D ultrasound density data for surface ex-traction. In both cases the extracted surface clearly resembles theactual surface. But notice two obvious problems. First the pres-ence of ultrasound speckle — inherent to all ultrasound images —results in a bumpy surface. Second, misalignment of some of theultrasound images creates vertical creases in the surface. The sec-ond problem is peculiar to the type of 3D ultrasound used in thisstudy. Freehand ultrasound relies upon the external position sensorsto align the images. Freehand ultrasound also takes many secondsto perform a sweep of an object. Any miscalibration of the systemor movement of the object during the sweep results in misalignmentof the images. For the duck phantom, miscalibration is the domi-nant error., but the hand examination contains both object motionand miscalibration. Improvements to calibration will likely reducethe misalignment significantly. Moreover, ultrasound scanner man-ufacturers are now researching 3D ultrasound acquisition systemswith two main differences from freehand systems: electronicallycontrolled image direction and real-time acquisition rates. The di-rect control of imaging direction eliminates the miscalibration prob-lem with external sensors and the faster acquisition rate (up to 16sweeps per second) reduces object motion error. Therefore thesesources of error can be expected to be eliminated in future 3D ul-trasound systems.

    Nevertheless, the RBF’s depict the actual surface. They havefilled the gaps between the sweeps (especially in the neck region)and filled the gaps along the boundary in a single image (especiallywhere the wing attaches to the torso). This is true for both the lowresolution (Fig. 6(c)) and high resolution (Fig. 6(d)) results. Noticethat the lower resolution result loses some small surface detail but,as expected, retains the larger features.

    Figures 6(d) - 6(f) show the effect of smoothing. One purposeof smoothing is to reduce the effect of speckle, and is easily imple-mented with the RBF formulation. The speckle size is a functionof the spatial resolution of the ultrasound images. The mediumlevel of smoothing was chosen to minimize speckle without losingspatial resolution. The axial resolution is approximately0.4mm(calculated using a two cycle pulse from the7MHz probe3), andthe lateral and elevational resolution are approximately1 − 2mm(depending on proximity to the focal point). The medium level ofsmoothing also avoids aliasing. Notice that the medium smoothingremoves speckle with a minimal loss of other features, such as thedetail of the features in the wings. Large level of smoothing startsto remove these small features, as this level of smoothing, begin toexceed the spatial resolution.

    Fig. 7 shows the ability to extract multiple surfaces from a singlevolumetric ultrasound data set. Here we attempt to extract the innersurface in addition to the outer surface (from Fig. 6). Since the toyduck is made from a constant thickness material, the inner surfaceis expected to be similar to the outer surface. Fig. 7(b) confirmsthis is the case.

    Fig. 8 shows the results of the hand examination. Again the

    3wavelengthλ = c/f . Wherec is the typical ultrasound propagatespeed in human tissues, which equals to1, 540m/s [12], andf is the fre-quency of ultrasound probe.

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  • (a) external surface (b) internal surface (c) two surfaces

    Figure 7: Reconstructed surface meshes of both external and internal surfaces from a single examination.

    surfaces clearly resemble the actual skin surface. Although bothspeckle and misalignment artifacts are visible, the RBF smoothingminimizes the effect of speckle. The misalignment is slightly morepronounced here because of small hand tremor during acquisition.As already mentioned, the misalignment errors can be eliminatedwith upcoming 3D ultrasound systems. Because the internal struc-tures are not distinguishible in the ultrasound images, extractionof interior surfaces was not attempted. Scanners with higher fre-quency, higher resolution probes should reveal the bony structuresmore clearly making their surface extraction feasible.

    It is worthwhile to compare our method to alternative methods.For example an approach with 3 steps: 1. Segment 2D images forboundary; 2. Triangulate directly on convex hull of these points;3. Apply a smoothing algorithm. This approach such as [1, 24, 9],used in 3D surface finding when data is already in form of scatteredpoints . But here, step 1 to extract boundary points is non trivial ingeneral. For reasons explained in the section 1, reliable extractionof boundary points is unlikely to be successful. The RBF’s workingdirectly on boundaryandnon-boundary pixels perform more robustsegmentation.

    A limitation that RBF fitting and isosurfacing shares with othersurface extraction methods is the fine-tuning of the various param-eters. In our case we selected the form of the RBF and adjusted thesmoothing or fitting accuracy, level of center reduction, the confi-dence we have in selecting the input pixels, resolution of surfaceand the surface isovalue. Fortunately, in these experiments, theresulting surfaces were not very sensitive to these parameters, al-though they need to be set manually.

    8 CONCLUSIONS

    We have shown, for the first time, how to represent the set of semi-structured ultrasound pixel data as a single function. From this wewere able to extract surfaces without first reconstructing the irreg-ularly spaced pixels into a regular 3D voxel array. We have shownthat the pixels data can be represented by a single function by usingfast fitting and evaluation techniques for radial basis functions. Wehave adapted recent advances in fast RBF’s to the special problemsencountered in medical 3D ultrasound. The process is demonstratedon both artificial phantoms, as well as human skin surfaces obtainedin vivo.

    Although we have readily extracted surfaces from water-tank ex-periments, the more general problem of extracting organ or bonesurfaces from a wide variety of 3D ultrasound scans is still unre-solved. Our use of RBF’s has resolved some issues such as resolu-tion loss in voxel array reconstruction and bridging gaps. yet otherartifacts such as speckle and shadow will still affect the qualify of

    the results.In future work, we expect to concentrate on these areas:

    • Larger data sets with improved resolution and calibration.

    • Both water-skin boundaries as well as soft-tissue and boneboundaries;

    • Other medical imaging modalities such as CT and MR;

    • Comparison to existing techniques;

    • Reconstruction of both front and back sides.

    REFERENCES

    [1] N.Amenta, M.Bern, M.Kamvysselis, A New Voronoi-BasedSurface Reconstruction Algorithm, SIGGRAPH’98, pp.415-421, 1998

    [2] R.K.Beatson, W.A.Light, S.Billings, Fast Solution of the Ra-dial Basis Function Interpolation Equations: Domain Decompo-sition Methods, 2000 Society for Industrual and Applied Math-ematic, Vol 22, pp.1717-1740, 2000

    [3] A.Blake, M.Isard, Active Contour. Springer Press, 1998

    [4] J.C.Carr, R.K.Beatson, J.B.Cherrie, T.J.Mitchell, W.R.Fright,B.C.McCallum, T.R.Evans, Reconstruction and Representationof 3D Objects with Radial Basis Functions, SIGGRAPH 2001

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  • (a) real hand (b) point cloud

    (c) no smoothing (d) medium smoothing(ω = 3) with 99,479points

    (e) large smoothing(ω = 9) with 99,479points

    Figure 8: Reconstructed hand surface.

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    [13] Lorensen W.E. and Cline H.E., ”Marching Cubes: A HighResolution 3-D Surface Reconstruction Algorithm”, ComputerGraphics, vol.21, no.4, 1987

    [14] T.McInerney, D.Tezopoulos, Topologically Adaptable Snake,Proceeding ICCV’95, pp840-845, 1995

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