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Transcript of Mcet Talk Dt-fdt
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Distance Transform and its Applications
Subhadip Basu, Ph.D.
Dept. of Computer Science and Engineering
Jadavpur University
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Contents
• Introduction to Digital Image
• Distance Transform(DT)
• Applications of DT
• Fuzzy Distance Transform(FDT)
• Fuzzy Connectivity
• Applications of FDT
• Reverse Fuzzy Distance Transform(RFDT)
• Applications of RFDT
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Introduction to a Digital Image
• A n-dimensional grid, or simply a grid, is represented by 𝒵n| 𝒵 is the set of integers.
• A grid point, often referred to as a point, is an element of 𝒵n .
• When 𝑛 = 2, each point in the 2-dimentional grid is referred to as a pixel.
• When 𝑛 = 3, each point in the 3-dimentional grid is referred to as a voxel and is represented by a triplet of integer coordinates.
• Standard 26-adjacency is used here, i.e., two voxels 𝑝 = 𝑥1, 𝑥2, 𝑥3 , 𝑞 = (𝑦1, 𝑦2, 𝑦3) ∈ 𝒵3
are adjacent if and only if max1≤𝑖≤3𝑥𝑖 − 𝑦𝑖 ≤ 1, where ∙ returns the absolute value.
• Two adjacent voxels are often referred to as neighbors of each other; the set of 26-neighboors of a voxel 𝑝 excluding itself is denoted by 𝒩∗(𝑝).
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Introduction to a Digital Image
• An object 𝒪 is a fuzzy subset (𝑝, 𝜇𝒪(𝑝)) 𝑝 ∈ 𝒵3 of 𝒵3, where 𝜇𝒪: 𝒵3 →
0,1 is the membership function. The support 𝛩(𝒪) of an object 𝒪 is the
set of all voxels with non-zero membership, i.e.,
𝛩 𝒪 = 𝑝 | 𝑝 ∈ 𝒵3 and 𝜇𝒪(𝑝) ≠ 0 ; 𝛩 𝒪 = 𝒵3 − 𝛩 𝒪 is the background.
• A 3-dimensional binary image is represented by 𝒵3| 𝒵 is in {0,1}.
• A binary object 𝒪 is a fuzzy subset (𝑝, 𝜇𝒪(𝑝)) 𝑝 ∈ 𝒵3 of 𝒵3, where
𝜇𝒪: 𝒵3 → {0,1} is the hard limiting membership function. The background
points are defined accordingly
2-D Binary Image 4
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Distance Transform(DT) • The DT maps each image pixel into its smallest distance
to regions of interest [Rosenfeld and Pfaltz 1966].
• Given a distance metric, the DT of an image 𝒵2 is an assignment to each point x in 𝛩(𝒪)of the distance between x and the closest background point (𝛩 𝒪 ) in 𝒵2.
• Thus formally,
DT(x) = min{ d(x , y)|x ϵ 𝛩(𝒪) & y ϵ 𝛩 𝒪 }, where d is the given distance metric.
Euclidean Distance Transform in 2d 5
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Distance Transform(DT) • A generalized distance metric is represented by
dp(x , y) = ( ∑i |xi − yi|p )1/p, where x and y are k- tuples, xi and yi are the i-th
coordinates of x and y.
• The d1 and d2 metrics are known as the Manhattan or city-block distance and Euclidean distance respectively.
• Popular Chessboard Distance is computed by dChessboard(x , y) = max{|xi − yi|}, where 1 ≤ i≤ k .
• City-block and Chessboard Distance metric incurs less complexity in DT computation but the DT computed are inexact.
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Distance Transform(DT)
Euclidean Distance Transform in 2d
DT values represented in Grayscale
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Distance Transform(DT)
Euclidean Distance Transform in 2d
0
8
1 2 4
9 10 16
5
Squared DT values 8
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Distance Transform(DT)
Euclidean Distance
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Distance Transform(DT)
Chessboard Distance
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Application: Skeletonization
DT based skeletons are -
rotation invariant
robust than thinning algorithm
based skeletons
Algorithm:
Step1: Compute DT
Step2: Find Initial Skeleton by selecting
voxels x such that DT(x) ≥ DT( Nx ) ,
where Nx is x’s neighbouring voxel
Step3: Extend Initial Skeleton based on
DT values to get continuous skeleton.
Skeletonization of a 3d synthetic phantom 11
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Application: Finding Shortest-Path
Start Voxel(s)
End
Voxel
(e)
Shortest-Path in 3d synthetic phantom
Shortest-Path computation between two
image pixel / voxel
d(p , s)+d(p , e) < d(q , s)+d(q , e),
where voxel p is on the shortest path
but q is not
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Application: Finding Shortest-Path
Start Voxel(s)
End
Voxel
(e)
Shortest-Path computation between two
image pixel / voxel
d(p , s)+d(p , e) < d(q , s)+d(q , e),
where voxel p is on the shortest path
but q is not
Shortest-Path in 3d synthetic phantom 13
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Fuzzy Image & Fuzzy DT(FDT)
Unlike a Binary Image where each voxel in a Fuzzy Image has a certain membership value µ(x)→[0,1] of being included in 𝛩(𝒪).
• Let 𝑆 denote a set of voxels; a path 𝜋 in 𝑆 from 𝑝 ∈ 𝑆 to 𝑞 ∈ 𝑆 is a sequence 𝑝 = 𝑝1, 𝑝2, ⋯ , 𝑝𝑙 = 𝑞 of voxels in 𝑆 such that every two successive voxels on the path
are adjacent.
• A link is a path 𝑝, 𝑞 consisting of exactly two adjacent voxels. The length of a path 𝜋 = 𝑝1, 𝑝2, ⋯ , 𝑝𝑙 in a fuzzy object 𝒪, denoted by П𝒪(𝜋), is defined as the sum of lengths of all links along the path, i.e.,
• П𝒪 𝜋 = 1
2𝑙−1𝑖=1 µ𝒪 𝑝𝑖 + µ𝒪 𝑝𝑖+1 ∥ 𝑝𝑖 − 𝑝𝑖+1 ∥,
• where ∥ 𝑝 − 𝑞 ∥ denotes the Euclidean distance between two voxels 𝑝, 𝑞.
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Fuzzy Distance Transform
• The fuzzy distance between two voxels 𝑝, 𝑞 ∈ 𝒵3 in an object 𝒪, denoted by 𝜔𝒪(𝑝, 𝑞), is the length of one of the shortest paths from 𝑝 to 𝑞, i.e.,
• 𝜔𝒪 𝑝, 𝑞 = min𝜋∈𝒫(𝑝,𝑞)
П𝒪 𝜋 ,
• where 𝒫(𝑝, 𝑞) is the set of all paths from 𝑝 to 𝑞. The fuzzy distance transform
or FDT of an object 𝒪 is an image 𝑝, 𝛺𝒪 𝑝 | 𝑝 ∈ 𝒵3 , where 𝛺𝒪: 𝒵3 → ℜ+| ℜ+
is the set of positive real numbers including zero, is the fuzzy distance from the background. i.e.,
• 𝛺𝒪 𝑝 = min𝑞∈𝛩 𝒪
𝜔𝒪(𝑝, 𝑞) .
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Fuzzy Connectivity
• Fuzzy morpho-connectivity strength of a path 𝜋 = 𝑝1, 𝑝2, ⋯ , 𝑝𝑙 in a fuzzy object 𝒪, denoted by 𝛤𝒪(𝜋), is defined as the minimum FDT value along the path:
• 𝛤𝒪 𝜋 = min1≤𝑖≤𝑙
𝛺𝒪(𝑝𝑖) .
• Fuzzy morpho-connectivity between two voxels 𝑝, 𝑞 ∈ 𝒵3, denoted by 𝛾𝒪(𝑝, 𝑞), is the strength of one of the strongest morphological paths between p and q, i.e.,
• 𝛾𝒪 𝑝, 𝑞 = max𝜋∈𝒫(𝑝,𝑞)
𝛤𝒪 𝜋 .
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Illustration of FDT and FC
A
Low FDT value High FDT value
Strongest path between A and B.
FDT value of the weakest point is
higher than the other path
Not the strongest path
between the A and B
SA SB
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Application to object separation
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Application to object separation
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Application to CTA Image Segmentation
Phase 1: After thresholding intensity
i.e. a voxel x is not removed only if
Th1 ≤ Intensity(x) ≤ Th2 ,
Segmentation of overlapping arteries
and soft tissues in 3d MRI image of brain
ACA
Bassilary
ICA
ICA
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Phase 2: After noise pruning
based on FDT.
Segmentation of overlapping arteries
and soft tissues in 3d MRI image of brain
ACA
Bassilary
ICA
ICA
Application to CTA Image Segmentation
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Phase 3: After noise pruning
using Gaussian Filter .
Segmentation of overlapping arteries
and soft tissues in 3d MRI image of brain
ACA
Bassilary
ICA
ICA
Application to CTA Image Segmentation
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Phase 4: After noise pruning
based on Connectivity of voxels.
Segmentation of overlapping arteries
and soft tissues in 3d MRI image of brain
ACA
Bassilary
ICA
ICA
Application to CTA Image Segmentation
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Subsequent Result
Detailed discussion in the next lecture
Segmentation of overlapping arteries
and soft tissues in 3d MRI image of brain
ACA
Bassilary
ICA
ICA
Application to CTA Image Segmentation
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Reverse Distance Transform(RDT)
• Unlike DT (or FDT), RDT(or its fuzzy counterpart RFDT) of a point is the minimum distance from the core or Skeletal point (S), i.e. RDT(x) = min{ d(x , y)|x ϵ 𝛩(𝒪) & y ϵ S }, where d is a distance metric.
Binary 2d image
Object Point
Background Point
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Reverse Distance Transform(RDT)
• Unlike DT (or FDT), RDT(or its fuzzy counterpart RFDT) of a point is the minimum distance from the core or Skeletal point (S), i.e. RDT(x) = min{ d(x , y)|x ϵ 𝛩(𝒪) & y ϵ S }, where d is a distance metric.
RDT values represented in Grayscale
RDT of the 2d image 26
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Reverse Distance Transform(RDT)
• Unlike DT (or FDT), RDT(or its fuzzy counterpart RFDT) of a point is the minimum distance from the core or Skeletal point (S), i.e. RDT(x) = min{ d(x , y)|x ϵ 𝛩(𝒪) & y ϵ S }, where d is a distance metric.
RDT of the 2d image
2
1
3
5
9
10
0
Squared RDT values +1
6
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Reverse Distance Transform(RDT)
• A synthetic 3d phantom
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Reverse Distance Transform(RDT)
• RDT of the 3d phantom
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Rectangle Block Corresponding FDT map
Application of FDT/RFDT in Object Localization
Key Observation: High FDT value towards the center of the object 30
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Image
Binary Image
FDT image
Text localization using FDT 31
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Image FDT image
Binary Image Text localization using FDT 32
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Allele detection from segmented confocal microscopic images of cell nuclei
• Initially
Image has too many noise
Element borders are not crisp
Disparity in no of allele
• Ideally
It should have 4 type of voxel-
Black background, Red and Green allele and Blue nucleus
No overlapping between elements
It should have 2 red allele and 2 green allele
Initial image (z-stack animation)
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Intensity based labelling
• Voxels with
1. higher red intensity => R (red allele)
2. higher green intensity => G (green allele)
3. higher blue intensity => B (blue nucleus)
4. low intensity => BL (black background)
5. everything else is unwanted noise
I. higher red and blue intensity => RB
II. higher red and green intensity => RG
III. higher green and blue intensity => GB
IV. higher red, green and blue intensity => RGB
Post labelling 3D snap
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Density based noise filtration
• Apply on noise voxels only
• Derive possible labels of a noise voxel
E.g.- RB ∈ {R || B}
• Calculate density of various labelled
voxel among neighbours
• Map noise point to most probable
possible label
Post filtration 3D snap
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Allele detection
• Use single-linkage hierarchical clustering
to detect the allele clusters
Single-linkage: min {d(a, b) : a∈A, b∈B}
• Keep largest four (2+ 2) clusters only
• Measure various allele properties
Eg- center position, size, distance to
background etc.
• Mark center voxel as different label
3D snap of alleles with marked center
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Distance calculation using DT
• Apply Distance Transformation (DT) to
every voxel inside nucleus
• Distance Transformation (DT):
distance to nearest background (BL) voxel
• For border voxels, distance to nearest
background is 0
• For other voxels, search in connected 26
neighbour to get the nearest background voxel
Various allele properties of a image
# Allele type CX CY CZ Size Distance to
Background
1 Red 24 94 13 204 5.477226
2 Red 110 47 14 226 9.055385
3 Green 115 46 4 160 4
4 Green 31 56 6 172 3
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Conclusion
• FDT and RFDT plays a key role in digital image analysis
• Elegant solutions to many challenging problems: Multi-scale opening of conjoined objects in shared intensity space
Applications to cerebrovascular segmentation in Cerebral CTA
Applications to Artery/Vein segmentation in Pulmonary CT
And many more…
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Acknowledgments
o Prof. Punam K Saha, Dept. of ECE, Univ. of Iowa, USA
o Dr. Dariusz Plewzcynski, ICM, University of Warsaw, Poland
o Dr. Jakub Wlodarc, Nencki Institute of Experimental Biology, Poland
o Prof. Eric Hoffman, Dept. of Radiology, Univ. of Iowa
o Prof. M. L. Raghavan, Dept. of BME, Univ. of Iowa
o Dr. Robert E. Harbaugh, Penn State Hershey Medical Center
Students
Azharuddin Mollah, Shauvik Paul, Ayan Paul, Pranati Rakshit and many others.
• My visit to the Structural Imaging Laboratory, Univ. of Iowa, USA, was funded by the BOYSCAST
fellowship (SR/BY/E-15/09), Dept. of Science and Technology, Govt. of INDIA.
• This study is supported in part by the FASTTRACK grant (SR/FTP/ETA-04/2012) by DST, Govt. of India.
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Thank you