mcet talk dt-fdt

Distance Transform and its Applications

Subhadip Basu, Ph.D.

Dept. of Computer Science and Engineering

Jadavpur University

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

2

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

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

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

7


Euclidean Distance Transform in 2d

0

8

1 2 4

9 10 16

5

Squared DT values 8


Euclidean Distance

9


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

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

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) .

15

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 .

16

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

17

Application to object separation

18

Application to object separation

19

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

20

Phase 2: After noise pruning

based on FDT.



ACA

Bassilary

ICA

ICA


21


using Gaussian Filter .



ACA

Bassilary

ICA

ICA


22


based on Connectivity of voxels.



ACA

Bassilary

ICA

ICA


23

Subsequent Result

Detailed discussion in the next lecture



ACA

Bassilary

ICA

ICA


24

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

25



RDT values represented in Grayscale

RDT of the 2d image 26



RDT of the 2d image

2

1

3

5

9

10

0

Squared RDT values +1

6

27


• A synthetic 3d phantom

28


• RDT of the 3d phantom

29

Rectangle Block Corresponding FDT map

Application of FDT/RFDT in Object Localization

Key Observation: High FDT value towards the center of the object 30

Image

Binary Image

FDT image

Text localization using FDT 31

Image FDT image

Binary Image Text localization using FDT 32

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)

33

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

35

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

36

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

37

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…

38

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.

39

Thank you

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