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PDE methods for Image Segmentation and Shape Analysis:

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PDE methods for Image Segmentation and Shape Analysis:

From the Brain to the Prostate and Backpresented by John Melonakos – NAMIC Core 1 Workshop – 30/May/2007

- Bhattacharyya Segmentation
- Segmentation Results
---------------------------------------

- Shape Analysis
- Shape-Driven Segmentation

- Georgia Tech-
- Yogesh Rathi, Sam Dambreville, Oleg Michailovich, Jimi Malcolm, Allen Tannenbaum

- S. Dambreville, Y. Rathi, and A. Tannenbaum. A framework for Image Segmentation using Shape Models and Kernel Space Shape Priors. IEEE Transactions on Pattern Analysis and Machine Intelligence, (to appear 2007).
- O. Michailovich, Y. Rathi, and A. Tannenbaum. Image Segmentation using Active Contours Driven by Informaion-Based Criteria. IEEE Transactions on Image Processing, (to appear 2007).
- Y. Rathi, O. Michailovich, and A. Tannenbaum. Segmenting images on the Tensor Manifold. In CVPR, 2007.
- Eric Pichon, Allen Tannenbaum, and Ron Kikinis. A statistically based surface evolution method for medical image segmentation: presentation and validation. In International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI), volume 2, pages 711-720, 2003. Note: Best student presentation in image segmentation award.
- Y. Rathi, O. Michailovich, J. Malcolm, and A. Tannenbaum. Seeing the Unseen: Segmenting with Distributions. In Intl. Conf. Signal and Image Processing, 2006.
- J. Malcolm, Y. Rathi, A. Tannenbaum. Graph cut segmentation with nonlinear shape priors. In Intl. Conf. Signal and Image Processing, 2007.

Threshold-based

Edge-based

Region-based

Parametric methods

Implicit methods

Parameterized representation of the curve (shape)

Implicit representation of the curve using level sets

Image

Use Calculus of variations

- Segmentation by separating intensity based probability distributions (not just intensity moments as in previous works).
- Novel formulation of the Bhattacharyya distance in the level set framework so as to optimally separate the region inside and outside the evolving contour.

The Bhattacharyya distance gives a measure of similarity between two distributions:

where z Z is any photometric variable like intensity, color vector or tensors.

- B can also be thought of as the cosine of the angle between two vectors.

Let x R2 specify the co-ordinates in the image plane and I : R2 Z be a mapping from image plane to the space of photometric variable Z. Then the pdf is given by:

This is the nonparametric density estimate of the pdf of z given the kernel K.

The pdf’s written in terms of the level set function is given by :

For segmentation purposes, we would like to minimize the Bhattacharyya distance. This is achieved using calculus of variations, by taking the first variation of B as follows :

The first variation of Pin and Pout is given by :

where,

Plugging in all the components, we get the following PDE (partial differential equation) for separating the distributions :

In numerical experiments, an additional regularizing term is added to the resulting PDE that penalizes the length of the contour making it smooth. Thus, the final PDE is given by:

- Bhattacharyya Segmentation
- Segmentation Results
---------------------------------------

- Shape Analysis
- Shape-Driven Segmentation

Template Image

Generated Image

Toy example: Region inside and outside was obtained by sampling from a Rayleigh distribution with the same mean and variance.

Starting distribution

Final distribution

Actual distribution

- Intensity is not enough to segment several types of images.
- Diffusion Tensor MRI images have become common, where at each pixel a tensor is computed from a set of gradients.

Color coded Fractional Anisotropy image

- Structure tensor reveals many features like edges, corners or texture of an image.
- A structure tensor for a scalar valued image I is given by: (K is a Gaussian kernel)

Color structure tensor is given by:

- The space of n x n positive definite symmetric matrices, is not a vector space, but forms a manifold (a cone).
- Many past methods by Wang-Vemuri, Lenglet et.al., have however assumed the tensor space to be Euclidean. The active contour based segmentation was performed under this assumption.

Structure tensor space for a typical image.

Euclidean distance d(A,B) = d(A,C) = d(C,B) = d1

Riemannian distance dr(A,B) = d(A,C) + d(C,B) = 2d1

Thus, under Euclidean manifold assumption, one obtains an erroneous estimate for mean and variance used in many active contour based segmentation algorithms.

M

Y

Log Map

x

TxM

Y’

For a tensor manifold (cone), TxM is the set

of all symmetric matrices and forms a vector space.

- A recent method proposed by Lenglet et.al. (2006), incorporates the Riemannian geometry of the tensor space and performs segmentation by assuming a Gaussian distribution of the object and background.
- By using the Bhattacharyya distance and taking into account the Riemannian structure of the tensor manifold, we propose to extend the above segmentation technique to any arbitrary and non-analytic probability distribution.

Compute Mean on the Riemannian Manifold

Map all points onto the Tangent Space TM at the mean

Perform Curve Evolution using the PDE described earlier.

Compute Target points or bins

Euclidean Space

Accepted for publication in IEEE CVPR 2007

Segmentation using Bhattacharyya flow, but using Riemannian metric

Segmentation using Bhattacharyya flow, but assuming Euclidean distance between tensors

Segmentation using Bhattacharyya flow, but using Riemannian metric

Segmentation using Bhattacharyya flow, but assuming Euclidean distance between tensors

Color structure tensor:

Segmentation assuming Riemannian metric

Segmentation assuming Euclidean metric

- No assumption on the distribution of the object or background.
- Computationally very fast, since we only need to update the probability distribution instead of having to map each point in the image from Riemannian space to tangent space after each iteration to compute the mean and variance (under a Gaussian assumption).

- Bhattacharyya Segmentation
- Segmentation Results
---------------------------------------

- Shape Analysis
- Shape-Driven Segmentation

- Georgia Tech-
- Delphine Nain, Xavier LeFaucheur, Yi Gao, Allen Tannenbaum

- D. Nain, S. Haker, A. Tannenbaum. Multiscale 3D shape representation and segmentation using spherical wavelets. IEEE Trans. Medical Imaging, 26 (2007). pp 598-618.
- D. Nain, S. Haker, A. Bobick, and A. Tannenbaum. Shape-Driven 3D Segmentation using using Spherical Wavelets. In Proceedings of MICCAI, Copenhagen, 2006. Note: Best Student Paper Award in the category Segmentation and Registration.
- D. Nain, S. Haker, A. Bobick, and A. Tannenbaum. Multiscale 3D Shape Analysis using Spherical Wavelets. In Proceedings of MICCAI, Palm Springs, 2005.

- 3D, Parametric, Data-driven prior

Caudate nucleus dataset

Prostate dataset

- K shapes in training set, N landmarks

- Rank of the covariance matrix DDTis at most K-1
- Small training set: only first K-1 major variations captured by shape prior
- E.g. Reconstruction, given new shape s

Ground Truth

Reconstructed with ASM shape prior

- Hierarchical decomposition: shape is represented at different scales [Davatzikos03]
- Learn variations at each scale

- Step 1: Find Landmarks
- [Conformal Mapping]

- Step 2: Multi-scale representation
- [Spherical Wavelets]

- Step 3: Find independent bands of variation
- [Spectral Graph Partitioning]

Nonlinear area-preserving mapping [Brechbuhler95], Conformal mapping [Haker04]

A function decomposed in wavelet

space is uniquely described by a

Weighted sum of scaling functions

and wavelet functions that are

localized in space and scale

Wavelet, level 1

Wavelet, level 2

Wavelet, level 3

Scaling, level 0

Spherical scaling and wavelet functions are defined on a multi-resolution grid

is signal on the sphere

Analysis:

Synthesis:

In matrix notation:

j=0

j=1

j=2

j=3

After the registration step, all shapes have the required mesh structure

Given a shape Sk, we find a 3 1D signals:

We take the wavelet transform of

each signal and represent the shape as:

Shape representation using a weighted combination of the lowest resolution scaling functions and wavelet functions up to jth resolution

Original Shape

- At each scale, we can truncate least significant coefficients based on spectrum analysis of population
- Results in local compression at each scale

Compression: from 2562 to 649 coefficients, mean error 5.10-3

- Previous approach [Davatzikos 03]

We propose a more principled approach where

for each scale, we cluster highly correlated coefficients

into a band, with the constraint that coefficients

across bands have minimum cross-correlation

1 2 3 4 5 6

2 3 6 1 4 5

1

2

3

4

5

6

2

3

6

1

4

5

Covariance Matrix

level 1

Rearranged Covariance Matrix

level 1

- Spectral Graph Partitioning technique [Shi00]
- Fully connected graph G = (V,E) where nodes V are wavelet coefficients for a particular scale
- Weight on each edge: w(i, j) is covariance between coefficients i and j
- Stopping criterion: validating whether the subdivided band correspond to two independent distributions based on KL divergence

Color is influence of coefficients

in a band: red (high), blue (none)

- Assuming K shapes in training set, for each band, we obtain (K-1) eigenvectors
- In total we have B(K-1) eigenvectors, where B is number of bands

- Dataset of N samples randomly into T training samples and [N − T] testing samples, where T = [5, 10, 25]
- Reconstruction task:
- Test:
- Compare to ASM, other wavelet band decomposition
- Effect of noise
- Effect of truncation

GT

Noise

Noise

GT

WAV rec. from GT

WAV rec. from Noise

WAV rec. from GT

WAV rec. from Noise

ASM rec. from GT

ASM rec. from Noise

ASM rec. from GT

ASM rec. from Noise

Reconstruction from Ground Truth

Reconstruction from Noisy Shape

- Bhattacharyya Segmentation
- Segmentation Results
---------------------------------------

- Shape Analysis
- Shape-Driven Segmentation

End goal is to derive a parametric surface evolution equation by evolving the wavelet coefficients directly so that we can include the shape prior directly in the flow

Shape Representation

- Probability Distribution
- Learn shape space (evectors U)
- Learn bounds within shape space

Pose: Rotations,

Translations, Scaling

shape

Segmentation

Likelihood

Evolve , p

Shape Prior

Constrain , p

Segmented Shape

- Region-based energy

Region inside evolving

surface

Data term

[Rousson05]

Points on

Evolving surface

Points on

Evolving surface

- Update equations

step size

Run until

- Start with lowest resolutions
- Run for 1 iteration
- Constrain

- Add next resolution s when

- Coarse to fine evolution

- Evolution with PDM shape prior (Active Shape Model)
- Evolution with WDM shape prior
- Two types image input:
- Using Ground Truth image data (binary): to test convergence
- Using real image data

- Quantitative measurements:
- Compare to Ground truth (manually segmented)

- Details:
- caudate nucleus shapes from MRI scans
- training set of 24 shapes, testing set of 5 shapes
- 4 subdivision levels, 16 bands in the shape prior
- Start with mean shape, mean position

- Image data is binary

GT in red; ASM in blue

GT in red; Mscale in blue

GT in red; ASM in yellow

GT in red; Mscale in blue

- MRI image intensity

GT in red; ASM in blue

GT in red; Mscale in blue

- Geodesic tractography (tomorrow)
- Fast non-rigid registration (tomorrow)
- Estimation and filtering techniques from tracking?