A State of Art Seminar on

Variational P.D.E. based Image Processing with

Applications to HDBT

Vijayakrishna Rowthu(Y9108070)

Under the guidance of Prof.B.V.Rathish Kumar

Dept. of Mathematics & Statistics

contents

What is an Image? Defects in Images and Recovery PDEs in Image Processing Variational/PDE Models from Axioms and

Results DTI and Brain Fibre Tracking Existing Methods and comparison Proposed work References

What is an Image?

An Image is a 2-D view (projection in a specific direction ) of the surface of an object (or a scene) from the space we live-in(3D) that is formed by the light rays reflected by the object.

Basics of an Image

Analogue Image(Continuous Image): Digital Image(Discrete Image): Obtained from

an analogue Image by sampling and quantization. --- Represented as A 2-Dimensional array of pixels with assigned brightness values.

Resolution: The number of Rows and Columns in an image.

The higher the resolution , the closer the digital image is to the physical world.

Mathematical Representation

Define an analogue Image(u) as a measurable ,bounded function:

Where Ω is a connected,bounded domain from ℝ2 , And ℝd is an intensity valued Range set.

In case of A Digital(Discrete) Image(I ):

For Grey Images : d=1 ; RGB images :d=3

Digital Image Processing

Image processing is a Mathematical procedure involving computer algorithms where the input is an image ,And , the output may be either an image or a set of characteristics or parameters related to the image.

Most image-processing techniques involve treating the image as a two-dimensional signal and applying standard signal-processing techniques to it.

Some Rough Idea on Edges

Typically, an Image represents a scene containing a background and several objects.

It is common that , inside the background and in each object the values are smoothly varying, but are discontinuous or change abruptly across object boundaries (edges), thus creating large image gradients at these locations.

We define edges as the locations where the gradient | f| is large due to a discontinuity or ∇a sharp transition in the image intensity.

Common Problems and Known Remedies

Noise (Unwanted information) , is a defect, occurs during the recording & communication process. Local Averaging is a simple solution. Local Median for impulse noise.

Components Extraction(Segmentation): By Properly detecting the contours (edge sets) , one can achieve this goal, but the contour finding in-general is not an easy task.

(cont.)Common Problems and Known Remedies

Deconvolution for Restoration, : Restoration attempts to reconstruct or recover an image that has been degraded by using a priori knowledge of the degradation phenomenon. g(x,y)=H(f(x,y))+n(x,y)

The main Difficulty arises when the SNR is low.The results are highly sensitive to noise ,sometimes losing the basic Structure of the image , even.

The Least Square Error Filter(wiener) is a popular Solution

Drawbacks

It is difficult to say which one is more natural approach than the other.

They rely on –Filters –Spectral Analysis –Basic Concepts of Statistics & Probability

Image Processor

Abstractly, image processing can be considered as an input-output system Q

0 → Image Processor T → Q

The input data Q0 can represent an observed

single image , and the output Q = (q1,q2,···) contains all the targeted image features.

Typical image processors(T): denoising,deblurring,segmentation, compression, or inpainting.

Models of Some Image Processors

Current Tools of Generation

The current day sophisticated tools emerged from 3-directions. 1.Stochastic Modeling :(evolution of some random value, or system, over time) 2.Wavelets:(small waves (or) brief oscillations) 3.Partial Differential Equations

Why PDEs ?

PDEs are closely related to the physical world. Reasoning in a continuous Framework makes

the understanding of physical realities easier and provides the intuition to propose new models.

Main Interest in using PDEs is that the Mathematical Theory is well established.

(cont.)Why PDEs

PDEs are written in a continuous setting , referring to analogue images, and once the existence and the uniqueness have been proven , we need to discretize them in order to find a numerical solution.

Mathematical Axioms for Image transformation allow us to develop a framework for good prosperities in PDE design

Most of the methods are derived from the support of Calculus of Variations and Differential Geometry.

Capability of PDEs

Image Restoration Enhancement (or) Denoising Image In-painting Edge Detection

Segmentation

In computer Vision ,analyzing images at different resolutions(scale) is necessary to extract Various Descriptions of an Image. As a primitive scale-parametrization, the Gaussian convolution is attractive for its "well-behavedness":

The Gaussian is symmetric and strictly decreasing about the

mean, and therefore the weighting assigned to signal values decreases smoothly with distance.

The Gaussian convolution behaves well near the limits of the scale parameter, t, approaching the un-smoothed Image for small t, and approaching the Mean value of Image for large t.

The Gaussian is also readily differentiated and integrated.

Gaussian Smoothing[witkin]

The first step in this direction is Scale Space Filtering for Smoothing

[Witkin,1983]:(Scale Space) The essential idea of this approach is to, Embed the original image in a family of derived images I(x, y, t);t≥0 obtained by convolving the original image I

0(x,y) with a Gaussian kernel

G(x,y; t) of variance t. I(x,y,t) = I

0(x, y)*G(x,y;t)

Larger values of t (the scale-space parameter), correspond to images at coarser resolutions.

Variance

Scaled Images from Gaussian convolution

For Scale parameter, t=0,1,4,16

Scale-Space

Define a multi-scale analysis(Scale Space)as a family of operators T

t;t ≥ 0

which, applied to the original image u0(x), yield

to a sequence of images u(t,x) = (T

tu

0)(x)

For simplicity, we suppose for all t ≥ 0, T

t: C∞

b(R2) → C

b(R2) where C∞

b(R2) is the

space of bounded functions having derivatives at any order.

Relating to the Heat Diffusion Equation (Linear Diffusion)

[Koenderink and Hummel ]: These one parameter family of derived (scale-space) images may equivalently be viewed as the solution of the heat diffusion equation.

It=ΔI =Ixx+Iyy (Isotropic Diffusion) with the initial condition, I(x,y,0)=I0(x, y), the original image and with no intensity loss at boundaries.

Primitive Axioms for Scale Space(Linear Filters)

[Koenderink] motivates the diffusion equation formulation by stating two criteria.

1) Causality: Any feature at a coarse level of resolution is required to possess a (not necessarily unique) “cause” at a finer level of resolution although the reverse need not be true. i.e. no spurious detail should be generated when the resolution is diminished. T

0= Id and s < t, T∀ ∃

(s,t): T

t= T

(s,t) T

s ;

(cont.)Primitive Axioms for Scale Space

Homogeneity and Isotropy: The blurring is required to be space invariant.(?) For R, rotation in R2, (Ru)(x) = u(Rx): T

t(Ru) = RT

t(u);

The Diffusion Equation ,suits well to these two criteria , It=cΔI =c(Ixx+Iyy )

where c is a positive diffusion constant.

Anisotropic diffusion for denoising [Perona & Malik,1987]: Observed that

isotropic diffusion is causing a serious damage to the visibility of Edges as diffusion progresses.

Found that , directionally dependent smoothing is the right choice for preserving edges with better Image denoising (enhancement / smoothing).

Chose, the diffusion factor c(x,y) ,in such a way that the diffusion will be relatively faster at locations of low Gradient than at the locations of High Gradients(Edges).

(cont.)Anisotropic diffusion for denoising

Now the problem of denoising becomes ,solving The second-order PDE,

Successful choices for c(x,y) have been,

(contd.)Theoretical Explanation

Associate the Second order PDE with an Energy Functional. Where Ω is the image support, and f(.)≥0 is an increasing function associated with the diffusion coefficient as Anisotropic diffusion is then shown to be an energy-dissipating process that seeks Min E(u).

Flaws in second order [Whittaker]:

Since the Laplacian of an image at a pixel is zero if the image is planar in its neighborhood, these PDEs attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image.

Piecewise planar images look more natural than step images(blocky) which anisotropic diffusion uses to approximate an observed image.

This effect is visually unpleasant and is likely to cause a computer vision system to falsely recognize as edges the boundaries of different blocks that actually belong to the same smooth area in the original image.

It's extensions to fourth order

[You,Kaveh,2000]: Fourth order linear diffusion dampens oscillations at high frequencies much faster than second order diffusion.

To avoid blocky effects while achieving good tradeoff between noise removal and edge preservation.

(Cont.) Fourth Order

Consider the Energy Functional, f(.)≥0 is an increasing function f '(.)>0, so that the functional will be an increasing function with respect to the smoothness of the image as measured by |2u|. Therefore,the minimization of the functional is equivalent to smoothing the image.

(contd.)Fourth order

Min E(u) is achieved by Euler Equation ,

The Euler equation may be solved through the following gradient descent procedure:

A dynamical scheme by making the function u(x,y) depend on an artificial

parameter (the time) t ≥ 0

The Alvarez-Guichard-Lions-Morel scale space theory

The remarkable work of [Alvarez et al] establishes the connection between scale space analysis and PDEs , rigorously.

Starting from a very natural filtering axiomatic (based on desired image properties) they prove that the resulting filtered image must necessarily be the viscosity solution of a PDE.

Some Basic Axioms that are very natural from Image Processing Perspective are:

Viscocity solution of a 2nd order PDE

These axioms and invariance properties are quite natural from an image analysis point of view. A1 means that a coarser analysis of the original image can be deduced from a finer one without any dependence upon the original picture. A2 states a continuity assumption of Tt.A3 means that (Ttu)(x) is determined by the behaviour of u near x. A4 expresses the idea that if an image v is brighter than another image u, this ordering is preserved across scale. Finally, I1 and I2 state respectively that no a priori assumption is made on the range of the brightness and that all points are equivalent.

Classical definition concerning the well-posedness of a

minimization problem or a PDE. Definition[Hadamard]:

When a minimization problem or a PDE admit a unique solution which depends continuously on the data, we say that the minimization problem or the PDE are well-posed.

If one of the following conditions: existence, uniqueness or continuity fails, we say that the minimization problem or the PDE are ill-posed.

Existence of a PDE based Filter

Proof in brief

Existence of a Linear Filter

Note: These theorems are very interesting since they express that the Alvarez et al theory is a very natural extension of the linear theory (Theorem 3.3.3) but also because the multi-scale axiomatic leads to new nonlinear filters (Theorem 3.3.4).

Curvature dependent Non-Linear Filter

Edge-Detector from Anisotrpic Diffusion[Perona & Malik]

[Canny,1986] Edge Detector:The image is convolved with the directional derivatives of a Gaussian. u→ *( ( , ,σ))|u G x y | Requires a number of convolutions ,involve blurring And complexity of combining outputs of filters at multiple scales.

Anisotropic E.detector : The complication of multiple scale, multiple orientation filters is avoided by locally adaptive smoothing.

In this, the edges are made sharp by the diffusion process, so that edge thinning and linking are almost unnecessary.

(cont.)Anisotropic Edge Detector

Deconvolution

Image Restoration The classic Model of a Restoration Problem is,

u0=Ru+η

where η stands for a white additive Gaussian noise and R is a linear operator representing the blur (usually a convolution).

Given u0, the problem(inverse) is then to

reconstruct u(x,y), the problem is ill-posed and we are only able to carry out an approximation of u.

The Energy Method supposing that η is a white Gaussian noise,

and according to the maximum likelihood principle, to find an approximation of u by solving the least square problem: The Minimizer (if exists) satisfies the equation,

This is an ill-posed problem as, R*R is not always one-to-one & may have small eigenvalues.

Regularisation of the problem(ill-posed)

[Tikhonov and Arsenin,1977]:(data fidelity+smoothing) in Functional Space,

Solution characterized by the Euler-Lagrange equation is,

Total Variation (L1 -) Minimization

As Laplacian Operator has very strong isotropic smoothing properties , edges are not preserved.

The Lpnorm with p = 2 of the gradient allows to remove the noise but penalizes much of the gradients(edges).

[Rudin, Osher and Fatemi]: Decrease p in order to keep as much as possible the edges.

To find ϕ(|u|) Find the properties on ϕ(s) so that the solution of

the minimization problem is close to a piecewise constant image, that is formed by homogeneous regions separated by sharp edges.

Euler-Lagrange Equation:

Decomposing the divergence term using the local image structures(isophotes) to see clearly the action of the operators in directionsT and N.

(cont.)To find ϕ(|u|)

Imposing , gives rise to, (a uniformly elliptic equation having strong regularizing properties in all direction.)

To diffuse along C (isophote)(in the T-direction) and not across it , Annihilate, the coefficient of u

NN (strong

gradients) and set the coefficient of uTT

not to vanish.

Performance Metrics[wang, et. al]

Figure of Merit Metric (FOM): (Edge preserving measure) where N^ and N

ideal

are the total numbers of detected and original edge pixels, respectively;

di is the Euclidean distance between the ith detected edge pixel and

the nearest original edge pixel; λ is a constant typically set to 1/9. The dynamic range of FOM is between the processed image and the

ideal image.

Performance Metrics[wang, et. al]

Structural Similarity Metric(SSIM): Let x=x

i; y=y

i be the original and the test

images. This quality index models any distortion as a combination of 3 different factors: loss of correlation, luminance distortion, and contrast distortion.

The dynamic range of SIMM is [-1,1]

Performance Metrics[wang, et. al]

Mean Square Error Metric (MSE): The smaller the MSE value, the better is the denoising performance.

SNR Metric: when the denoised image has a large SNR it will be closer to the original image and will have a better quality.

Image Inpainting

Image Inpainting u

0denotes the observed(noisy or blurry) portion

of u, on a sub domain D. The goal of inpainting is to recover u on the entire image domain Ω.

A simple Geometric Model is , with blurring followed by noise degradation and spatial restriction. where K is a continuous blurring kernel (linear,shift-invariant), and η is an additive white noise field assumed to be close to Gaussian for simplicity &the information [u

0]

Ω\D

is missing.

(cont.)Fidelity term for deconvolution

Define a error measure for the Quality of Fit : measures how well the observation u

0 fits

if the original image is indeed u and |D| is its area or cardinality for Discrete case.

(cont.)Regularity Condition E[u]

Regularity of the new Image(u) is enforced through The “energy” functionals: E[u]

Then the Image Inpainting becomes a constrained optimization problem: min E[u] over all u such that E[u

0|u] ≤ σ2

Here σ2 (variance of the white noise), is assumed to be known by proper statistical estimators.

Examples of E[u]

Sobolev norm E[u] =∫Ω| u|∇ 2dx,

The total variation (TV) model E[u] =∫Ω|Du| of

Rudin, Osher, and Fatemi, and The Mumford-Shah free-boundary model

E[u,Γ] =∫Ω\Γ

| u|∇ 2dx + βH1(Γ),

where H1 denotes the one-dimensional Hausdorff measure.

TV is better than Sobolev Norm

For The TV norm case ,The space of functions ( images ) of bounded total variation: BV(Ω) is considered. It does the removal of spurious oscillations, while sharp signals are preserved.(where as The Sobolev energy blows up.)

The Minimization Model with control parameters is,

TV Inpainting

The object-edge model [Ems

][Mumford and Shah]

An image u is understood as a combination of both the geometric feature Γ and the piecewise smooth “objects” u

i on all the

connected components Ωi of Ω \ Γ , assuming

Γ to be Lipschitz. And the smoothness of the “objects” characterized by Sobolev Norm.

(cont.)Higher-order geometric Image Models

[Esedoglu and Shen]: For large-scale inpainting problems, high-order image models which incorporate the curvature information become necessary for more faithful visual effects. Replacing length energy by Euler's elastica Energy:

The curvature is given by

Euler Elastica Model----->

Mumford-Shah Inpainting

Image Segmentation Images are the proper 2-D projections of the 3-D world

containing various objects. Segmentation is to identify the regions in images that

correspond to individual objects. Denote by u

0 an observed image on a 2-D Lipschitz

open and bounded domain Ω. Segmentation means finding a visually meaningful

edge set Γ that leads to a complete partition of Ω. Each connected component Ω

i of Ω\Γ should

correspond to at most one real physical object or pattern in our 3-D world, for example, the white matter in brain images.

Piecewise-smooth Mumford-Shah Model (2-phase Segmentation)

This method tries to find the segmented “objects” to have smoothly varying intensities(Homogenous Regions).

The Problem of Minimization will be,

The ideal image u(x) is segmented to u±(2-phases) by the level set function . ( )=u x u+( ) (x H ( ))+x u-( )( - (x 1 H ( ))).x

(First Stage) Denoising the regions

Assuming, both u+ and u- are C1functions up to the boundary = 0 ,Minimize the Energy ,

First, with fixed, the variation on E[u+,u-,|u

0] leads to the two Euler-Lagrange equations

for u± separately. These act as denoising operators on the homogeneous regions only,not on edge set=0

(Second stage)Contour evolution Next, keeping the functions u+ and u- fixed

and minimizing E[u+,u-,|u0] with respect to , we obtain the motion of the zero-level set with some initial guess (t=0,x). This single model, which includes both the original energy formulation And the elliptic and evolutionary PDEs ,naturally combining all three image processors —active contour, segmentation, and denoising.

(cont.)Evolution of contours &region smoothing

Extension to 4-Phase Four-Phase Formulation:(For 4 disjoint

segments) 1>0,2>0

Another Model :If we allow the segmented regions by Piecewise-constant , then the Minimization Problem will be of much simpler form for computations.

Piecewise-constant M-S (Texture) Segmentation

Active contours without Edges(snakes)[The Kass-Witkin-Terzopoulos]

Unlike the Mumford and Shah functional, the aim is no longer to find a partition of the image but to automatically detect contours of objects,starting from a initial guess and g(s),an Edge-detector function. Working: Boundary detection consists in matching a deformable model to an image by means of energy minimization.

Detection of a simulated minefield And Segmentation of an MRI brain image.Interior boundaries are also automatically detected.

(cont.)Evolution of Contours

(cont.)

The first-order term makes the curve act like a membrane and the second-term makes it act like a thin plate. Setting β = 0 allows second order discontinuities as corners. The third term, the external energy, attracts the curve towards the edges of the objects. As Ω is bounded, the energy J(c) admits at least a global minimum in the Sobolev space (W 2,2(a,b))2.

(cont.)

In practice, to solve numerically the problem we embed the E-L conditions into a dynamical scheme by making the curve depend on an artificial parameter (the time) t ≥ 0. where c

0(q) is an initial curve surrounding

the object to be detected.

Human Brain Anatomy

Neurons & Signal Transmission

Neurons receive information either at their dendrites or cell bodies. The axon of a nerve cell is, in general, responsible for transmitting information over a relatively long distance. Therefore, most neural pathways are made up of axons. If the axons have myelin sheaths, then the pathway appears bright white because myelin is primarily lipid. If most or all of the axons lack myelin sheaths , then the pathway will appear a darker beige color, which is generally called gray,

White matter

What is Tractography? A neural pathway(tract),

connects one part of the nervous system with another and usually consists of bundles of elongated, myelin-insulated neurons, known collectively as white matter.

Neural pathways serve to connect relatively distant areas of the brain or nervous system, compared to the local communication of grey matter.

DT-MRI[Denis Le Bihan] THE BASIC PRINCIPLES of diffusion MRI were

introduced in the mid-1980s [Taylor et al,1985]; they combined NMR imaging principles with those introduced earlier to encode molecular diffusion effects in the NMR signal by using bipolar magnetic field gradient pulses .

Molecular diffusion refers to the random translational motion of molecules, also called Brownian motion, that results from the thermal energy carried by these molecules.

These random, diffusion driven displacements molecules probe tissue structure at a microscopic scale well beyond the usual image resolution:

Diffusion tensor

‖Dan

Measures the deviation of diffusion Tensor ‖from Isotropy

Diffusion Anisotropy Measures

Geometrical Measures of Diffusion

Lattice index (LI)

Existing Approaches for Fibre Tracking

Using principle eigen direction. Navier stokes-GGVF-Active contours(snakes) Using Heat Diffusion Equation-GGVF-Snakes

Fibre Assignment by Continuous Tracking(FACT)

Each Voxel at postion (x,y,z) is characterized by Second order diffusion Tensor D which represents the local 3D anisotropic Gaussian diffusion process

To infer continuity of Fibre orientation from voxel to voxel and to reconstruct the connections between the brain regions [Basser et al,2000] , a 3D-arc length parametrized trajectory r(s) ,has been proposed as , and is solved for r(s) ,starting from a seed position r

0=r(s

0).

Reconstructed wild type mouse brain fibers from in vivo DT-MRI

Source: http://www.bruker-biospin.com/dti-apps.html

Using Navier Stokes equations simulate the flow of an artificial fluid governed

by the Navier–Stokes equations.

Using Finite volume method to approximate the solution ,solution of the steady state is found by,

Gradient Vector Flow(GVF)(new external force for Snakes )

[Xu et. al,1998]-Define the gradient vector flow field to be the vector field Φ(x,y)=[u(x,y),v(x,y)] that minimizes the energy functional , where f is the fluid volume & when |f| is large, the 2nd term dominates the integrand, and is minimized by setting Φ= ,f producing the desired effect of keeping Φ(x,y) nearly equal to the gradient of the edge map and also forcing the field to be slowly-varying in homogeneous regions.

(cont.)

Solve the E-L Equations , for Φ(x,y) that minimizes The Energy functional ,

(cont.)Computing Probable connection

The most likely connection path is then estimated using a generalized gradient vector flow (GGVF)based approach to compute the trajectory through the fluid velocity vector field .

Tract connecting the ROI's using GGVF instead of GVF

(cont.)

Heat Diffusion Equation based Tractography

Generate The Heat flow vector Field , using :

Trial on a Helical Phantom

Metric of Tract Fidelity(to compare the methods)

Average fractional Anisotropy: The principal eigen vector, defines the main direction of diffusion of water molecules in that voxel. Chose a path that gives Maximum of the integral of innerproduct (path and principal direction) multiplied by FA.

Maximum Inner product colour coded Map:

On HARDI data DT-MRI implicitly assumes that the diffusion is

Gaussian everywhere when estimating intra-voxel diffusion configurations where more than one single fiber direction predominates.

DTI cannot model crossing or kissing fibers but it also estimates wrong directions in the case of multiple fiber configurations

High Angular Resolution Diffusion Imaging (HARDI) comes as an interesting alternative as it samples the diffusion signal only on the single sphere following discrete gradient directions;.

Proposed work To take advantage of adaptive grids in computations

from Finite element methods. To work on 3D-Image data (stacks) with the

knowledge from Differential Geometry. Applications to Brain tractography segmentation for

HARDI. To Frame, segmentation problems using cahn-hilliard

equation (reverse diffusion) for binary fluids. Enhancing the performance of ill-posed problems by

suitable addition of arguments. Test various models with the Norm of L1+ϵ

Cahn-hilliard Evolution of random initial data γ=0.5 and C=0,

demonstrating phase separation

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