Non-Parametric Mixture Model Based Evolution of Level Sets and Application to

Medical Images

Niranjan Joshi and Michael Brady

Presented by Lu Ren

Dec. 19, 2010

Outline

• Problem & Motivation• Introduction• Non-Parametric Estimation of PDFs• Non-Parametric Mixture Model-ICLS• Curve Evolution Methods• NPMM Based Evolution of Curves• Results and Discussion

ProblemImage Segmentation

modelling the data spatial regularization

Modelling the data: Gaussian density vs. non-parametric probability density

GMM vs. non-parametric mixture model (NPMM)

Spatial regularization:

learn region statistics gradually within spatial contiguity

Medical applications:analysis of MRI

MotivationIndividual class PDFs do not follow Gaussian distribution

Especially true for MR images due to the bias field distortion

residual bias at the top

heterogeneous nature in the central partasymmetric nature

IntroductionSingle class PDF estimation

histogram approximation, kernel density approach, NP windows

Block diagram describing the NP windows PDF estimator

Partial volume effect for medical images

downsampling a high resolution image partial tissue classes

PMFs for the pure tissue classes basis PMFs of all tissures

Curve evolution methodslevel set embed the curve into a higher dimensional functionevolve the curve along the normal direction

NP-Windows MethodEstimate the PDF of images using a continuous representation

A simple 1-d signal case:

Consider a 2D image: bilinear interpolation

Suppose that is related to the positional variables by bilinear interpolation over a piecewise section

over

NP-Windows Method

NP-Windows Method

The PDFs obtained over each tessellated bilinear section are summed and normalized to develop the estimated PDF of the given image.

NPMM-ICLSICLS: inequality constrained least square

estimate of prior PMF of tissue classes

: the indices of pixels: the observed intensity image: the possible intensity levels: the underlying partial volume (PV) segmentation

: all possible tissue class labels: the number of pure tissue classes: the total number of tissue classes

: the contributing tissue fractions

NPMM-ICLS: the high resolution image

: the underlying segmentation

Assume

given the PMFs in high resolution image

Estimate the basis PMFs of all tissues in low resolution image

Calculate the tissue class weight via ICLS

Calculate with MAP

Curve EvolutionThe curves coincide with the boundaries of the segmentation

Level Set Methods evolve the curve C along the normal direction

, where called a level set function

C is always given by

Geodesic Active Contours:

region competition weighted length of the curve

Minimizing the above energy function to get the level set update equation

NPMM based Evolution of CurvesThe NPMM algorithm provides the following information:

: boundary detector

Consider K separate level set functions ,one for each pure class

with a little modification:

inhibition term

NPMM based Evolution of Curves

Experiment Results

Initial level set contour Final zero contour

Individual class distribution

Fitted distribution vs. overall intensity

distribution

Initial level set contour Final zero contour

Individual class distribution

Fitted distribution vs. overall intensity

distribution

Results on natural and simulated images

2 class segmentation

Quality evaluation: visual inspection & closeness of the NPMM fitted distribution and the overall intensity distribution

Experiment Results

Initial level set contour Final zero contour Individual class distribution

Fitted distribution vs. overall intensity

distributioncompare with GMM:

Experiment ResultsPartial volume segmentation

synthetic image (PSNR=26dB)

level set segmentation

tissue fractions estimated along the row 64, all the

rows and the ideal values

squared error in estimating partial tissue fractions

performance plot of mean squared error in estimating tissue fractions

against PSNR in dB

Experiment ResultsSimulated brain MR image segmentation

1mm slice thickness 3mm slice thickness

Dice index: compares intersection of the set of segmented pixels with the set of ground truth pixels to the addition of total number of pixels in both the sets

Experiment ResultsSimulated lung PET image segmentation

one slice of the 3D phantom and the simulated PET volume

red region: lung blue: heart

green: other tissues 3D zero level set surface compare various distributions

Experiment ResultsMR image segmentation

slice 1 slice 2

compare various distributionsslice 3 slice 4

Conclusions

Primary contributions:

(i) estimation of probability distribution using the NP-windows method

(ii) NPMM-ICLS modeling of the image histogram

(iii) accommodation of the partial volume effect

Discussions:

(i) the NPMM estimates the intensity values of various classes

(ii) level set framework provides spatial continuity