Post on 20-Mar-2021
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