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![Page 1: Medical Image Analysis - pages.stat.wisc.edupages.stat.wisc.edu/.../MIA.lecture09.Gaussian.Smoothing.mar.01.2007.pdf · Gaussian Kernel Smoothing We will study basic properties of](https://reader030.fdocuments.in/reader030/viewer/2022041201/5d4a8eca88c993d7648b8e31/html5/thumbnails/1.jpg)
Medical Image AnalysisInstructor: Moo K. [email protected]
Lecture 09.Gaussian Kernel Smoothing
March 01, 2007
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Gaussian Kernel Smoothing
We will study basic properties ofGaussian kernel smoothing and
numerical implementation issues.
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Kernel Smoothing, Convolution, Linear Filter
inputoutput kernel
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2D example
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Motivation for image smoothing: Improveperformance of PDE based segmentation - level set
No image filtering = More manual correction
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Malladi & Sethian’s Min/Max Flowdone by Thomas Hoffmann.This is basically a PDE smoother.
Original
Gaussian
Min/Max Flow
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Shape of kernelUnimodal, Symmetric (Isotropic), normalized
1D and 2D Gaussian kernel
Quiz: The cross section of 2D Gaussian kernel ?
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1D Brownian motion
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Brownian motion simulation ---> Gaussian kernel
# random walk hitting a target voxel Probability = ----------------------------------------------- # total random walk
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Constructing n-dimensional Gaussian Kernel
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Red= Gaussian kernel smoothingBlue = Diffusion smoothing after 5, 25 and 50 iterations
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MATLAB codeK=inline('exp(-(x.^2+y.^2)/2/sig^2)');>>KInline function:K(sig,x,y) = exp(-(x.^2+y.^2)/2/sig^2)
[dx,dy]=meshgrid([-2:2]);weight=K(0.5,dx,dy)/sum(sum(K(0.5,dx,dy)));>>weight
0.0000 0.0000 0.0002 0.0000 0.00000.0000 0.0113 0.0837 0.0113 0.00000.0002 0.0837 0.6187 0.0837 0.00020.0000 0.0113 0.0837 0.0113 0.00000.0000 0.0000 0.0002 0.0000 0.0000
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weight=K(1,dx,dy)/sum(sum(K(1,dx,dy)));
>>weight =
0.0030 0.0133 0.0219 0.0133 0.00300.0133 0.0596 0.0983 0.0596 0.01330.0219 0.0983 0.1621 0.0983 0.02190.0133 0.0596 0.0983 0.0596 0.01330.0030 0.0133 0.0219 0.0133 0.0030
Y=conv2(X,weight,'same');
Quiz: Why there is no sqrt(2)*sigma term in thecomputation?
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2D simulation results
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Better Algorithm
2D approach = 1D approach x 2
Perform 1D version of kernel smoothing in each coordinate
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Gaussian Kernel estimator
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obervation = signal + noise
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Signal
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Prediction
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PredictionSignal
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Optimal bandwidthchoose sigma that minimizes the
integrated squared error
Many technique uses some sort of cross-validation
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Iterated kernel smoothing
smoothedX=X;for i=1:100 smoothedX=conv2(smoothedX, weight,'same');end;Y=smoothedX;
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Covariance function ofrandom field
White noise = random field with Dirac-delta function asthe covariance function.
Gaussian white noise = Gaussian + white noise
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Dirac-delta function
This is not really a function intraditional mathematical sense.
How you construct numerically?
Let the bandwidth of isotropicGaussian kernel goes to zero.
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How to simulate Gaussian field
Smooth field Gaussian white noise
How?
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Numerical Implementation
e=normrnd(0,0.4,101,101);smooth_e=e;for i=1:10 smooth_e=conv2(smooth_e,K,'same'); figure;imagesc(smooth_e);colorbar;end;
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Simulating Gaussian field
N(0, 0.4^2) Gaussian white noiseIterative kernel smoothing with sigma=0.4 and 1,4, 9 iterations
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Lecture 10 Topics
Random Field TheoryMultiple Comparison Corrections