CSE 5780 Medical ImagingFlorida Institute of Technology

Fall 2012

Frequency Spectrum◦ Be basically the frequency components

(spectral components) of that signal◦ Show what frequencies exists in the signal

Fourier Transform (FT) ◦ One way to find the frequency content◦ Tells how much of each frequency exists in a

signal

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Different in Time Domain

Frequency: 2 Hz to 20 Hz Frequency: 20 Hz to 2 Hz

Same in Frequency Domain

At what time the frequency components occur? FT can not tell!

At what time the frequency components occur? FT can not tell!

Wavelet decompostion Matlab code : clc; clear all; close all; for indx=1:72 [Y] = dicomread('demo', 'frames',indx); figure(1) subplot(2,3,1),imshow(Y,[]); xlabel('Orginal DICOM image'); A = (fft2(double(Y))); [P,Q]=size(Y);

% step 2: Filtering the image in F-D by gaussian filter H = gausslfilter(P,Q,5*i);% A = H.*single(A); % step 3: Display the results in F-D subplot(2,3,4), imshow(log(1+abs(A)),[]); xlabel('H X Image spectrum (2D fft)'); subplot(2,3,3), imshow(log(1+abs(H)),[]); xlabel('Gaussian Filter)'); % step 4: apply 2D IFFt on each frame(takes the image from frequeency domain to % spatial domain ) A = ifft2(A); % step 5: Display the results in spacial-D subplot(2,3,5), imshow(abs(A),[]); xlabel('Restored DICOM image (2D ifft)'); end pause(5) end

Orginal DICOM image Image spectrum (2D fft)

H X Image spectrum (2D fft)

Gaussian Filter)

Restored DICOM image (2D ifft)

Orginal DICOM image Image spectrum (2D fft)

H X Image spectrum (2D fft)

Gaussian Filter)

Restored DICOM image (2D ifft)

Orginal DICOM image Image spectrum (2D fft)

H X Image spectrum (2D fft)

Gaussian Filter)

Restored DICOM image (2D ifft)

Orginal DICOM image Image spectrum (2D fft)

H X Image spectrum (2D fft)

Gaussian Filter)

Restored DICOM image (2D ifft)

For image processing applications we need wavelets that are two-dimensional.

This problem reduces down to designing 2D filters.

We will focus on a particular class of 2D filters: separable filters (Daubechies wavelets)

LH

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high pass

high pass high pass

We can interpret the decomposition as a breakdown of the signal into spatially oriented frequency channels.

Decomposition of frequency support

Arrangement of wavelet representations

Three levels of 2D wavelet decompositions:[Y] = dicomread('demo', 'frames',indx)[a1,h1,v1,d1]=dwt2((Y),'db2'); plotting(a1,h1,v1,d1); [a2,h2,v2,d2]=dwt2((a1),'db2'); plotting(a2,h2,v2,d2);[a3,h3,v3,d3]=dwt2((a2),'db2'); plotting(a3,h3,v3,d3);

d1=zeros(size(d1)); For the first level d2=zeros(size(d2)); For the second level d3=zeros(size(d3)); For the third level

a2=idwt2(a3,h3,v3,d3,'db2'); a1=idwt2(a2,h2,v2,d2,'db2'); recImage=idwt2(a1,h1,v1,d1,'db2');

orginal RECONSTRUCTION