Final Project Part I MATLAB Session ES 156 Signals and Systems 2007 SEAS Prepared by Frank Tompkins

Outline Discrete Cosine Transform (DCT) Basic Image Processing Review of DFT/FFT 1-D and 2-D Basic Image Processing 2-D LTI systems Impulse response Filtering kernels and fspecial Thresholding

DFT Review Finite length discrete time signal x[n] Discrete fourier transform fft, ifft, fftshift, ifftshift in MATLAB

DCT Discrete Cosine Transform Discrete Fourier Transform Note the similarity DCT is roughly real part of DFT

2-D Discrete Time Signals x[m,n], finite length in both dimensions Can think of it as a matrix Doesn’t have to be square An image is one example How to do frequency analysis of a 2-D signal? Just do two Fourier transforms; one for each dimension

DCT (2-D) Each signal/function can be thought of as a matrix We can treat x[m,n] as a function of m for each n Define xn[m] = x[m,n] Compute DCT of xn[m], call it Xn[k] Treat Xn[k] as a function of n for each k Define xk[n] = Xn[k] Compute DCT of xk[n], call it Xk[l] Then we define the (2-D) DCT of x[m,n] to be X[k, l] = Xk[l]

DCT Overall formula Note that signal doesn’t need to be square Compute with MATLAB commands dct2 and idct2

DCT Example

2-D LTI Systems In 1-D, LTI systems is characterized entirely by impulse response h[n] Input x[n] yields output y[n] = h[n] * x[n] For 2-D LTI systems, we have a similar result Input x[m,n] yields output y[m,n] = h[m,n] ** x[m,n] This is a 2-D convolution

2-D Convolution MATLAB command conv2

But what is h[m,n]? Analogous to 1-D case In 1-D, h[n] is output when d[n] is input In 2-D, h[m,n] is output when d[m,n] = d[m]d[n] is input

fspecial MATLAB command to generate some common 2-D filters (aka kernels) Gaussian Sobel Prewitt

Filtering Example Prewitt kernel This filter emphasizes horizontal edges in an image H = fspecial('prewitt'); filt = conv2(H, X); imshow(filt); XT = dct2(filt); imshow(XT);

Filtering Example Note that the left side of the DCT is for horizontal edges

Thresholding Set lowest 78% (or whatever) image values to zero and the rest to white to emphasize edges frac = 0.78; thresh = max(max(filt))*frac + min(min(filt))*(1-frac); result = filt > thresh; imshow(result);