Final Project Part I MATLAB Session

1. Final Project Part IMATLAB Session ES 156 Signals and Systems 2007 SEAS Prepared by Frank Tompkins

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

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

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

5. 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

6. 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]

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

8. DCT Example

9. 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 outputy[m,n] = h[m,n] **x[m,n] • This is a 2-D convolution

10. 2-D Convolution • MATLAB command conv2

11. 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 whend[m,n] = d[m]d[n] is input

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

13. 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);

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

15. 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);