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

2. 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

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 output y[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 when d[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);