Final Project Part IMATLAB Session

ES 156 Signals and Systems 2007

SEAS

Prepared by Frank Tompkins

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

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

Xkn0

N1xnj2 k nN

Xk wkn0

N1xncosk 2n1

2N

wk 1N if k 02N if k 0

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

Xk, L wkwLm0

M1n0

N1xm, ncosk 2m 1

2McosL 2n 1

2N

wt 1N if t 02N if t 0

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 [n] is input– In 2-D, h[m,n] is output when

[m,n] = [m][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

frac = 0.78;

thresh = max(max(filt))*frac + min(min(filt))*(1-frac);

result = filt > thresh;

imshow(result);

• Set lowest 78% (or whatever) image values to zero and the rest to white to emphasize edges