Homework 5 (Due: 21st Jan.)

(1) What are the vanish moments of (a) the sinc wavelet and (b) the 12-point symlet? (10 scores)

(2) Why the complexity of the 1-D discrete wavelet transform is linear with N? (10 scores)

(3) (a) What is the advantage of the symlet when compared with the Daubechies wavelet? (b) What is the advantage of the 2x2 structure when compared with the original structure for the DWT? (c) What is the advantage of the curvelet and the bandlet when compared with the original 2-D wavelet? (20 scores)

(4) Why the wavelet transform can be used for (a) edge and corner detection and (b) filter design, and (c) image compression? (15 scores)

(5) For a two-point wavelet filter, if g[0] = a, g[1] = b, and g[n] = 0 otherwise, (a) What are the constraints of a, b if g[n] is a quadratic mirror filter?

(b) What are the constraints of a, b if g[n] is an orthonormal filter? (15 scores)

(6) (a) Write a Matlab program for the following 2-D discrete 8-point Daubechies wavelet.

The Matlab program should be mailed to me. (30 scores)

[x1L, x1H1, x1H2, x1H3] = wavedbc8(x)

x[m, n]

g[n]

h[n]

2

2

along n

along n

v1,L[m, n]

v1,H[m, n]

g[m]

h[m]

along m 2 x1,L[m, n]

2along m

x1,H1[m, n]

g[m]

h[m]

along m

along m

2

2

x1,H2[m, n]

x1,H3[m, n]

(b) Also write the program for the inverse 2-D discrete 8-point Daubechies wavelet transform.

x = iwavedbc8(x1L, x1H1, x1H2, x1H3)