ECE 547: Image Processing
Wavelet Homework
1. Given a binary image Abelow (i.e. a white square in the black background), compute the Haar
wavelet transform using the filter bank structure in the lecture notes with H0(z) = 1
√2(1 + z),
H1(z) = 1
√2(1 −z)), and two level of decomposition by hand.
A=
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 1 1 1 1 0 0
0 0 1 1 1 1 0 0
0 0 1 1 1 1 0 0
0 0 1 1 1 1 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
B=
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 1 1 1 1 0
0 0 0 1 1 1 1 0
0 0 0 1 1 1 1 0
0 0 0 1 1 1 1 0
0 0 0 0 0 0 0 0
2. Now let’s consider image B, a shifted version of A. Compute the Haar wavelet transform for
the new image B. With the results you get, comment on the shift-variant property of the
wavelet transform.
3. (Optional) Following the same method for deriving the basis signals for the one-dimensional
Haar wavelet transform given in the lecture notes, verify that the basis images for the 2D
Haar wavelet transform for 4×4 images are as shown in the lecture notes (with proper nor-
malization).
4. Matlab Problem:
(a) Write a Matlab program, which can load an image, decompose it by a 4-level wavelet
transform, and then reconstruct the image from the wavelet coefficients. Compare the
output with the input to check the perfect reconstruction property.
(b) Now before reconstruction, threshold the wavelet coefficients with a given threshold T
(i.e. set those coefficients whose absolute values are smaller than Tto zero). Compare
the reconstructed image with the original one. Do this with several different threshold
values Tthat keep only about 5% to 10% of wavelet coefficients for reconstruction.
(Hint: You can learn how to do wavelet-related computations in Matlab by typing help
wavelet. The functions that you might need are: wavedec2, waverec2, and wthresh. You
can choose any wavelet filter, their names are given by typing help wfilters.)
1