1. Multiresolution Analysis (Chapter 7)
CS474/674 – Prof. Bebis

2. Multiresolution Analysis
Many signals or images contain features at various levels of detail (i.e., scales).
Small size objects should
be examined at a high
resolution.
Large size objects should
be examined at a low

3. Multiresolution Analysis (cont’d)
Local image statistics
are quite different from
global image statistics.
Modeling entire image is
difficult or impossible.
Need to analyze images at multiple levels of detail.

4. Multiresolution Analysis (cont’d)
Remember the issue of choosing the “right” window size in Short-Time Fourier Transform (STFT)?
Multiresolution methods analyze a signal or image using windows of different size!
Multiresolution analysis provides information simultaneously the spatial and frequency (i.e., scale) domains.
Scale = 1/Frequency

5. Multiresolution Analysis and Wavelets
Mallat (1987) showed that wavelets unify a number of multiresolution techniques, including:
Pyramidal coding (image processing).
Subband coding (signal processing)
Quadrature mirror filtering (speech processing)

6. Image Pyramids
A collection of decreasing resolution images arranged in the shape of a pyramid.
low resolution
j=0
high resolution
j = J

7. Pyramidal coding Approximation and prediction residual pyramids:
Downsampling:
half resolution
averaging  mean pyramid
Gaussian  Gaussian pyramid
no filter  subsampling pyramid
nearest neighbor
biliner
bicubic
Upsampling:
double resolution
A variety of approximation/interpolation filters can be used!

8. Pyramidal coding (cont’d)
Approximation pyramid
(based on Gaussian filter)
Prediction residual pyramid
(based on bilinear interpolation)
Top levels are the same

9. Pyramidal coding (cont’d)
In the absence of quantization errors, the approximation pyramid
can be constructed from the prediction residual pyramid.

10. Subband coding
Decompose an image (or signal) into a set of bandlimited components (i.e., subbands).
Decomposition is performed so that the subbands can be reassembled to reconstruct the original image without error.
Decomposition/Reconstruction are performed using digital filters.

11. Digital Filter - Example
h(n): impulse
response.
FIR filter
(finite impulse
response)

12. Digital Filter – Example (cont’d)
oroginal
sign reversed
order reversal
order reversal
modulation
order reversal
and modulation

13. Subband coding (cont’d)
Example: 1D, 2-band decomposition
flp(n): approximation of f(n) 1D
fhp(n): detail of f(n)

14. Subband coding (cont’d)
There are many 2-band, real coefficient, FIR perfect reconstruction filter banks.
In all of them, the synthesis filters (g0(n) and g1(n)) are modulated versions of the analysis filters (h0(n), h1(n)).
One (and only one) synthesis filter is signed reversed.

15. Subband coding (cont’d)
For perfect reconstruction, h0(n), h1(n), g0(n) and g1(n) must be related in one of the following two ways:
Also, they are bio-orthogonal: or

16. Subband coding (cont’d)
Of special interest, are filters satisfying orthonormality conditions:
An orthonormal filter bank can be designed from a single prototype filter; all other filters are computed from the prototype (biorthogonal filters require two prototypes).

17. Subband coding (cont’d)
Example: orthonornal filters

18. Subband coding (cont’d)
Example: 2D, 4-band decomposition (using separable filters) 2D

19. Subband coding (cont’d)
horizontal detail
approximation
diagonal detail
vertical detail