Multiresolution Analysis (Section 7.1)

CS474/674 – Prof. Bebis

Multiresolution Analysis

• Many signals or images contain features at different scales or level of detail (e.g., people vs buildings)

• Analyzing information at the same resolution or at a single scale will not be sufficient!

Multiresolution Analysis (cont’d)

• Many signals or images contain features at different scales or level of detail (e.g., people vs buildings)

Small size objects shouldbe examined at a high resolution

Large size objects shouldbe examined at a low resolution

Equivalent to using windows of multiple size!

High resolution(high frequencies)

Low resolution(low frequencies)

Intermediate resolution

Multiresolution Analysis (cont’d)

• We will review two techniques for representing multiresolution information efficiently:– Pyramidal coding

– Subband coding

high resolution j = J

low resolution j=0

A collection of decreasing resolution images arranged in the shape of a pyramid.

Image Pyramid

(low scale)

(high scale)

If N=256, the pyramid will have 8+1=9 levels

Pyramidal coding

Two pyramids: approximation and prediction residual

averaging mean pyramidGaussian Gaussian pyramidno filter subsampling pyramid

nearest neighborbilinerbicubic

Pyramidal coding (cont’d)

Approximation pyramid(based on Gaussian filter)

Prediction residual pyramid(based on bilinear interpolation)

Last level same as that in theapproximation pyramid

Pyramidal coding (cont’d)

In the absence of quantization errors, the approximation pyramidcan be re-constructed from the prediction residual pyramid.

Keep the prediction residual pyramid only!Much more efficient to represent!

Subband coding

• Decompose an image (or signal) into a set of

different frequency bands (analysis step).

• Decomposition is performed so that the subbands can be re-assembled to reconstruct the original image without error (synthesis step)

• Analysis/Synthesis are performed using appropriate filters.

Subband coding – 1D Example

flp(n): approximation of f(n)

fhp(n): detail of f(n)

2-band decomposition

Subband coding (cont’d)

• For perfect reconstruction, the synthesis filters (g0(n) and g1(n)) must be modulated versions of the analysis filters (h0(n), h1(n)):

or

original modulated

Subband coding (cont’d)

• Of special interest, are filters satisfying orthonormality conditions:

• An orthonormal filter bank can be designed from a single prototype filter:

(non-orthonormal filters require two prototypes)

Subband coding (cont’d)Example: orthonornal filters

Subband coding – 2D Example

4-band decomposition (using separable filters)

approximation

horizontal detail

vertical detail

diagonal detail

LL

LH

HL

HH

Subband coding (cont’d)

approximation

vertical detail

horizontal detail

diagonal detail