1. DESCRIPTION
1.1. Introduction
1.2. Image Statistics
1.3. GSM
1.4. Plus Noise
1.5. BLS-GSM
1.6. Results
1.7. References
2. EXAMPLES
3. TEST IMAGES
4. SOFTWARE
1.2. Image statistics in the wavelet domain
Natural images have an interesting behavior when represented using wavelets or, in general, when seen through a set of orientation and scale selective band-pass filters. First, typically, most responses of the filters (corresponding to uniform soft texture areas - gray areas of the subband below) have a close to zero value, whereas a few of them (corresponding to the responses to edges, lines, corners and other localized salient features - black and white features on the image below, on the left panel) have comparatively very large amplitude responses. Therefore, if we look at the histogram of a subband of a natural image it typically has a strong peak at zero and long heavy tails (figure below, right panel):
Left: A wavelet subband of a typical image Right: Typical subband histogram
Note the logarithmic scale of the ordinates. This feature was already observed by [Field 1987] and first modeled by [Mallat 1989], using a generalized Gaussian (see also [Simoncelli 1996, Buccigrossi 1997]). It is often referred to as the "high kurtosis", "leptokurtosis behavior" or "sparseness" of the wavelet coefficients of a subband.
The second interesting feature observable in images is that the amplitude responses of neighbor coefficients (neighbor in space for the same subband, or in orientation, for the same scale and spatial position, or in scale, for the same orientation and spatial position) are strongly coupled: when one observed coefficient is close to zero, then its immediate neighbors will probably be also close to zero (see pyramid in absolute value below, left). However, if the observed coefficient has a large amplitude, then the neighbors may have both large or small amplitude responses, having a large conditional variance. Shown below (right) is a joint 2-D histogram with its columns normalized (black represents low density, white high density). This "bow-tie" shape can be interpreted as representing the conditional density of the neighbor for a given coefficient value. This joint feature was observed and commented explicitly first by [Simoncelli 1997]. This statistical coupling had already been used before, for instance in the zero-tree coding [Shapiro 1993], which assumes that a close-to-zero node will typically have close-to-zero children nodes in the quadtree. It has also been used for image compression explicitly [Simoncelli 1997, Buccigrossi 1997].
Left: Orthogonal wavelet pyramid Right: Typical conditional 2-D histogram (with
of a typical image, in amplitude normalized columns) of neighbor coefficients