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What is actual difference between scale-space and wavelet transform? It seems that wavelets require an orthonormal basis of kernels, whereas scale-space does not. Is it the only difference? Can scale-space be considered as particular case of wavelet transform?

UPDATE

Example. Suppose I convolve a 1d signal with several gaussian kernel of different width. I got:

Result of convolutions

What was it? Wavelet decomposition or scale-space? Are they technically the same?

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  • $\begingroup$ I added links. Type of wavelets? Some of Gaussian type. $\endgroup$
    – Leonid
    Sep 6, 2016 at 16:54
  • $\begingroup$ Yes, let's talk in 2D $\endgroup$
    – Leonid
    Sep 6, 2016 at 17:30

2 Answers 2

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Laurent's answer gives a good summary of the theory, so I will focus more on the practical side, with an emphasis on computer vision & image processing.

One key difference between the two multi-scale representations is in their goals. Wavelet decompositions give a complete description of the data, but typically they are aimed at a sparse representation. This allows truncation of the full wavelet transform (e.g. thresholding small values) while preserving most of the image information, for example wavelets are commonly used for compression.

On the other hand, the scale-space decomposition is over-complete (redundant) by design. As noted by Laurent, differences between scale-space levels (e.g. "Laplacian" image pyramids) have some similarity to the wavelet decomposition. Indeed these were first developed with compression in mind (i.e. Burt & Adelson 1983).

However, the primary goal of most practical methods that use scale-space decompositions (a.k.a. image pyramids) is robustness. In this context the "redundant" information in the scale-space representation is a virtue. For example, scale-space approaches are ubiquitous in feature detection/description (such as SIFT), as well as dense correspondence mapping (e.g. large-offset optical flow is typically done in coarse-to-fine fashion). Both of these applications (and others) are well described in Richard Szeliski's E-book, which I highly recommend.

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  • $\begingroup$ Please, could you comment on the update? $\endgroup$
    – Leonid
    Sep 7, 2016 at 6:02
  • $\begingroup$ @Leonidas your update would be scale space. In a discrete setting, the smoothing would typically be followed by a downsampling. This construction is called a Gaussian Pyramid. $\endgroup$
    – GeoMatt22
    Sep 7, 2016 at 12:40
  • $\begingroup$ But why this is not a wavelet transform? Yes, such downsampling is a feature of scale-space. But before we downsample we actually perform a wavelet transform, yes? $\endgroup$
    – Leonid
    Sep 7, 2016 at 14:50
  • $\begingroup$ @Leonidas my understanding is that in practical usage of the terms, it is not just the transform that distinguishes the two, but also the filter. Both transforms could be considered "multi-scale filter bank composed of self-similar filters obtained by re-scaling of a base filter". The base filter for scale-space is formally a Gaussian (but more generally a moving-average style symmetric non-negative kernel) with unit integral. The base filter for a wavelet transform (mother wavelet) is typically more like a derivative filter, with zero integral. $\endgroup$
    – GeoMatt22
    Sep 7, 2016 at 15:03
  • $\begingroup$ With continuous wavelets, and inserting derivative operators in the scale-space, the distinction between the two reduces. I would say space-space insists on similarity, while wavelets insist on variations. With the Witkin picture you gave, you could built an approximate wavelet decomposition only with subsampling, because Gaussians do not tolerate well the admissibility property central to wavelets $\endgroup$ Sep 7, 2016 at 19:46
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To make it very short:

The difference between scale-space and wavelets is ... the difference

The notions of scale-space theory and wavelets have strong links, although with different points of view. What follows is an heavy simplification of all the works related to those theories.

Scale-space

As shown in the images above, objects persist differently at different scales. The picture was taken by Dr. Károly Szatmáry, very kindly, at Szeged University, and honors A. Haar and F. Riesz (two important names in wavelets and functional analysis). It was subsequently processed with different Gaussian filters, to illustrate how some features persist across scale, and some don't. The white square persist, the grid and the bricks do not, albethey differently.

A scale space is representation of data at multiple resolution levels

The scale-space theory was axiomatized notably by T. Lindenberg, see Generalized axiomatic scale-space theory, 2013. It is generally associated with a continuous setting, related to differential equations, e.g. the heat equation, and approximation spaces.

The wavelet theory originates more from A. Haar work on unconditional bases, regularity and approximations. Basically, it is interested in differences between approximation spaces, and how one can "compact" information inside each of them. Wavelet allow sparse representations of regular data, ending with either orthogonal, biorthogonal or frames, that allow fast computations. To quote Wim Sweldens:

Wavelet are building blocks that can quickly decorrelate data

With orthogonal or tight bases, wavelets better accomodate Gaussian disturbances, and can be plugged in many sparse optimization algorithms, that lead to efficient denoising of some data, using statistical principles such as Stein's Unbiased Risk Estimation (SURE), see for instance A Nonlinear Stein-Based Estimator for Multichannel Image Denoising, Chaux et al., 2008.

In short again:

wavelets can be seen as subsampled versions of differences between close enough scale-space decompositions.

An overview of those multi-scale techniques is provided in: A panorama on multiscale geometric representations, intertwining spatial, directional and frequency selectivity, L. Jacques et al., 2011

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  • $\begingroup$ Laurent, I will look forward to reading your paper. I added an answer giving some of my own thoughts (though I am certainly an amateur compared to you!). I was curious if you have any thoughts on where multi-layer ("deep"?) ConvNets might fit in as a multi-scale representation? $\endgroup$
    – GeoMatt22
    Sep 6, 2016 at 21:04
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    $\begingroup$ Laurent, please, could you comment on the update? $\endgroup$
    – Leonid
    Sep 7, 2016 at 6:02
  • $\begingroup$ @GeoMatt22 I cannot speak about ConvNets. However, I am quite interested in the "wavelet" side explored by Stéphane Mallat with scattering networks, and recent extensions such as A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction $\endgroup$ Sep 7, 2016 at 20:04

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