I am interested in image analysis and am looking for an approachable tutorial to the Wiener filter. At some point I am interested in implementing such a filter but I would like to have a deeper understanding of the algorithms I'll be writing. Does anyone have recommendations for papers, web sites, etc. that present good coverage on the topic?
[Math] Wiener filter: A good tutorial
fourier analysisreference-requestsignal processing
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OK, I'm going to hijack this thread even though there's an answer as I haven't found any quality, localized information about multifractals.
As mentioned in the comments, I first heard about multifractals from a Google Tech Talk by Rogene M. Eichler West, which can be found, without sound, on YouTube, called "Multifractals: Theory, Algorithms, & Applications" . Unfortunately Google Video got discontinued after they bought out YouTube and I can't find the original video that had the sound included.
I still do not understand on a deep level what, how and why multifractals are doing, are better than another method or how they do it, but from what I understand the idea is to generalize the concept of spectrum to include functions that have a scale symmetry, where the scale symmetry can be on many different scales (thus multi-fractal, instead of just being fractal). Just as the Fourier spectra is constructing a profile of the translation invariances of a function, the multifractal spectra gives information about the scale invariances of a function.
The general methodology seems to be, for a given function $f(t)$:
- Find the Hölder exponent, $h(t)$, as a function of time, $t$
- Find the singularity spectrum, $D(\alpha)$
Where $D(\alpha) \stackrel{def}{=} D_F\{x, h(x) = \alpha\}$, and $D_F\{\cdot\}$ is the (Hausdorff?) dimension of a point-set.
I believe the idea is that for chaotic/fractal/discontinuous functions, at any point they can be characterized, locally, by the largest term of their Taylor expansion and the Hölder exponent is a way to characterize this. Once you have the function, $h(t)$, characterizing the Hölder exponent, you use that to construct the singularity spectrum. I believe the singularity spectrum is a synonym for the multi-fractal spectrum.
From what I can tell, the specifics of how to calculate $h(t)$ and $D(\alpha)$ in practice vary from approximating them outright by their definition or by using wavelets to approximate the Hölder exponent and then using a Legendre transform to approximate the multifractal spectrum.
From what I understand, $D(\alpha)$ tends to be (or is always?) concave. I have only the vaguest notion of why this is so. How one relates wavelet transforms to finding the Hölder exponent, how one uses the Legendre transform to find the multi-fractal spectrum, why the multi-fractal spectrum should be concave, what kind of intuitive feeling one should get about a function from viewing the spectrum, amongst many others, I still have no idea about.
The multiplicative cascade seems to be a canonical example of a multifractal process.
Online, "A Brief Overview of Multifractal Time Series" gives a terse run through of multifractals. They claim to be able to tell a healthy heart from one that is suffering from congestive heart failure (see here).
Here are some slides giving a brief overview of multifractals. Near the end of the slides, they give a wavelet transform of the Devil's staircase function and talk a bit about using Wavelet Transform Modulus Maxima Method (WTMM), which appears to be a standard tool when doing this type of analysis (anyone have any good links for this?).
Looking around, I found Wavelets in physics by J. C. van den Berg that had this section web accessible for a definition of the singularity spectrum.
Rudolf H. Riedi seems to have a few papers out there that describe multifractal processes. Here are a few:
- "Multifractal Processes"
- "Introduction to Multifractals"
- Along with Jacques L. Véhel "TCP traffic is multifractal: a numerical study."
While focused on finance, Laurent Calvet and Adlai Fisher have a lot of introduction to terminology in "Multifractility in asset returns: Theory and evidence".
And of course Mandelbrot, along with other authors, has many papers, some of which are:
Fractional Brownian Motion is also mentioned frequently, but I have no real idea of how they relate. Large Deviation Theory also seems to be mentioned, but I don't know how this relates to multifractals either. I believe I've also seen entropy, phase transitions and statistical mechanics mentioned here and there. I would be curious if and what the relation to these subjects and multifractals is.
I feel like I'm stumbling around trying to understand this subject and I have yet to find a cohesive text that brings together enough intuition, math and implementation details so that I feel like I have a firm grasp of what's going on. I would welcome any additional resources or corrections to this answer.
I really enjoyed reading Forman Acton's Numerical Methods that Usually Work. He writes with a lot of personality and presents interesting problems. I don't know that this is the best book to learn the field from because (1) it was written in 1970 and last revised in 1990 and (2) I am not an expert, so I simply don't know how to evaluate it. But it sounds like these might not be major obstacles from your perspective.
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Detection Estimation and Modulation Theroy Volume 1 by Harry Van Trees is a good start if you have a basic understanding of Probability and Random Variables. Any statistical signal processing book should also cover Wiener Filtering