In sparse recovery one aims to leverage the sparsity of some vector {u} to get a good reconstruction of this vector from a noisy and indirect measurement {u^{0}}. Sparsity can be expressed in two different ways: The analysis way and the synthesis way. For both of them you need a system of other vectors {v_{k}}.

  • The analysis way: Here you consider the inner products {\langle u,v_{k}\rangle} and the sparsity assumption is, that this sequence is sparse.
  • The synthesis way: Here you assume that there is sparse sequence of coefficients {\psi_{k}} such that {u=\sum_{k} \psi_{k}v_{k}}.

The first approach is called “analysis” since here one analyzes the vector {u} with the system of vectors. In the second approach, you use the vectors {v_{k}} to synthesize the vector {u}.

We will keep things very simple here and only consider a noisy observation (and not an indirect one). The recovery approach in the analysis way is then to find {u} as a solution of

\displaystyle \tfrac12\|u-u_{0}\| + \alpha R(\langle u,v_{k}\rangle)

while for the synthesis way you write {u = \sum_{k} \psi_{k}v_{k} = V\psi} (where we collected the vectors {v_{k}} as the columns of the matrix {V}) and solve

\displaystyle \tfrac12\|V\psi-u_{0}\| + \alpha R(\psi)

The functional {R} should enforce the sparsity of the vector of coefficients in the former case and the sparsity of {\psi} in the latter case. With the matrix {V} the analysis way reads as

\displaystyle \tfrac12\|u-u_{0}\| + \alpha R(V^{T}u)

One important observation is, that both approaches coincide if the matrix {V} is orthonormal: Since {V^{-1} = V^{T}} in this case, {u=V\psi} is equivalent to {\psi = V^{T}u}, i.e. {\psi_{k} = \langle u,v_{k}\rangle}. For other sets of vectors {v_{k}}, this is not true and it is not so clear, how the analysis and the synthesis approach are related. In this post I’d like to show one instance where both approaches do lead to results that are not even close to being similar.

1. The “ROF analysis” approach to denoising

If {u^{0}} is an image, the famous Rudin-Osher-Fatemi denoising model fits under the “analysis” framework: The vectors {v_{k}} are all the finite differences of neighboring pixels such that {V^{T}u = \nabla u} with the discrete gradient {\nabla u}. As sparsifying functional you take a mixed {2}{1} norm, i.e. {R(\nabla u) = \|\nabla u\|_{2,1} = \sum_{ij}\sqrt{(\partial_{1}u_{ij})^{2} + (\partial_{2}u_{ij})^{2}}}. The denoised {u} is the solution to

\displaystyle \min_{u} \tfrac12\|u-u^{0}\|^{2} + \alpha\|\nabla u\|_{2,1}

Thus, the model will lead to some denoised {u} with a sparse gradient. This sparse gradient is partly the reason for the success of the model in that it keeps edges, but is also responsible for the drawback of this approach: The denoised image will be mostly piesewise constant.

2. The “ROF synthesis” approach

As {V^{T} = \nabla}, the corresponding synthesis approach would be to solve

\displaystyle \min_{\psi}\tfrac12\|\nabla^{T}\psi - u^{0}\|^{2} + \alpha \|\psi\|_{2,1}

and the set {u = \nabla^{T}\psi}. In this approach, the denoised image will be a sparse linear combination of particular two-pixel images. If you think about that, it does not really sound like a good idea to approximate an image by a sparse linear combination of two-pixel images. (One technicality: The operator {\nabla^{T}} is not onto – {\nabla} has a non-trivial kernel, consisting of the constant images and hence, all images in the range of {\nabla^{T}} have zero mean. Thus, to get something remotely close to {u^{0}} one should subtract the mean of {u^{0}} before the reconstruction and add it back afterwards.)

Here are some results:

rof_analysis-_synthesis

 

Note that the results actually fit to what we would have predicted: For larger {\alpha} we get a “sparser gradient” (less edges, less variation) for the analysis approach, and “sparser two-pixel combinations” for the synthesis approach. In fact, for the synthesis approach to give something that is remotely close to the image, one needs quite a lot two-pixel images, i.e. a very low sparsity.

In conclusion, one should not consider the analysis and synthesis approach to be something similar.

Incidentally, the dual of the analysis approach

\displaystyle \min_{u} \tfrac12\|u-u^{0}\|^{2} + \alpha\|\nabla u\|_{2,1}

can be written as

\displaystyle \min_{\|\phi\|_{2,\infty}\leq\alpha}\tfrac12\|\nabla^{T}\phi-u^{0}\|^{2} = \min_{\phi}\tfrac12\|\nabla^{T}\phi-u^{0}\|^{2} + I_{\|\cdot\|_{2,\infty}\leq\alpha}(\phi)

which looks related to the synthesis approach (one basically replaces the {2,1}-norm by its conjugate function). Actually, a similar relation between the dual of the analysis approach and the synthesis approach always holds.

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Yesterday I uploaded the paper An extended Perona-Malik model based on probabilistic models by Lars Mescheder and myself to the arXiv. Recently I already blogged about this work, so I do not have to add that much. The main theme of the work was that if we have an image that is a blurred and noisy version of some true image, we formulate the reconstruction via Bayesian statistics. As a prior model for images we used a Gaussian scale mixture, i.e. we have a latent variable z (in every pixel x) and the joint prior for the image u and the latent variable z is

p(u,z) \propto \exp\Big(-\sum_x \frac{z(x)}2 |\nabla u(x)|^2 + v(z(x))\Big)

where x denotes the pixels, $\nabla$ is the discrete gradient of the image and v is some non-negative function defined on the non-negative reals. Besides algorithms to approximate the MAP estimate, Lars proposed a mean-field approximation which does not calculate the most probable image, but iteratively approximates the posterior distribution by distributions which factorize over the variables u and z. Using some further approximation (since the resulting algorithm in its plain form is not implementable) one arrives at an algorithm which some keeps more uncertainty in u and, in practice, gives a point estimate for the denoised image that seems more “representative”. Here is an example:

This is the blurred and noisy image (the blur is motions blur and implemented with periodic boundary conditions for simplicity):
perona-malik-conv-corrupted
The next image is the approximation of the MAP estimate we got and you see the usual drawbacks. Since the MAP neglects all uncertainty in u and maximizes the posterior probability, the image is “too sharp” in the way that smooth transitions (e.g. at the lighthouse) turn into piecewise constant stairs. Also the rocks appear very blocky.
perona-malik-conv-map
Here is the result from the mean-field approximation. Since uncertainty in u is taken into account, the image does not suffer from staircasing and also has a more natural appeal, most prominently in the part with the rocks.
perona-malik-conv-meanfield

The paper has some more examples and also shows another relation to Mumford-Shah denoising (loosely speaking, one uses a discrete latent variable z to serve as a binary variable to say if a pixel is an edge or not). Oh, by the way, the algorithms behind the mean-field approximation use some parts of more general duality between random variables that Lars and his co-authors develop in another paper.

Today I gave a talk at STOR colloquium at the University of North Carolina (UNC). I spoke about the Master’s thesis of Lars Mescheder in which he developed probabilistic models for image denoising. Among other things, he proposed a Gaussian scale mixture model as an image prior and developed several methods to infer information from the respective posterior. One curios result was that the Perona-Malik diffusivity (and variants thereof) popped up in basically every algorithm he derived. It’s not only that the general idea to have an edge dependent diffusion coefficient in a PDE or, formally equivalent, a concave penalty for the magnitude of the gradient in a variational formulation, but it also turned out that the very diffusion coefficient {\frac1{1+|\nabla u|^{2}}} appeared exactly in this form.

Besides the talk I got the chance to chat with many interesting people. I’d like to mention a chat about a background story on Perona-Malik which I find quite interesting:

Steven Pizer, Kenan Professor for computer science at UNC and active in a lot of fields for several decades, and in particular a driving for in the early days of digital image analysis, told that the very idea of the non-linear, gradient-magnitude dependent diffusion coefficient actually dates back to works of neuroscientists Stephen Grossberg and that he moderated a discussion between Stephan Grossberg, Pietro Perona and Jitendra Malik in which the agreed that the method should be attributed as “Grossberg-Perona-Malik” method. I could not find any more hints for this story anywhere else, but given the mathematical, psychological and biological background of Grossberg and a look at Grossbergs list of publications in the 1980’s and earlier make this story appear more than plausible.

It may also well be, that this story is just another instance of the effect, that good ideas usually pop up at more than one place…

I found this draft of a post in my backlog and decided that it would be better to finish it than to leave it unpublished. Actually, the story happened already over a year ago.

Some month ago I stumbled upon this piece of shadow art

by Fred Eerdekens and a few days later I received the information that my university was going to celebrate his yearly “open house event” this year as “TU Night”. The theme of the TU Night was “Night, Light, Energy” and all member were invited to submit ideas for talks, experiments and exhibits.

The piece of shadow art prompted the question “If this weired piece of metal can cast this sentence as a shadow, wouldn’t it be possible to produce another piece of metal that can produce two different sentences, when illuminated from different directions” Together with the announcement of the upcoming TU Night I thought if one could even produce an exhibit like this.

Since I am by no means an artists, I looked around at my university and found that there is a Department of Architecture. Since architects are much closer to being artist than I am, I contacted the department and proposed a collaboration and well, Henri Greil proposed to have a joint seminar on this topic. Hence, this summer term I made the experience and worked with students of architecture.

In the end, the student produced very nice pieces of shadow art:

IMG_5932

IMG_5941

IMG_5940

Although the exhibits produced interesting and unexpected shadows, no group of students could make it and produce two different shadows out of the same object.

However, some nice effects can be produced pretty easy:

The basic idea is that moving one object around will move around both shadows rather independently. Well this is not totally true but what you can do is to “zoom” one shadow while moving the other sideways (just move the object straight towards one light source). See this movie for a small illustration:

I also did my best to produce a more complex object. While it is theoretically not very difficult to see that some given shadows are possible in some given projection geometry, it is not at all straight forward to produce the object theoretically (not to speak of the real world problems while building the piece). It tried hard but I could not do better than this:

I am at the SSVM 13 and the first day is over. The conference is in a very cozy place in Austria, namely the Schloss Seggau which is located on a small hill near the small town Leibnitz in Styria (which is in Austria but totally unaffected by the floods). The conference is also not very large but there is a good crowd of interesting people here and the program reads interesting.

Schloss Seggau

The time schedule is tight and I don’t know if I can make it to post something everyday. Especially, I will not be able to blog about every talk.

From the first day I have the impression that two things have appeared frequently at different places:

  • Differential geometry: Tools from differential geometry like the notion of geodesics or Riemannian metrics have not only been used by Martin Rumpf to model different types of spaces of shapes but also to interpolate between textures.
  • Primal dual methods: Variational models should be convex these days and if they are not you should make them convex somehow. Then you can use primal dual methods. Somehow the magic of primal dual methods is that they are incredibly flexible. Also they work robustly and reasonably fast.

Probably, there are more trends to see in the next days.

Geometry, Imaging and Computing

A short note to myself: There is the new journal “Geometry, Imaging and Computing” published by International Press which looks interesting for papers inbetween computer vision and computer graphics.

Image

I have an open position for a Scientific Assistant/PhD student available. The salary is according to TV-L EG 13. (Don’t know what that means? Have a look here.). The position starts at 01.09.2013 (earlier start is possible) and is initially limited to two years; further extension is possible and for a PhD student, at least three years are planned.

Candidates should

  • have a degree (Masters or Diploma) in mathematics above average,
  • have very good knowledge in numerical mathematics and functional analysis,
  • have good knowledge in scientific computing or optimization,
  • know German well and
  • have strong interest in applied mathematics. Also, bring a high commitment for scientific research.

The responsibilities include

  • participation in teaching and
  • independent but supervised research in field of applied mathematics (especially mathematical imaging and inverse problems).

Please send applications including CV, copies of certificates and letters of recommendation (if any) in electronic form directly to me. Deadline is the 30.06.2013.

If you would like to post the job advertisement at you bulletin board, here’s the pdf file.