**Igor Carron**

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**Reference pages include The Big Picture in Compressive Sensing (and learning Compressive Sensing), the Advanced Matrix Factorization Jungle Page and the Reproducible Research page**

## Saturday, March 24, 2007

### Driving on a manifold: unveiling our strategy

Our entry in DARPA Urban Challenge will feature compressed sensing as a way to reduce the dimensionality of our vision sensor. We will then have to infer the connection between our GPS track (RNDF and MDF) and the reduced parameters obtained from random projections.

## Friday, March 23, 2007

### Damaris is in the news.

Damaris is featured on Florida Today. The article refers to Starnav and Columbia. The story of the experiment was featured elsewhere.

## Thursday, March 22, 2007

### Compressed Sensing, Primary Visual Cortex, Dimensionality Reduction, Manifolds and Autism

In a previous entry, I mentioned the potential connection between compressed sensing, the primary cortex and cognition deficit diseases like autism without much explanation. Here is an attempt at filling the holes.

When David Field and Bruno Olshausen showed that the primary cortex was getting inputs from our eyes as a set of sparse functions that looked like ridgelets and curvelets, it became obvious that one result was missing: If natural images are sparse and our eye system has sparse receptors, is there a way our brain finds a sparse decomposition of the world in a way that works in a linear fashion? The thinking goes that our brain is really capable of understanding scenes without an iteration process (an iteration process is nonlinear and has a high cost in terms of energy). When Emmanuel Candes and David Donoho showed that in fact, non-adaptive schemes using curvelets could decompose natural images it became obvious that a good parallel could be made between the physiology of the primary cortex and this new type of decomposition. But how do you do this decomposition ? While an m-term curvelet expansion of a scene can be thresholded and can rival with complex adaptive approximation schemes, it does not answer how the primary cortex eventually comes up with that m number.

The state of the art on our thinking about the primary cortex can be found here in this review by Graham and Field on sparse coding in the neocortex. It specifically addresses the bounds on the primary cortex induced by the metabolic constraints:

Well it's nice to acknowledge we have physical limitations, but to assume that a linear code cannot exist simply because we currently do not have a model for it, is probably assuming too much. So what do we know ? the primary cortex is a low energy system which basically removes from consideration any complex system that requires resources (like an iteration system). This situation favors a linear system but so far, we have not found a good model for that. There is something deeper still. Even if we knew much about the sparsity of a scene, understanding the brain is really about understanding how large amount of information is reduced when traveling from the eye into the brain. In other words, we need to reduce the amount of data that our megapixel sensor called the eye is bringing in, and we must do this very fast (30 times a second). To put things in perspective, let us take an example: let us imagine we are seeing a scene where somebody waves a hand. If we were to take a video of this scene, we would probably get a 40 MB avi file (uncompressed). That file could then be compressed to 1 MB using MPEG for instance. While the compression is impressive, it is not impressive enough. In effect, when our brain sees this video, it can remember the hand and how it moved. The movement is probably two dimensional and so the brain really remembers the two parameters needed to produce a hand that waves in the manner that is shown in the video. In other words, the brain is probably not storing 1 MB of information when it stores this hand waving activity, it is most probably storing how two parameters changed over time, which in many occasion is much less than 1 MB of data. The real question becomes: Is there a way to reduce that the 1MB information further ? We are not asking ourselves if the input or the receptor are sparse (it is a necessary condition), we are interested in answering the question on how sparser we can make this information by using the connection between these sparse elements. Can we reduce the dimensionality of the signal further and exploit it ?

Enters Dimensionality reduction: Ever since the discovery of dimensionality reduction schemes (LLE, Isomap, Laplacian-diffusion..) that are taking high dimensionality data and and are able to map them into low dimensionality manifolds, researchers have been trying to extend these techniques to wider sets of problems. In the cognition world for instance, Jonathan Pillow and Eero Simoncelli perform dimensionality reduction applied to neural models but it is not obvious how these techniques directly translate into a specific functionality of the primary cortex even if they take an example of a V1 cell. It is also not obvious how some of these techniques are robust to noise. But as stated earlier, there are different ways to go about dimensionality reduction. One of the most intriguing which has robustness built into it is Compressed Sensing. Compressed Sensing has the ability to produce a robust decomposition of a manifold. Mike Wakin looked into that during his dissertation and found that smooth manifolds can be readily compressed using Compressed Sensing thereby making it a very simple solution to dimensionality reduction (see R. G. Baraniuk and M. B. Wakin in Random Projections of Smooth Manifolds ) but as Donoho and Grimes had shown earlier, sharp objects such as arms, legs have edges and that makes the manifold non-differentiable. This is a problem because it means that one cannot easily extract parameters from a video if we have these sharp edges. In order take care of that problem, one can be inspired by Biology i.e. to smooth these images using Gabor wavelets as in the human vision system (Object Recognition with Features Inspired by Visual Cortex by Thomas Serre, Lior Wolf and Tomaso Poggio) and then use the random projection of smooth manifolds to eventually figure out the parameters of the movements ( for more information on how to do that see High-resolution navigation on non-differentiable image manifolds or the Multiscale Structure of Non-Differentiable Image Manifolds)

[in Object Recognition with Features Inspired by Visual Cortex by Thomas Serre, Lior Wolf, Tomaso Poggio one may note that one can only be struck by the pain the algorithm goes through into in order to be robust.

]

Besides Wakin, Donoho, Baraniuk and others collaborating with them, few have made that connection. Yves Meyer makes a passing reference to the use of compressed sensing in physiology (in Perception et compression des images fixes.) Gabriel Peyre however is a little more specific here.

If bandelets provide a recognizable Faces nonlinearly with 500 bytes, one only needs 5 x 500 bytes = 2.5 KB random samples (within the meaning of Compressed sensing) of that Face to be able to reconstruct it. 2.5KB is better than 10 MP or 1 MB. The number 5 is the bound for compressed sensing (more specific asymptotic laws/results can be found in the summary of Terry Tao)

However, the strongest result so far is the one found by Mike Wakin on random projection on a manifold where he uses neighborhood criteria in the compressed sensing space to permit an even better reconstruction than just assuming sparsity. In the figure below, one can compare the 5 random projection results using the Manifold based recovery from the traditional algorithms used like Orthogonal Matching Pursuit and Basis Pursuit.

Naturally, an extension of this is target detection as featured in the Smashed Filter article from Rice. With this type of result/framework, I am betting we can go lower than 2.5 KB of universal samples to characterize a face. The connection between cognition and compressed sensing and autism is simple: Face processing seem deficient for people affected with autism and we don't know why. A model based on the elements I just mentioned might give an insight into this issue.

When David Field and Bruno Olshausen showed that the primary cortex was getting inputs from our eyes as a set of sparse functions that looked like ridgelets and curvelets, it became obvious that one result was missing: If natural images are sparse and our eye system has sparse receptors, is there a way our brain finds a sparse decomposition of the world in a way that works in a linear fashion? The thinking goes that our brain is really capable of understanding scenes without an iteration process (an iteration process is nonlinear and has a high cost in terms of energy). When Emmanuel Candes and David Donoho showed that in fact, non-adaptive schemes using curvelets could decompose natural images it became obvious that a good parallel could be made between the physiology of the primary cortex and this new type of decomposition. But how do you do this decomposition ? While an m-term curvelet expansion of a scene can be thresholded and can rival with complex adaptive approximation schemes, it does not answer how the primary cortex eventually comes up with that m number.

The state of the art on our thinking about the primary cortex can be found here in this review by Graham and Field on sparse coding in the neocortex. It specifically addresses the bounds on the primary cortex induced by the metabolic constraints:

We conclude our discussion by returning to the issue of metabolic constraints. Could we argue that primary evolutionary pressure driving towards sparse coding is one related to the metabolic costs of neural firing? As noted earlier, both Attwell and Laughlin (2001) and Lennie (2003) argue that there are not enough resources to achieve anything but a low-activity system. Moreover, when we find sparse activity in frontal cortex (Abeles et al., 1990), it is more difficult to argue that the sparse activity must arise because it is mapping the sparse structure of the world. Even at early levels, if sparseness were metabolically desirable, there are a number of ways of achieving sparseness without matching the structure of the world. Any one of a wide variety of positively accelerating nonlinearities would do. Simply giving the neurons a very high threshold would achieve a sparse code, but the system would lose information. We argue that the form of sparse coding found in sensory systems is useful because such codes maintain the information in the environment, but do so more efficiently. We argue that the evolutionary pressure to move the system towards a sparse code comes from the representational power of sparse codes. However, we do accept that metabolic constraints are quite important. It has been demonstrated that at the earliest levels of the visual system, ganglion cells (Berry, Warland and Meister, 1997) and lateral geniculate nucleus cells (Reinagel and Reid, 2000) show sparse (non-Gaussian) responses to temporal noise. A linear code, no matter how efficiently it was designed would not show such sparse activity, so we must assume that the sparseness is at least in part due to the nonlinearities in the system and not due to the match between the receptive fields and the sparse structure of the input. As with the results show sparse responses in non-sensory areas, we must accept that metabolic may also be playing a significant role.

Well it's nice to acknowledge we have physical limitations, but to assume that a linear code cannot exist simply because we currently do not have a model for it, is probably assuming too much. So what do we know ? the primary cortex is a low energy system which basically removes from consideration any complex system that requires resources (like an iteration system). This situation favors a linear system but so far, we have not found a good model for that. There is something deeper still. Even if we knew much about the sparsity of a scene, understanding the brain is really about understanding how large amount of information is reduced when traveling from the eye into the brain. In other words, we need to reduce the amount of data that our megapixel sensor called the eye is bringing in, and we must do this very fast (30 times a second). To put things in perspective, let us take an example: let us imagine we are seeing a scene where somebody waves a hand. If we were to take a video of this scene, we would probably get a 40 MB avi file (uncompressed). That file could then be compressed to 1 MB using MPEG for instance. While the compression is impressive, it is not impressive enough. In effect, when our brain sees this video, it can remember the hand and how it moved. The movement is probably two dimensional and so the brain really remembers the two parameters needed to produce a hand that waves in the manner that is shown in the video. In other words, the brain is probably not storing 1 MB of information when it stores this hand waving activity, it is most probably storing how two parameters changed over time, which in many occasion is much less than 1 MB of data. The real question becomes: Is there a way to reduce that the 1MB information further ? We are not asking ourselves if the input or the receptor are sparse (it is a necessary condition), we are interested in answering the question on how sparser we can make this information by using the connection between these sparse elements. Can we reduce the dimensionality of the signal further and exploit it ?

Enters Dimensionality reduction: Ever since the discovery of dimensionality reduction schemes (LLE, Isomap, Laplacian-diffusion..) that are taking high dimensionality data and and are able to map them into low dimensionality manifolds, researchers have been trying to extend these techniques to wider sets of problems. In the cognition world for instance, Jonathan Pillow and Eero Simoncelli perform dimensionality reduction applied to neural models but it is not obvious how these techniques directly translate into a specific functionality of the primary cortex even if they take an example of a V1 cell. It is also not obvious how some of these techniques are robust to noise. But as stated earlier, there are different ways to go about dimensionality reduction. One of the most intriguing which has robustness built into it is Compressed Sensing. Compressed Sensing has the ability to produce a robust decomposition of a manifold. Mike Wakin looked into that during his dissertation and found that smooth manifolds can be readily compressed using Compressed Sensing thereby making it a very simple solution to dimensionality reduction (see R. G. Baraniuk and M. B. Wakin in Random Projections of Smooth Manifolds ) but as Donoho and Grimes had shown earlier, sharp objects such as arms, legs have edges and that makes the manifold non-differentiable. This is a problem because it means that one cannot easily extract parameters from a video if we have these sharp edges. In order take care of that problem, one can be inspired by Biology i.e. to smooth these images using Gabor wavelets as in the human vision system (Object Recognition with Features Inspired by Visual Cortex by Thomas Serre, Lior Wolf and Tomaso Poggio) and then use the random projection of smooth manifolds to eventually figure out the parameters of the movements ( for more information on how to do that see High-resolution navigation on non-differentiable image manifolds or the Multiscale Structure of Non-Differentiable Image Manifolds)

[in Object Recognition with Features Inspired by Visual Cortex by Thomas Serre, Lior Wolf, Tomaso Poggio one may note that one can only be struck by the pain the algorithm goes through into in order to be robust.

- S1: Apply a battery of Gabor ﬁlters to the input image. The ﬁlters come in 4 orientations θ and 16 scales s (see Table 1). Obtain 16×4 = 64 maps (S1)sθ that are arranged in 8 bands (e.g., band 1 contains ﬁlter outputs of size 7 and 9, in all four orientations, band 2 contains ﬁlter outputs of size 11 and 13, etc).
- C1: For each band, take the max over scales and positions: each band member is sub-sampled by taking the max over a grid with cells of size NΣ ﬁrst and the max between the two scale members second, e.g., for band 1, a spatial max is taken over an 8 ×8 grid ﬁrst and then across the two scales (size 7 and 9). Note that we do not take a max over different orientations, hence, each band (C1)Σcontains 4 maps.
- During training only: Extract K patches Pi=1,...K of various sizes ni × ni and all four orientations (thus containing ni × ni × 4 elements) at random from the (C1)Σ maps from all training images.
- S2: For each C1 image (C1)Σ, compute: Y = exp(−γ||X − Pi||2) for all image patches X (at all positions) and each patch P learned during training for each band independently. Obtain S2 maps (S2)Σi .
- C2: Compute the max over all positions and scales for each S2 map type (S2)i (i.e., corresponding to a particular patch Pi) and obtain shift- and scale-invariant C2 features (C2)i , for i = 1 . . .K.

]

Besides Wakin, Donoho, Baraniuk and others collaborating with them, few have made that connection. Yves Meyer makes a passing reference to the use of compressed sensing in physiology (in Perception et compression des images fixes.) Gabriel Peyre however is a little more specific here.

Analogies in physiology:I am mentioning Gabriel Peyre's work because he works on bandelets. I had mentioned bandelets before in the context of an announcement made by the company Let it wave (headed by Stephane Mallat ) where they showed that faces could be compressed down to 500 bytes or the size of a bar code.

This compressed sampling strategy could potentially lead to interesting models for various sensing operations performed biologically. Skarda and Freeman have proposed a non-linear chaotic dynamic to explain the analysis of sensory inputs. This chaotic state of the brain ensures robustness toward unknown events and unreliable measurements, without using too many computing resources. While the theory of compressed sensing is presented here as a random acquisition process, its extension to deterministic or dynamic settings is a fascinating area for future research in signal processing.

If bandelets provide a recognizable Faces nonlinearly with 500 bytes, one only needs 5 x 500 bytes = 2.5 KB random samples (within the meaning of Compressed sensing) of that Face to be able to reconstruct it. 2.5KB is better than 10 MP or 1 MB. The number 5 is the bound for compressed sensing (more specific asymptotic laws/results can be found in the summary of Terry Tao)

However, the strongest result so far is the one found by Mike Wakin on random projection on a manifold where he uses neighborhood criteria in the compressed sensing space to permit an even better reconstruction than just assuming sparsity. In the figure below, one can compare the 5 random projection results using the Manifold based recovery from the traditional algorithms used like Orthogonal Matching Pursuit and Basis Pursuit.

Naturally, an extension of this is target detection as featured in the Smashed Filter article from Rice. With this type of result/framework, I am betting we can go lower than 2.5 KB of universal samples to characterize a face. The connection between cognition and compressed sensing and autism is simple: Face processing seem deficient for people affected with autism and we don't know why. A model based on the elements I just mentioned might give an insight into this issue.

## Tuesday, March 20, 2007

### Traduction de Compressed Sensing en Francais

Apres en avoir parle avec Emmanuel Candes, il semble qu'une traduction francaise judicieuse de "Compressed Sensing" pourrait etre: Acquisition Comprimée

After talking to Emmanuel Candes, a good french translation of "Compressed Sensing" should probably be: Acquisition Comprimée.

It looks as though "Compressed Sensing" and "Compressive Sampling" have been used for the same technique. Now a new term has surfaced "Compressive Sensing".

After talking to Emmanuel Candes, a good french translation of "Compressed Sensing" should probably be: Acquisition Comprimée.

It looks as though "Compressed Sensing" and "Compressive Sampling" have been used for the same technique. Now a new term has surfaced "Compressive Sensing".

### Compressed Sensing video presentations

It is one thing to read papers and presentations, it is also interesting to watch video presentations of the compressed sensing (or compressive sampling) idea. Richard Baraniuk has two:

If you know of another one, please drop me a line.

[ Update October 2007: there is a flurry of them here in this entry: Compressed Sensing Videos: Where Is My Popcorn ? ]

[Update August 2008: I have gathered most videos on the subject of Compressive Sensing on this CS Videos site ]

If you know of another one, please drop me a line.

[ Update October 2007: there is a flurry of them here in this entry: Compressed Sensing Videos: Where Is My Popcorn ? ]

[Update August 2008: I have gathered most videos on the subject of Compressive Sensing on this CS Videos site ]

## Saturday, March 17, 2007

### l1_ls: Simple Matlab Solver for l1-regularized Least Squares Problems (compressed sensing)

Make that four codes available to perform reconstruction in the compressed sensing setting. Kwangmoo Koh, Seung-Jean Kim, and Stephen Boyd just made available l1_ls. According to the authors:

l1_ls: Simple Matlab Solver for l1-regularized Least Squares Problems

l1_ls is a Matlab implementation of the interior-point method for l1-regularized least squares described in the paper, A Method for Large-Scale l1-Regularized Least Squares Problems with Applications in Signal Processing and Statistics. l1_ls solves an optimization problem of the form

,

where the variable is and the problem data are , and .

The solver l1_ls is developed for large problems. It can solve large sparse problems with a million variables with high accuracy in a few tens of minutes on a PC. It can also efficiently solve very large dense problems, that arise in sparse signal recovery with orthogonal transforms, by exploiting fast algorithms for these transforms.

## Thursday, March 15, 2007

### A beautiful mind can save your life

Terry Tao, the Fields medalist and the prodigy who developed compressed sensing with Emmanuel Candes and David Donoho, has an autistic brother:

The Taos had different challenges in raising their other two sons, although all three excelled in math. Trevor, two years younger than Terry, is autistic with top-level chess skills and the musical savant gift to play back on the piano a musical piece — even one played by an entire orchestra — after hearing it just once. He completed a Ph.D. in mathematics and now works for the Defense Science and Technology Organization in Australia.The youngest, Nigel, told his father that he was “not another Terry,” and his parents let him learn at a less accelerated pace. Nigel, with degrees in economics, math and computer science, now works as a computer engineer for Google Australia.

This is interesting in many ways. A particular way to look at it, is that Terry has provided a technique (Compressed Sensing) that has the potential to revolutionize the way people model the primary cortex: i.e how the brain decomposes, processes and understands information. A trait of autism is the inability to treat correctly information and subsequently act on it. This type of story where somebody discovers something that eventually he is affected by afterwards, reminds me of a story Glenn Seaborg told us when he came to campus six months before he passed away. During his talk, he reminded us that some colleagues of his came to his lab once and asked if he could find a radioactive element that had a specific half life that was not too long and not too short. Seaborg obliged and discovered Iodine-131. According to this entry on wikipedia:

Livingood and Seaborg collaborated to create an important isotope of iodine, iodine-131 (I-131) which is still used to treat thyroid disease. (Many years later, it was credited with prolonging the life of Seaborg's mother.)

### All you wanted to know about Compressed Sensing

## Wednesday, March 14, 2007

### Implementing Compressed Sensing in Applied Projects

We are contemplating using Compressed Sensing in three different projects:

- The Hyper-GeoCam project: This is a payload that will be flown on the HASP platform in September. Last year, we flew a simple camera that eventually produced a 105 km panorama of New Mexico. We reapplied for the same program and have been given the OK for two payloads. The same GeoCam will be re-flown so that we can produce a breath taking panorama from 36 km altitude. The second payload is essentially supposed to be a hyperspectral imager on the cheap: i.e. a camera and some diffraction gratings allowing a fine decomposition of the reflected sun light from the ground. The project is called Hyper-GeoCam and I expect to implement a random lens imager such as the one produced at MIT. Tests will be performed on the SOLAR platform.
- The DARPA Urban Challenge: We have a car selected in the track B: We do not have Lidars and need to find ways to navigate in an urban settings with little GPS availability. The autonomous car is supposed to be navigating in a mock town and follow the rules of the California traffic laws, that includes passing other cars.
- Solving the Linear Boltzmann equation using compressed sensing techniques:The idea is that this equation has a known suite of eigenfunctions (called Case eigenfunctions) and because they are very difficult to use and expand from, it might be worth a try to look into the compressed sensing approach to see if it solves the problem more efficiently.

## Tuesday, March 13, 2007

### Compressive Sensing as a way to solve Integral Equations ?

While reading Compressive Radar Imaging by Richard Baraniuk and Philippe Steeghs, I realized that something very different is happening with Compressed Sensing. I mentioned earlier that in the past, the solving of integral equations implies the use of weak formulations in order to obtain a moment based description of that integral equation. Hence, the general consensus with using an L_2 norm approach (or a least square reduction of the approximation error) is to use trial functions that are the same of the basis functions (in neutron transport, the situation is in fact a little different but nobody knows really why. ) This fact leads most researchers into projecting kernels onto sets of functions for which they become sparse like the wavelets. In other words, there is an expectation that the projected kernel will be very sparse and that matrix-vector operations will be reduced so that the overall method becomes very competitive with moment based method using non-sparse kernels. However, when the kernel reflects a geometry, it is difficult to know in advance which of the component of that kernel will have to be retained and one is therefore led to the nonlinear operation of computing them all first and then getting rid of the ones that are too small. This is a little bit what compressed sensing is avoiding when sampling a signal.

As it turns out, the integral equation that is being solved by a CS-Radar is different, the trial functions are specifically tailored to be incoherent with the basis functions. The big question would be to figure out how to do several iterations instead of just one. In neutron transport, we call some of these methods "source iteration". If we are to push the parallel further, on the one hand the Green's function is generally a description of the medium in which the neutron lives whereas the source is really the neutron distribution, hence there is no reason to believe that both types of functions should come from the same family and bear the same decomposition.

The neutron example goes further. Emmanuel Candes made a presentation last year on compressed sensing and showcased an example using neutron radiography at NIST on a fuel cell. It turns out, this is an experiment I did a long time ago at the Nuclear Science Center. The main issue being that in order to see exactly what is going on inside the fuell cell (which has a lot of graphite and copper), neutrons are the only way to see through and detect water or water vapor at the same time.

The reason for this type of investigation is simple: If water gets stuck at one place in the groves of the fuel cell, it will stop the fuel cell from working. It is therefore important to know exactly the hydrodynamics of the water in the cell while the cell is working. What is interesting is that the neutron scatters when in presence of water, so that the color on the graph of his presentation represents roughly the density of water in the cell. The neutron beams are used as in CT tomography with the fuel cell being hit from different angles. The compressed sensing formalism allows for fewer number of angles. What is interesting though is that many neutrons are deflected or diffused and therefore the problem at hand cannot be rigorously be seen as a CT scan. It is good enough for observation, but it would not be appropriate for neutron flux measurements for instance. The scattering issue should be solved using an integral equation called the linear Boltzmann equation!

## Thursday, March 08, 2007

### Debris Scattering to GEO

With regards to the Chinese test: I have not seen this being articulated in any of the websites I have seen on the subject but while LEO is indeed a very sensitive location at least for ISS, has anybody quantified how much of these debris flux would eventually migrate to GEO ? My point is that with debris of similar sizes, collision is likely to produce elements with a large angular deviation (like in neutron transport) and might produce a flux to GEO. GEO would be a problem for everybody not just the ISS or low earth orbit countries.

If one looks at this 1999 UN report, one can see that there are in fact few models that take into account both LEO and GEO and therefore I am expecting that very few have coupling between LEO and GEO models within these codes.

The LEGEND model developed at JSC seems to address this issue. LEGEND was produced...

..To continue to improve our understanding of the orbital debris environment, the NASA Orbital Debris Program Office initiated an effort in 2001 to develop a new model to describe the near-Earth debris environment. LEGEND, a LEO-to-GEO Environment Debris model, is a full-scale three-dimensional debris evolutionary model. It covers the near-Earth space between 200 and 50,000 km altitude, including low Earth orbit (LEO), medium Earth orbit (MEO), and geosynchronous orbit (GEO) regions. ... The main function of the LEGEND historical component is to reproduce the historical debris environment (1957 to present) to validate the techniques used for the future projection component of the model. The model utilizes a recently updated historical satellite launch database (DBS database), two efficient state-of-the-art propagators (PROP3D and GEOPROP), and the NASA Standard Breakup Model. .... A key element in the LEGEND future projection component is a three-dimensional collision probability evaluation model. It provides a fast and accurate way to estimate future on-orbit collisions from LEO to GEO. Since no assumptions regarding the right ascensions of the ascending node and arguments of perigee of objects involved are required, this model captures the collision characteristics in real three-dimensional physical space. It is a critical component of a true three-dimensional debris evolutionary model.

The typical projection period in LEGEND is 100 years. Due to uncertainties involved in the process (e.g., future launch traffic, solar activity, explosions, collisions), conclusions are usually drawn based on averaged results from 30 Monte Carlo simulations.

J-C Liou one of the researcher involved in the development of LEGEND explained a year ago that

The current debris population in the LEO region has reached the point where the environment is unstable and collisions will become the most dominant debris-generating mechanism in the future...Even without new launches, collisions will continue to occur in the LEO environment over the next 200 years, primarily driven by the high collision activities in the region between 900- and 1000-km altitudes, and will force the debris population to increase.

Downscatter from MEO to LEO seems to be well taken into account but I still wonder about upscattering from MEO to GEO or from LEO to GEO.

### Compressed Sensing Hardware Implementations

[Update Nov. '08: Please find the Compressed Sensing Hardware page here ]

The one pixel camera made by Richard Baraniuk and his group at Rice is the one that received the most press. In order to build this new type of camera ones needs a DMD controlled board (at 6K$), one pixel and you're set. That one pixel detector could really be a photodiode, a radiation detector or other Teraherz receiver.

However, the concept of compressed sensing does not really need hardware transformation/implementation as impressive as this one. At one end, of the spectrum, one can just change the way sampling is done as in MRI work ( Sparse MRI: The application of compressed sensing for rapid MR imaging by Michael Lustig, David Donoho, and John M. Pauly), where sampling in the k-space is done below the Nyquist criterion. Current technology already allows for the gathering of signals with less than pure frequency content. Using the ability to put several frequencies together in one spike, allows one to retrieve directly compressed samples thereby enabling substantial savings in acquisition time.

In between the ends of that spectrum are several set-ups that are trying to use current hardware with some slight modification to provide lower sampling sets. I have come across two: Random Lens Imagers and Compressive Sampling Spectrometers.

* The random lens imaging technique developed at MIT by Rob Fergus, Antonio Torralba, and William T. Freeman is where one uses a normal DSLR camera but removes the lens and replaces it with a transparent material in which mirrors where included at random. The point they are making is the following: The image obtained from this system are the compressed sensing measurements. In order to reconstruct the original image, they have to calibrate the new camera set-up. The only way to do this is by having say a laser light shone on the camera and see how the ray is being imaged. When you have a 10 Mp camera, this means that you will shine that laser light from ten millions locations in the field of view in order to have the ten million unit response to each laser light. When that is done, you solve a linear algebra problem (a simple matrix inversion) to obtain the calibration/response matrix. Then every time, you take a picture, you multiply your result with that matrix and obtain the picture you were looking for. It is pretty obvious that the calibration step could be improved by removing the need to shine a laser light ten million times. In other words, you have too many unknowns for too few equations (being lazy you will not shine that laser light ten millions times). The compressed sensing theory really says that you can solve that problem having too few equations (or calibration image ) so that you can find the inverted matrix with much fewer trials. The advantage is the potential ability to provide more information from current CMOS/CCDs. In effect, by using many calibration images, one could potentially obtain superresolution information (smaller than the pixel size of the DSLR) or depth information. The ability to use the current very large CMOS capability (instead of one pixel) has a very real potential. Consider this: if a 1 MP camera can produce 30 KB images in JPEG, there is 30 000 information allow a good representation of a 2-D scene (about 170 information per dimension.) Therefore, a 3-D scene would require about 4.9 MB for a good description. Clearly one is already gathering that much information from normal cameras.

* Compressive sampling imager: With most hyperspectral cameras there is the need to reduce the amount of information gathered while it is being gathered not after. For instance, the Hyperion camera on EO-1 was designed with the bandwidth for transmitting the information down in mind (the TDRSS system cannot handle more than 6 MBit/s) that is why people try to compress the data after it has been gathered in order to allow efficient transmission of signals. But on a spacecraft like EO-1 you don't really have that much computational power and you are really looking for a way to acquire the minimum amount of information in the first place. The current interesting undertaking in this field seems to be about compressed sensing spectrometers or compressive sampling spectrometers by the DISP group at Duke (David J. Brady, Mike Gehm, Scott McCain, Zhaochun Xu, Prasant Potuluri, Mike Sullivan, Nikos Pitsianis, Ben Hamza, Ali Adibi).

A good presentation of their effort can be found here. The idea is that if you assume that:

• Measurements are expensive

• Photons are scarce and

• Spectra are sparse

you can modify your current set-up by introducing a mask between two gratings. So instead of having systems that respond to single spectral signals, group testing is used so that several spectral bands can provide one compressed measurement.

More advances on this type of hardware can be found in this fascinating article where it shows why Nyquist and Golay sampling theorems were reduced to piece by current advances, this a must read.

The one pixel camera made by Richard Baraniuk and his group at Rice is the one that received the most press. In order to build this new type of camera ones needs a DMD controlled board (at 6K$), one pixel and you're set. That one pixel detector could really be a photodiode, a radiation detector or other Teraherz receiver.

However, the concept of compressed sensing does not really need hardware transformation/implementation as impressive as this one. At one end, of the spectrum, one can just change the way sampling is done as in MRI work ( Sparse MRI: The application of compressed sensing for rapid MR imaging by Michael Lustig, David Donoho, and John M. Pauly), where sampling in the k-space is done below the Nyquist criterion. Current technology already allows for the gathering of signals with less than pure frequency content. Using the ability to put several frequencies together in one spike, allows one to retrieve directly compressed samples thereby enabling substantial savings in acquisition time.

In between the ends of that spectrum are several set-ups that are trying to use current hardware with some slight modification to provide lower sampling sets. I have come across two: Random Lens Imagers and Compressive Sampling Spectrometers.

* The random lens imaging technique developed at MIT by Rob Fergus, Antonio Torralba, and William T. Freeman is where one uses a normal DSLR camera but removes the lens and replaces it with a transparent material in which mirrors where included at random. The point they are making is the following: The image obtained from this system are the compressed sensing measurements. In order to reconstruct the original image, they have to calibrate the new camera set-up. The only way to do this is by having say a laser light shone on the camera and see how the ray is being imaged. When you have a 10 Mp camera, this means that you will shine that laser light from ten millions locations in the field of view in order to have the ten million unit response to each laser light. When that is done, you solve a linear algebra problem (a simple matrix inversion) to obtain the calibration/response matrix. Then every time, you take a picture, you multiply your result with that matrix and obtain the picture you were looking for. It is pretty obvious that the calibration step could be improved by removing the need to shine a laser light ten million times. In other words, you have too many unknowns for too few equations (being lazy you will not shine that laser light ten millions times). The compressed sensing theory really says that you can solve that problem having too few equations (or calibration image ) so that you can find the inverted matrix with much fewer trials. The advantage is the potential ability to provide more information from current CMOS/CCDs. In effect, by using many calibration images, one could potentially obtain superresolution information (smaller than the pixel size of the DSLR) or depth information. The ability to use the current very large CMOS capability (instead of one pixel) has a very real potential. Consider this: if a 1 MP camera can produce 30 KB images in JPEG, there is 30 000 information allow a good representation of a 2-D scene (about 170 information per dimension.) Therefore, a 3-D scene would require about 4.9 MB for a good description. Clearly one is already gathering that much information from normal cameras.

* Compressive sampling imager: With most hyperspectral cameras there is the need to reduce the amount of information gathered while it is being gathered not after. For instance, the Hyperion camera on EO-1 was designed with the bandwidth for transmitting the information down in mind (the TDRSS system cannot handle more than 6 MBit/s) that is why people try to compress the data after it has been gathered in order to allow efficient transmission of signals. But on a spacecraft like EO-1 you don't really have that much computational power and you are really looking for a way to acquire the minimum amount of information in the first place. The current interesting undertaking in this field seems to be about compressed sensing spectrometers or compressive sampling spectrometers by the DISP group at Duke (David J. Brady, Mike Gehm, Scott McCain, Zhaochun Xu, Prasant Potuluri, Mike Sullivan, Nikos Pitsianis, Ben Hamza, Ali Adibi).

A good presentation of their effort can be found here. The idea is that if you assume that:

• Measurements are expensive

• Photons are scarce and

• Spectra are sparse

you can modify your current set-up by introducing a mask between two gratings. So instead of having systems that respond to single spectral signals, group testing is used so that several spectral bands can provide one compressed measurement.

More advances on this type of hardware can be found in this fascinating article where it shows why Nyquist and Golay sampling theorems were reduced to piece by current advances, this a must read.

It is pretty obvious that any of these elements for adopting compressed sensing fit pretty well with radiation measurements where one can direct radiation beams. I am also thinking of implementing some of these systems for the HASP 2007 flight.

## Wednesday, March 07, 2007

### Simply Knowing Yourself

While reading this paper on the current illusion set forth by "better" classifiers, I could not help but remember Sebastian Thrun's presentation on winning DARPA first Grand Challenge.

While classifying is one thing, maybe quantifying one's state of knowledge or your state of ignorance might be more relevant.

While classifying is one thing, maybe quantifying one's state of knowledge or your state of ignorance might be more relevant.

### A refreshing look at power law distribution

So it looks like I am not the only one to think of the issue of finding a power law distribution is very weak on drawing inference from empirical data. Surely there must be the same level of skepticism for Benford's law and the other interesting laws. Mark Newman shows why this should be the case for different physical and sociological processes (like the several scenarios modeling extinction). The terrorism paper is very close to an analysis Paul Nelson and I wanted to get funded some years ago. I am glad somebody did it.

## Tuesday, March 06, 2007

### It's the palm cooling, stupid.

When I was reading Tony Tether's interview on the cool glove, I could not shake the thought that it is connected to several areas of interest I have. In this Stanford paper, it is shown that cooling through the palms of your hand is really important for most physical exhaustive activity as well as for people who suffer from MS. The principle is that palms are the main radiators for the body.

What Heller and Grahn were seeing was the return trip: when externally applied heat shocked open the radiators in the cold palms of anesthesia patients, warmed blood was returned straight to the heart, and the body was reheated from the inside out. Applying a mild vacuum to the hand intensified this effect.

But this part of the entry stuck me

Grahn’s latest homemade version features soft vinyl against the hand instead of metal. One design challenge is obvious—how to create a vacuum-bearing glove flexible enough so that its wearers can use their hands, not just sit cooling their palms.

what he is describing is an element of an reversed advanced spacesuit.

This quote

Heller and Grahn have found in the lab that the temperature under which the radiators shut down in humans is highly individual.

strucks me as requiring some type of system to evaluate the radiator capacity for every potential customer. The RTX device using this concept is currently made by Avacore.

While reading this, I could not shake the fact that it was doing the reverse of the heat pipe glove and wonder how Bejan's work can be used to figure out an optimal cooling/heating solution that does not require a compressor.

### Darpa's ability to innovate

Noah Shachtman interviews Tony Tether, the head of DARPA in this WIRED blog entry. Here is the quote I like:

People come to me from all over the world, and they look at our track record and what we're doing, and they want to know how they can make an organization like ours. I tell them it's simple. You just have to make sure the people don't stay there very long.

## Monday, March 05, 2007

### Why Compressed Sensing is important when detecting movement

Richard Baraniuk in his presentation entitled Multiscale Geometric Analysis (the real player version of this presentation is here) makes a very good presentation of how compressed sensing works. What is interesting in this presentation are the questions of the audience at the end: They focus on the engineering of the mask/camera but in fact the new element is the mathematics (and that element is not a new wavelet basis). The progressivity aspect of the reconstruction is important and it is, in my view, shown very well in the recent bayesian paper on this. So in order to build a new camera ones needs a DMD controlled board (at 10K$), one pixel and you're set. The underlying reason why this concept is so important is when it brings large advantage over current solutions. For instance, movement detection. In traditional approach one needs to do some type of optic flow computation between two frames to evaluate changes.

With compressed sensing, since events shown on a camera are highly correlated to each other over time, Baraniuk shows that instead of having to deal/compute with 36 millions bits per second (for example using a traditional optic flow solution) one only needs to deal with 24 bits per second to evaluate changes. Another case in point is shown in the presentation on Intelligent Motion Detection Using Compressed Sensing by Heather Johnston, Siddharth Gupta, Grant Lee, Veena Padmanabhan

.

### Better Super-resolution

The science of using several frames in order to produce super-resolution just improved with the new paper from Minh Do and Hat Nguyen entitled Optimal super-resolution without input assumptions. I am waiting for their Matlab implementation.

## Saturday, March 03, 2007

### Smashed filters and Bayesian approach to Compressed Sensing among others

Found in the fantastic Compressed Sensing Resource at Rice. In terms of application, two fascinating papers just surfaced:

* The smashed filter for compressive classification and target recognition by Mark Davenport, Marco Duarte, Michael Wakin, Jason Laska, Dharmpal Takhar, Kevin Kelly, and Richard Baraniuk it is an extension of the Matched Filter theory to compressed measurements. The interesting part is that the ability to do classification is based in part on the dimension of the underlying manifold (generally it is low).

* Bayesian compressive sensing by Shihao Ji, Ya Xue, and Lawrence Carin, where there is description of a reconstruction scheme that relies on Bayesian inversion. The difference between this scheme and others is the ability to figure out how your convergence is doing during the signal reconstruction computations.

* Sparse MRI: The application of compressed sensing for rapid MR imaging.

Michael Lustig, David Donoho, and John M. Pauly, where compressed measurements are directly taken by "randomly" undersampling in the k-fourier space thereby enabling substantial savings in acquisition time.

* and the ever fascinating paper on The Johnson-Lindenstrauss Lemma Meets Compressed Sensing by Richard Baraniuk, Mark Davenport, Ronald DeVore, and Michael Wakin.

* The smashed filter for compressive classification and target recognition by Mark Davenport, Marco Duarte, Michael Wakin, Jason Laska, Dharmpal Takhar, Kevin Kelly, and Richard Baraniuk it is an extension of the Matched Filter theory to compressed measurements. The interesting part is that the ability to do classification is based in part on the dimension of the underlying manifold (generally it is low).

* Bayesian compressive sensing by Shihao Ji, Ya Xue, and Lawrence Carin, where there is description of a reconstruction scheme that relies on Bayesian inversion. The difference between this scheme and others is the ability to figure out how your convergence is doing during the signal reconstruction computations.

* Sparse MRI: The application of compressed sensing for rapid MR imaging.

Michael Lustig, David Donoho, and John M. Pauly, where compressed measurements are directly taken by "randomly" undersampling in the k-fourier space thereby enabling substantial savings in acquisition time.

* and the ever fascinating paper on The Johnson-Lindenstrauss Lemma Meets Compressed Sensing by Richard Baraniuk, Mark Davenport, Ronald DeVore, and Michael Wakin.

## Thursday, March 01, 2007

### Current use of EO data for Search And Rescue operations (SAR)

The current use of Earth Observation data is directed toward using wind scattometer data and include them directly into the drift modeling used for Search and Rescue Operations of known objects (objects for which we know the original location but want to know more about drift).

What is interesting is the apparent mismatch between the model data and the EO actual observations as shown by Michel Olagnon from IFREMER in the photo below (the picture shows the model color with satellite swath lines providing real information, the mismatch show the inaccuracy of the model).

With regards to data fusion in the current SAROPS software by the USCG, it looks like the current configuration only include overlays of low resolution.

it does not seem to address the ability to image directly the objects of interest (either a boat lost at sea or drifting containers) .

What is interesting is the apparent mismatch between the model data and the EO actual observations as shown by Michel Olagnon from IFREMER in the photo below (the picture shows the model color with satellite swath lines providing real information, the mismatch show the inaccuracy of the model).

With regards to data fusion in the current SAROPS software by the USCG, it looks like the current configuration only include overlays of low resolution.

it does not seem to address the ability to image directly the objects of interest (either a boat lost at sea or drifting containers) .

### Drifting behavior while searching

This series of images comes from the presentation of Art Allen (USCG) on SAROPS. One can clearly sees that drift is an important component of the search activity while the search is underway. Another question begs to be answered: how come the search grid is not uniform in order to provide efficient information ?

### Tracking multiple targets at sea: not a question of if.

With the ever increasing world reliance on large cargo containers, it would seem that tracking multiple targets like 20/40 feet containers should be an on-going concern. The most recent occurence is that of the MSC Napoli close to the English coast.

While the modeling of Marc le Boulluec at Ifremer on the hydrodynamics of drifting containers is a good start [the study reminded me of the issue of space debris falling back into the atmosphere and the probable need to go probabilistic on these laws],

it is also important to realize that the tracking of multiple targets is a difficult issue that includes data fusion especially if one uses different data acquisition systems (satellite imagery, airplane sightings, human eyewitnesses.....) at different times. The other issue is finding out when these drifting containers are not a danger anymore as they are not designed to stay afloat but will certainly wreck any unsuspecting small boats

In line with that, I would be interested to see how one has summarized the knowledge on head-sea parametric rolling and integrated it into an algorithm used for predicting trajectories. I hope it goes beyond a simple formula.

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