The AstroStat Slog » multiscale

[ArXiv] Sparse Poisson Intensity Reconstruction Algorithms

hlee — Thu, 07 May 2009 16:14:39 +0000

One of [ArXiv] papers from yesterday whose title might drag lots of attentions from astronomers. Furthermore, it’s a short paper.
[arxiv:math.CO:0905.0483] by Harmany, Marcia, and Willet.

Estimating f under “Sparse Poisson Intensity” condition is an frequently appearing topic in high energy astrophysics data analysis. Some might like to check references in the paper, which offer solutions to compressed sensing problems with different kinds of sparsity, minimization approaches, and constraints on f.

Apart from the technical details, the first two sentences from the conclusion,

We have developed computational approaches for signal reconstruction from photon-limited measurements – a situation prevalent in many practical settings. Our method optimizes a regularized Poisson likelihood under nonnegativity constraints

tempt me to study and try their algorithm.

[MADS] multiscale modeling

hlee — Thu, 11 Dec 2008 19:46:05 +0000

A few scientists in our group work on estimating the intensities of gamma ray observations from sky surveys. This work distinguishes from typical image processing which mostly concerns the point estimation of intensity at each pixel location and the size of overall white noise type error. Often times you will notice from image processing that the orthogonality between errors and sources, and the white noise assumptions. These assumptions are typical features in image processing utilities and modules. On the other hand, CHASC scientists relate more general and broad statistical inference problems in estimating the intensity map, like intensity uncertainties at each point and the scientifically informative display of the intensity map with uncertainty according to the Poisson count model and constraints from physics and the instrument, where the field, multiscale modeling is associated.

As the post title [MADS] indicates, no abstract has keywords multiscale modeling. It seems like that just the jargon is not listed in ADS since “multiscale modeling” is practiced in astronomy. One of examples is our group’s work. Those CHASC scientists take Bayesian modeling approaches, which makes them unique to my knowledge in the astronomical society. However, I expected constructing an intensity map through statistical inference (estimation) or “multiscale modeling” to be popular among astronomers in recent years. Well, none came along from my abstract keyword search.

Wikipedia also shows a very brief description of multiscale modeling and emphasized that it is a fairly new interdisciplinary topic. wiki:multiscale_modeling. TomLoredo kindly informed me some relevant references from ADS after my post [MADS] HMM. He mentioned his search words were Markov Random Fields which can be found from stochastic geometry and spatial statistics in addition to many applications in computer science. Not only these publications but he gave me a nice comment on analyzing astronomical data, which I’d rather postpone for another discussion.

Quantifying Doubt and Confidence in Image “Deconvolution” by Connors, Alanna; van Dyk, D.; Chiang, J.; CHASC
Blind Bayesian restoration of adaptive optics telescope images using generalized Gaussian Markov random field models by Jeffs, Brian D.; Christou, Julian C.
Segmenting Chromospheric Images with Markov Random Fields (paper in SCMA II) Turmon, Michael J.; Pap, Judit M.
Bayesian deconvolution methods in astronomy by Molina, R.; Katsaggelos, A. K.; Mateos, J
Compound Gauss-Markov random fields for astronomical image restoration by Molina, R.; Katsaggelos, A. K.; Mateos, J.; Abad, J
Markov random field applications in image analysis by Jain, A. K.; Nadabar, S. G (I bet “Jain” is the author of many celebrated papers in image processing and machine learning. I often find that well known computer scientists involve in astronomical researches ).

The reason I was not able to find these papers was that they are not published in the 4 major astronomical publications + Solar Physics. The reason for this limited search is that I was overwhelmed by the amount of unlimited search results including arxiv. (I wonder if there is a way to do exclusive searches in ADS by excluding arxiv:comp, arxiv:phys, arxiv:math, etc). Thank you, Tom, for providing me these references.

Please, check out CHASC website for more study results related to “multiscale modeling” from our group.

[Added] Nice tutorials related to Markov Random Fields (MRF) recommended by an expert in the field and a friend (all are pdfs).

Mexican Hat [EotW]

vlk — Wed, 28 May 2008 17:00:38 +0000

The most widely used tool for detecting sources in X-ray images, especially Chandra data, is the wavelet-based wavdetect, which uses the Mexican Hat (MH) wavelet. Now, the MH is not a very popular choice among wavelet aficianados because it does not form an orthonormal basis set (i.e., scale information is not well separated), and does not have compact support (i.e., the function extends to inifinity). So why is it used here?

The short answer is, it has a convenient background subtractor built in, is analytically comprehensible, and uses concepts very familiar to astronomers. The last bit can be seen by appealing to Gaussian smoothing. Astronomers are (or were) used to smoothing images with Gaussians, and in a manner of speaking, all astronomical images already come presmoothed by PSFs (point spread functions) that are nominally approximated by Gaussians. Now, if an image were smoothed by another Gaussian of a slightly larger width, the difference between the two smoothed images should highlight those features which are prominent at the spatial scale of the larger Gaussian. This is the basic rationale behind a wavelet.

So, in the following, G(x,y;σ_x,σ_y,x_o,y_o) is a 2D Gaussian written in such that the scaling of the widths and the transposition of the function is made obvious. It is defined over the real plane x,y ε R² and for widths σ_x,σ_y. The Mexican Hat wavelet MH(x,y;σ_x,σ_y,x_o,y_o) is generated as the difference between the two Gaussians of different widths, which essentially boils down to taking partial derivatives of G(σ_x,σ_y) wrt the widths. To be sure, these must really be thought of as operators where the functions are correlated with a data image, so the derivaties must be carried out inside an integral, but I am skipping all that for the sake of clarity. Also note, the MH is sometimes derived as the second derivative of G(x,y), the spatial derivatives that is.

The integral of the MH over R² results in the positive bump and the negative annulus canceling each other out, so there is no unambiguous way to set its normalization. And finally, the Fourier Transform shows which spatial scales (k_x,y are wavenumbers) are enhanced or filtered during a correlation.

[ArXiv] 3rd week, Apr. 2008

hlee — Mon, 21 Apr 2008 01:05:55 +0000

The dichotomy of outliers; detecting outliers to be discarded or to be investigated; statistics that is robust enough not to be influenced by outliers or sensitive enough to alert the anomaly in the data distribution. Although not related, one paper about outliers made me to dwell on what outliers are. This week topics are diverse.

[astro-ph:0804.1809] H. Khiabanian, I.P. Dell’Antonio
A Multi-Resolution Weak Lensing Mass Reconstruction Method (Maximum likelihood approach; my naive eyes sensed a certain degree of relationship to the GREAT08 CHALLENGE)
[astro-ph:0804.1909] A. Leccardi and S. Molendi
Radial temperature profiles for a large sample of galaxy clusters observed with XMM-Newton
[astro-ph:0804.1964] C. Young & P. Gallagher
Multiscale Edge Detection in the Corona
[astro-ph:0804.2387] C. Destri, H. J. de Vega, N. G. Sanchez
The CMB Quadrupole depression produced by early fast-roll inflation: MCMC analysis of WMAP and SDSS data
[astro-ph:0804.2437] P. Bielewicz, A. Riazuelo
The study of topology of the universe using multipole vectors
[astro-ph:0804.2494] S. Bhattacharya, A. Kosowsky
Systematic Errors in Sunyaev-Zeldovich Surveys of Galaxy Cluster Velocities
[astro-ph:0804.2631] M. J. Mortonson, W. Hu
Reionization constraints from five-year WMAP data
[astro-ph:0804.2645] R. Stompor et al.
Maximum Likelihood algorithm for parametric component separation in CMB experiments (separate section for calibration errors)
[astro-ph:0804.2671] Peeples, Pogge, and Stanek
Outliers from the Mass–Metallicity Relation I: A Sample of Metal-Rich Dwarf Galaxies from SDSS
[astro-ph:0804.2716] H. Moradi, P.S. Cally
Time-Distance Modelling In A Simulated Sunspot Atmosphere (discusses systematic uncertainty)
[astro-ph:0804.2761] S. Iguchi, T. Okuda
The FFX Correlator
[astro-ph:0804.2742] M Bazarghan
Automated Classification of ELODIE Stellar Spectral Library Using Probabilistic Artificial Neural Networks
[astro-ph:0804.2827]S.H. Suyu et al.
Dissecting the Gravitational Lens B1608+656: Lens Potential Reconstruction (Bayesian)