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Research Curvelet Curvelets form a 1-tight frame which is near
optimal in representing curved singularities. Each element is called a
curvelet. Curvelets have different scales, orientations and locations. Unlike
wavelets, which are tensor product of 1D wavelets, curvelets are
intrinsically defined in 2D (and 3D). Most importantly, curvelets are
needle-shaped atoms with a lot of orientations. Thus, 2D curvelets are
efficient in representing curved singularities and 3D curvelets are efficient
in capturing surface singularities. As we know that edges are the main
characteristics for an image, curvelets thus have applications in image
denoising, compression and quality assessment, etc. Another nice property of curvelets is that they
remain coherent waveforms under wave equation in a smooth medium. There is
also research going on in related area, such as wave computations, seismic
imaging, etc. Reference: H. Douma, Maarten V. de Hoop, `Leading-order seismic imaging using curvelets', Geophysics (2007) in print. J.-L. Starck, M.K. Nguyen and F. Murtagh, `Wavelets and
Curvelets for Image Deconvolution: a Combined Approach', 2003. Sparse Approximation When solving an underdetermined system of
equations, there are infinite many solutions if no criterion is applied. Only
by adding some kind of constraint, one could possible narrow down the
solution to a unique one. Sparse constraint is found to benefit so many
modern applications in signal and image processing. Sparse approximation is
the problem of approximating an input signal by a linear combination of atoms
of an over-complete dictionary with one kind of sparsity constraint. Sparsity
is usually measured by ell-0 norm, the number of non-zero items, which
results in NP hard optimization problems. As a result, relaxed forms of
minimizations involving ell-1 or ell-p (p<1) norm come into a play.
Traditional methods such as interior point methods are accurate but slow.
Greedy methods are fast but may fail badly. Facing large scale problems,
researchers are looking for new algorithms which are efficient and also accurate.
On the other hand, we would also like to know under what conditions, the
solution is unique. Both theoretical and numerical aspects are under
investigation. Reference: Afred M. Bruckstein, David L. Donoho and Michael
Elad, `From Sparse Solutions of Systems of
Equations to Sparse Modeling of Signals and Images', SIAM Review, 2009. Other Image Quality Assessment (IQA) is the method of
quantifying the quality of natural images. In most cases, images are
distorted by one or several filters. IQA has applications in data management,
comparison among digital camera lens, evaluation of image processing
algorithms such as denoising, deblurring algorithms, etc. Since noise, blur
affect the edges more than smooth parts in an image, we expect to design new
algorithms based on curvelet transforms. Reference: Ji Shen, Qin Li, Gordon Erlebacher, `Curvelet based
No-Reference Objective Image Quality Assessment', PCS, Chicago, 2009 Currently, I am doing research on using curvelet as
a sub-dictionary for separating 2D and 3D data into parts carrying different
kinds of information. The algorithms I try to apply on are from the family of
shrinkage algorithms. The problems are how efficient and stable the
algorithms are, what the criterions are for good separation and under what
condition, we can perfectly separate the information as we desire. Furthermore, I am interested in IRLS-based
algorithm for sparse approximation. There are variations on the traditional
IRLS algorithm in order to suit for sparse approximation problems. However, in
spite of the fact that the methods were shown to be accurate in several
applications numerically, not many theoretical results have been established
for this kind of methods. Inspired by a recent paper on the IRLS for sparse
recovery, I try to prove the convergence of an IRLS method for
error-constrained sparse approximation. The other paper our group is working on is the
curvlet-based no reference method for IQA. The log-pdf of curvelet
coefficients for fine scales forms Generalized Gaussian Distribution for
natural images. This statistic fact leads to our IQA method based on
curvelet. |
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