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Hyper-pyramid-adapted shearlet transform with application to compressive level set estimation
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DOI：10.1088/1361-6420/ae10d8.png)
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Abstract
En 中文
Sparse approximation properties are key to the thriving area of compressed sensing which requires a sparsifying system for the data. The shearlet-based signal representation systems offer optimal sparsity of piecewise smooth functions while maintaining a direct relationship with the traditional wavelet-based systems in the sense that shearlets constitute an affine system that stems out from a square-integrable group representation. In this paper, we propose and examine the hyper-pyramid-adapted shearlets in the d-dimensional Euclidean space; the usual cone-adapted shearlet system is a specific case for d = 2. Our use case of the associated hyper-pyramid-adapted shearlet transform (Hpast) is in the process of formulating a shearlet-based algorithm for the problem of compressive level set estimation. The level set of a function f:[0,1](d)-> R is a region S & lowast; in its domain over which the function exceeds a certain critical value nu; that is, S & lowast;={x is an element of[0,1](d): f(x)>nu}. We aim to estimate the level set when the function is not directly accessible but is acquired through compressive measurements, which may additionally be corrupted by additive noise. Moreover, our main goal is to estimate the level set directly without intermediate function reconstruction. This is a problem of interest and has an important role in medical imaging, astronomy, and remote sensing. The procedure that we use for estimating the level set in such a scenario entails a recursive partitioning of the signal domain [0,1](d) in a computationally tractable manner and a search for an 'optimal' partition. The notion of optimality of the partition combines a computable loss function and a measure of level set regularity in terms of Hpast coefficients. These Hpast coefficients have the prime advantage of being inherently effective only on the boundary of the level set because their decay is rapid both within and outside the boundary. Besides presenting the explicit performance bounds for our algorithm, we also demonstrate its efficacy via several numerical experiments.
Keywords:
shearlet transform
compressive measurements
level set estimation
Journal
I
IF:
2.1
Papers:
97
Citations:
8.4K