Please cite the publication as :
Getreuer,
P.,
"RudinOsherFatemi Total Variation Denoising using Split Bregman",
Image Processing On Line
, 2012.
Please cite the companion website as :
Getreuer, P., "RudinOsherFatemi Total Variation Denoising using Split Bregman", RunMyCode companion website, http://www.execandshare.org/CompanionSite/Site148
Variable/Parameters  Description, constraint  Comments 

NoisyImage  The input noisy image.  
NoiseModel  The noise model: Gaussian, Laplace, or Poisson  
NoiseSigma  Noise standard deviation 
Variable/Parameters  Description  Visualisation 

NoisyImage  
NoiseModel  
NoiseSigma 
Computing Date  Status  Actions 

Pascal Getreuer
Yale University, Math Department
United States
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Consider an underdetermined system of linear equations y = Ax with known d*n matrix
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A there is a threshold phenomenon: if the sparsest solution is sufficiently sparse, it can be
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completely equivalent to the statement that, whenever y = Ax has a nonnegative solution
with at most k nonzeros, it is the nonnegative solution to y = Ax having minimal sum.
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First, because many [d/2]neighborly polytopes are known, there are many systems
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optimization  provided the answer has fewer nonzeros than half the number of equations.
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imply that, if A is a typical uniformlydistributed random orthoprojector with n = 2d
and n large, the sparsest nonnegative solution to y = Ax can be found by linear programming
provided it has fewer nonzeros than 1/8 the number of equations.
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x with k nonzeros are uniquely determined by y = Ax. As a corollary of recent work
counting faces of random simplices, it is known that most polytopes P generated by large n
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Donoho,
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A common approach in model reduction is balanced truncation, which is based on gramian matrices classifiying certain attributes of states or parameters of a given dynamic system.
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An accepted model reduction technique is balanced truncation, by which negligible states of a linear system of ODEs are determined by balancing the systems controllability and observability gramian matrices. To be applicable for nonlinear system this method was enhanced through the empirical gramians, while the cross gramian conjoined both gramians into one gramian matrix. This work introduces the empirical cross gramian for square MultipleInputMultipleOutput systems as well as the (empirical) joint gramian. Based on the cross gramian, the joint gramian determines, in addition to the Hankel singular values, the parameter identifiability allowing a combined model reduction, concurrently reducing state and parameter spaces. Furthermore, a controllability and an observability based combined reduction method are presented and the usage of empirical gramians is extended to parameter reduction in (Bayesian) inverse problems. All methods presented are evaluated by numerical experiments.
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A common approach in model reduction is balanced truncation, which is based on gramian matrices classifiying certain attributes of states or parameters of a given dynamic system.
Initially restricted to linear systems, the empirical gramians not only extended this concept to nonlinear systems, but also provide a uniform computational method.
This work introduces a unified software framework supplying routines for six types of empirical gramians.
The gramian types will be discussed and applied in a model reduction framework for multipleinputmultipleoutput (MIMO) systems.
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We consider the problem of finding optimally stable polynomial approximations to the exponential for application to onestep integration of initial value ordinary and partial differential equations. The objective is to find the largest stable step size and corresponding method for a given problem when the spectrum of the initial value problem is known. The problem is expressed in terms of a general least deviation feasibility problem. Its solution is obtained by a new fast, accurate, and robust algorithm based on convex optimization techniques. Global convergence of the algorithm is proven in the case that the order of approximation is one and in the case that the spectrum encloses a starlike region. Examples demonstrate the effectiveness of the proposed algorithm even when these conditions are not satisfied.
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Abstract
In compressed sensing, one takes n < N samples of an N dimensional vector x0 using an n × N matrix A, obtaining undersampled measurements y = Ax0 . For random matrices with Gaussian i.i.d entries, it is known that, when x0 is ksparse, there is a precisely determined phase transition: for a certain region in the (k/n, n/N )phase diagram, convex optimization min x_1 subject to y = Ax, x ∈ X^N typically ﬁnds the sparsest solution, while outside that region, it typically fails. It has been shown empirically that the same property – with the same phase transition location – holds for a wide range of nonGaussian random matrix ensembles. We consider speciﬁc deterministic matrices including Spikes and Sines, Spikes and Noiselets, Paley Frames, DelsarteGoethals Frames, Chirp Sensing Matrices, and Grassmannian Frames. Extensive experiments show that for a typical ksparse object, convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian matrices. In our experiments, we considered coefﬁcients constrained to X^N for four different sets X ∈ {[0, 1], R_+ , R, C}. We establish this ﬁnding for each of the associated four phase transitions.
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