Fast projection onto the simplex and the $$\pmb {l}_\mathbf {1}$$ l 1 ball

Condat, Laurent

doi:10.1007/s10107-015-0946-6

Cited by 322 publications

(300 citation statements)

References 20 publications

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“…Again, the proximal operator for the simplex projection is computationally efficient, see [25]. The complete optimization scheme is given in Algorithm 1.…”

Section: Optimization Schemementioning

confidence: 99%

Hyperspectral Super-Resolution with Spectral Unmixing Constraints

2017

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Hyperspectral sensors capture a portion of the visible and near-infrared spectrum with many narrow spectral bands. This makes it possible to better discriminate objects based on their reflectance spectra and to derive more detailed object properties. For technical reasons, the high spectral resolution comes at the cost of lower spatial resolution. To mitigate that problem, one may combine such images with conventional multispectral images of higher spatial, but lower spectral resolution. The process of fusing the two types of imagery into a product with both high spatial and spectral resolution is called hyperspectral super-resolution. We propose a method that performs hyperspectral super-resolution by jointly unmixing the two input images into pure reflectance spectra of the observed materials, along with the associated mixing coefficients. Joint super-resolution and unmixing is solved by a coupled matrix factorization, taking into account several useful physical constraints. The formulation also includes adaptive spatial regularization to exploit local geometric information from the multispectral image. Moreover, we estimate the relative spatial and spectral responses of the two sensors from the data. That information is required for the super-resolution, but often at most approximately known for real-world images. In experiments with five public datasets, we show that the proposed approach delivers up to 15% improved hyperspectral super-resolution.

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“…Again, the proximal operator for the simplex projection is computationally efficient, see [25]. The complete optimization scheme is given in Algorithm 1.…”

Section: Optimization Schemementioning

confidence: 99%

Hyperspectral Super-Resolution with Spectral Unmixing Constraints

2017

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“…For this simplex projection, a recent, computationally efficient algorithm is available (Condat, 2014). We set γ1 = γ2 = 1.01 for the experiments.…”

Section: Proposed Solutionmentioning

confidence: 99%

“…Thus, we treat LMM as a coupled, constrained matrix factorization of the two input images into endmembers and their abundances. On the technical level, we employ efficient state-of-the-art optimization algorithms to tackle the resulting constrained least-squares problem (Bolte et al, 2014, Condat, 2014. Experimental results on several different image pairs show a consistent improvement compared to several other fusion methods.…”

Section: Introductionmentioning

confidence: 99%

Advances in Hyperspectral and Multispectral Image Fusion and Spectral Unmixing

Lanaras

Baltsavias

Schindler

2015

Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci.

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ABSTRACT:In this work, we jointly process high spectral and high geometric resolution images and exploit their synergies to (a) generate a fused image of high spectral and geometric resolution; and (b) improve (linear) spectral unmixing of hyperspectral endmembers at subpixel level w.r.t. the pixel size of the hyperspectral image. We assume that the two images are radiometrically corrected and geometrically co-registered. The scientific contributions of this work are (a) a simultaneous approach to image fusion and hyperspectral unmixing, (b) enforcing several physically plausible constraints during unmixing that are all well-known, but typically not used in combination, and (c) the use of efficient, state-of-the-art mathematical optimization tools to implement the processing. The results of our joint fusion and unmixing has the potential to enable more accurate and detailed semantic interpretation of objects and their properties in hyperspectral and multispectral images, with applications in environmental mapping, monitoring and change detection. In our experiments, the proposed method always improves the fusion compared to competing methods, reducing RMSE between 4% and 53%.

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“…So, we consider a convex relaxation, obtained by replacing A by its convex hull, which is the simplex ∆, i.e. the set of vectors with nonnegative elements whose sum is 1 [23]. Let us introduce the ball B = {s ∈ R M×Ω : s 1,∞ ≤ K}, and the convex indicator function ı E of a convex set E, which takes the value 0 if its variable belongs to E and +∞ else.…”

Section: Convex Relaxation Of the Problemmentioning

confidence: 99%

A Convex Approach to K-Means Clustering and Image Segmentation

Condat

2018

Lecture Notes in Computer Science

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Abstract.A new convex formulation of data clustering and image segmentation is proposed, with fixed number K of regions and possible penalization of the region perimeters. So, this problem is a spatially regularized version of the K-means problem, a.k.a. piecewise constant Mumford-Shah problem. The proposed approach relies on a discretization of the search space; that is, a finite number of candidates must be specified, from which the K centroids are determined. After reformulation as an assignment problem, a convex relaxation is proposed, which involves a kind of l1,∞ norm ball. A splitting of it is proposed, so as to avoid the costly projection onto this set. Some examples illustrate the efficiency of the approach.

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Fast projection onto the simplex and the $$\pmb {l}_\mathbf {1}$$ l 1 ball

Cited by 322 publications

References 20 publications

Hyperspectral Super-Resolution with Spectral Unmixing Constraints

Hyperspectral Super-Resolution with Spectral Unmixing Constraints

Advances in Hyperspectral and Multispectral Image Fusion and Spectral Unmixing

A Convex Approach to K-Means Clustering and Image Segmentation

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