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Recently convex optimization models were successfully applied for solving various problems in image analysis and restoration. In this paper, we are interested in relations between convex constrained optimization problems of the form argmin{Φ(x) subject to Ψ(x) ≤ τ } and their penalized counterparts argmin{Φ(x) + λΨ(x)}. We recall general results on the topic by the help of an epigraphical projection. Then we deal with the special setting Ψ := L · with L ∈ R m,n and Φ := ϕ(H·), where H ∈ R n,n and ϕ : R n → R ∪ {+∞} meet certain requirements which are often fulfilled in image processing models. In this case we prove by incorporating the dual problems that there exists a bijective function such that the solutions of the constrained problem coincide with those of the penalized problem if and only if τ and λ are in the graph of this function. We illustrate the relation between τ and λ for various problems arising in image processing. In particular, we point out the relation to the Pareto frontier for joint sparsity problems. We demonstrate the performance of the constrained model in restoration tasks of images corrupted by Poisson noise with the I-divergence as data fitting term ϕ and in inpainting models with the constrained nuclear norm. Such models can be useful if we have a priori knowledge on the image rather than on the noise level.

Abstract. This paper considers supervised multi-class image segmentation: from a labeled set of pixels in one image, we learn the segmentation and apply it to the rest of the image or to other similar images. We study approaches with p-Laplacians, (vector-valued) Reproducing Kernel Hilbert Spaces (RKHSs) and combinations of both. In all approaches we construct segment membership vectors. In the p-Laplacian model the segment membership vectors have to fulfill a certain probability simplex constraint. Interestingly, we could prove that this is not really a constraint in the case p = 2 but is automatically fulfilled. While the 2-Laplacian model gives a good general segmentation, the case of the 1-Laplacian tends to neglect smaller segments. The RKHS approach has the benefit of fast computation. This direction is motivated by image colorization, where a given dab of color is extended to a nearby region of similar features or to another image. The connection between colorization and multi-class segmentation is explored in this paper with an application to medical image segmentation. We further consider an improvement using a combined method. Each model is carefully considered with numerical experiments for validation, followed by medical image segmentation at the end.

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