Image Labeling by Assignment

Åström, Freddie; Petra, Stefania; Schmitzer, Bernhard; Schnörr, Christoph

doi:10.1007/s10851-016-0702-4

Cited by 55 publications

(120 citation statements)

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“…A smooth dynamical system for supervised image labeling was proposed by [1]. Given a manifold-valued input image F , the distances between the data F i at each pixel i ∈ [n] and prototypes (labels) from a pre-specified codebook {G 1 , .…”

Section: Motivation and Preliminariesmentioning

confidence: 99%

“…The first component V 1 (W, G) determines the spatially regularized label assignments W (t) by means of the assignment flow that, more generally than the original formulation of [1], involves a time-varying distance matrix depending on the evolving prototypes G(t). The first component V 1 (W, G) determines the spatially regularized label assignments W (t) by means of the assignment flow that, more generally than the original formulation of [1], involves a time-varying distance matrix depending on the evolving prototypes G(t).…”

Section: Coupled Flow For Label Learning and Assignmentmentioning

confidence: 99%

“…A smooth dynamical system for supervised image labeling was proposed by [1]. In particular, we detail a numerical update scheme for the evolution of prototypes on the Riemannian manifold of symmetric positive definite matrices.…”

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confidence: 99%

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Spatially Regularized Geometric Assignment for Unsupervised Label Learning on Manifolds

Zern

Zisler

Petra

et al. 2019

Proc Appl Math and Mech

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We introduce a smooth coupled system of Riemannian flows for simultaneously learning a dictionary of manifold-valued prototypes and assigning these prototypes to the image data in a spatially coherent way. In particular, we detail a numerical update scheme for the evolution of prototypes on the Riemannian manifold of symmetric positive definite matrices. The scheme is applied to covariance descriptors are extracted from image data in order to demonstrate our approach. A smooth dynamical system for supervised image labeling was proposed by [1]. Given a manifold-valued input image F , the distances between the data F i at each pixel i ∈ [n] and prototypes (labels) from a pre-specified codebook {G 1 , . . . , G c } are measured and collected into the distance matrix. Geometrically averaging and evolving local assignments based on these distances enforces spatial regularization and in turn defines the assignment flow. The flow evolves on the assignment manifold, which is the set of row-stochastic matrices with full support denoted by W := {W ∈ R n×c : W ij > 0, W 1 c = 1 n } and endowed with the Fisher-Rao (information) metric. Sparse numerical updates based on the affine e-connection of information geometry converge to (unique) assignments of a label to each pixel i in terms of the probability vector W i (t), for sufficiently large t.In [2], the assignment flow was extended to unsupervised image labeling in order to determine proper features as labels from data, which enable to classify pixelwise image data into different categories using the assignment flow while taking into account spatial context. The approach couples two geometric flows, the assignment flow and label evolution on a feature manifold, in order to adapt labels to representative feature values during the spatially regularized assignment process. Since the inference of the assignments is separately represented by W (t) ∈ W, the approach applies to arbitrary manifold-valued data as long as the corresponding Riemannian means are well-defined and computationally feasible. In this work, we consider as a case study label evolution on the matrix manifold of symmetric positive definite matrices P s := {X ∈ R s×s : X = X, X 0} together with the affine invariant Riemannian metric [3]. Scenarios relying on features from P s include classifications of texture or faces in images, and action recognition in videos, using covariance descriptors X ∈ P s .We sketch the approach [2] in Section 2 and provide an illustrative numerical example in Section 3. In addition, we refer the reader to recent follow-up work [4], where the present approach has been generalized in order to learn from scratch representative labels from given raw data, without any supervision. This work is supplemented by a linearized flow and a variational formulation in the supervised case, see references in this volume [5]. Coupled Flow for Label Learning and AssignmentFollowing [2], the coupling of the assignment flow and label evolution is given bywhere the defining vector field V has two compone...

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Section: Motivation and Preliminariesmentioning

confidence: 99%

Section: Coupled Flow For Label Learning and Assignmentmentioning

confidence: 99%

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Spatially Regularized Geometric Assignment for Unsupervised Label Learning on Manifolds

Zern

Zisler

Petra

et al. 2019

Proc Appl Math and Mech

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“…The replicator operator R W and the similarity matrix S(W ), both introduced in [2], define the assignment flow aṡ Endowed with the Fisher-Rao (information) metric on the tangent space T 0 , it becomes a Riemannian manifold.…”

Section: Motivation and Preliminariesmentioning

confidence: 99%

Exponential Integration of the Linear Assignment Flow

Zeilmann

Savarino

Petra

et al. 2019

Proc Appl Math and Mech

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We introduce the linear assignment flow as an approximation of the full nonlinear assignment flow, which is a method for contextual data labeling on arbitrary graphs. The linear assignment flow is a dynamical system evolving on the tangent space of a statistical manifold. It is numerically determined using exponential integrators and Krylov subspace approximation, for which we provide error estimates. The approximation property of the linear assignment flow is illustrated by a numerical experiment. This work is supplemented by two papers on variational modeling and unsupervised labeling [1].

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“…Recent applications include astronomy [9, 18, 19], biomedical sciences [3, 25–27, 77, 81, 82, 88, 89], colour transfer [14, 17, 49, 62, 63], computer vision and graphics [7, 44, 60, 65, 68, 74, 75], imaging [36, 40, 64], information theory [78], machine learning [1, 15, 20, 34, 37, 48, 76], operational research [69] and signal processing [54, 58]. …”

Section: Introductionmentioning

confidence: 99%

A Transportation $$L^p$$ Distance for Signal Analysis

et al. 2017

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Transport based distances, such as the Wasserstein distance and earth mover'sdistance, have been shown to be an effective tool in signal and image analysis. The success of transport based distances is in part due to their Lagrangian nature which allows it to capture the important variations in many signal classes. However these distances require the signal to be nonnegative and normalized. Furthermore, the signals are considered as measures and compared by redistributing (transporting) them, which does not directly take into account the signal intensity. Here we study a transport-based distance, called the TLp distance, that combines Lagrangian and intensity modelling and is directly applicable to general, non-positive and multi-channelled signals. The distance can be computed by existing numerical methods. We give an overview of the basic properties of this distance and applications to classification, with multi-channelled non-positive one-dimensional signals and two-dimensional images, and color transfer.

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Image Labeling by Assignment

Cited by 55 publications

References 50 publications

Spatially Regularized Geometric Assignment for Unsupervised Label Learning on Manifolds

Spatially Regularized Geometric Assignment for Unsupervised Label Learning on Manifolds

Exponential Integration of the Linear Assignment Flow

A Transportation $$L^p$$ Distance for Signal Analysis

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