Artiom Kovnatsky

Coupled quasi‐harmonic bases

The use of Laplacian eigenbases has been shown to be fruitful in many computer graphics applications. Today, state‐of‐the‐art approaches to shape analysis, synthesis, and correspondence rely on these natural harmonic bases that allow using classical tools from harmonic analysis on manifolds. However, many applications involving multiple shapes are obstacled by the fact that Laplacian eigenbases computed independently on different shapes are often incompatible with each other. In this paper, we propose the construction of common approximate eigenbases for multiple shapes using approximate joint diagonalization algorithms, taking as input a set of corresponding functions (e.g. indicator functions of stable regions) on the two shapes. We illustrate the benefits of the proposed approach on tasks from shape editing, pose transfer, correspondence, and similarity.

Functional correspondence by matrix completion

In this paper, we consider the problem of finding dense intrinsic correspondence between manifolds using the recently introduced functional framework. We pose the functional correspondence problem as matrix completion with manifold geometric structure and inducing functional localization with the L1 norm. We discuss efficient numerical procedures for the solution of our problem. Our method compares favorably to the accuracy of state-of-the-art correspondence algorithms on non-rigid shape matching benchmarks, and is especially advantageous in settings when only scarce data is available.

Photometric heat kernel signatures

In this paper, we explore the use of the diffusion geometry framework for the fusion of geometric and photometric information in local heat kernel signature shape descriptors. Our construction is based on the definition of a diffusion process on the shape manifold embedded into a high-dimensional space where the embedding coordinates represent the photometric information. Experimental results show that such data fusion is useful in coping with different challenges of shape analysis where pure geometric and pure photometric methods fail.

MADMM: A Generic Algorithm for Non-smooth Optimization on Manifolds

Numerous problems in computer vision, pattern recognition, and machine learning are formulated as optimization with manifold constraints. In this paper, we propose the Manifold Alternating Directions Method of Multipliers (MADMM), an extension of the classical ADMM scheme for manifold-constrained non-smooth optimization problems. To our knowledge, MADMM is the first generic non-smooth manifold optimization method. We showcase our method on several challenging problems in dimensionality reduction, non-rigid correspondence, multi-modal clustering, and multidimensional scaling.

Multimodal Manifold Analysis by Simultaneous Diagonalization of Laplacians

We construct an extension of spectral and diffusion geometry to multiple modalities through simultaneous diagonalization of Laplacian matrices. This naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real examples of manifold learning, object classification, and clustering, showing that the joint spectral geometry better captures the inherent structure of multi-modal data. We also show the relation of many previous approaches for multimodal manifold analysis to our framework.

Laplacian colormaps: a framework for structure-preserving color transformations

Mappings between color spaces are ubiquitous in image processing problems such as gamut mapping, decolorization, and image optimization for color‐blind people. Simple color transformations often result in information loss and ambiguities, and one wishes to find an image‐specific transformation that would preserve as much as possible the structure of the original image in the target color space. In this paper, we propose Laplacian colormaps, a generic framework for structure‐preserving color transformations between images. We use the image Laplacian to capture the structural information, and show that if the color transformation between two images preserves the structure, the respective Laplacians have similar eigenvectors, or in other words, are approximately jointly diagonalizable. Employing the relation between joint diagonalizability and commutativity of matrices, we use Laplacians commutativity as a criterion of color mapping quality and minimize it w.r.t. the parameters of a color transformation to achieve optimal structure preservation. We show numerous applications of our approach, including color‐to‐gray conversion, gamut mapping, multispectral image fusion, and image optimization for color deficient viewers.

Affine-invariant photometric heat kernel signatures

In this paper, we explore the use of the diffusion geometry framework for the fusion of geometric and photometric information in local shape descriptors. Our construction is based on the definition of a modified metric, which combines geometric and photometric information, and then the diffusion process on the shape manifold is simulated. Experimental results show that such data fusion is useful in coping with shape retrieval experiments, where pure geometric and pure photometric methods fail. Apart from retrieval task the proposed diffusion process may be employed in other applications.

Stable spectral mesh filtering

The rapid development of 3D acquisition technology has brought with itself the need to perform standard signal processing operations such as filters on 3D data. It has been shown that the eigenfunctions of the Laplace-Beltrami operator (manifold harmonics) of a surface play the role of the Fourier basis in the Euclidean space; it is thus possible to formulate signal analysis and synthesis in the manifold harmonics basis. In particular, geometry filtering can be carried out in the manifold harmonics domain by decomposing the embedding coordinates of the shape in this basis. However, since the basis functions depend on the shape itself, such filtering is valid only for weak (near all-pass) filters, and produces severe artifacts otherwise. In this paper, we analyze this problem and propose the fractional filtering approach, wherein we apply iteratively weak fractional powers of the filter, followed by the update of the basis functions. Experimental results show that such a process produces more plausible and meaningful results.

Gamut mapping with image Laplacian commutators

In this paper, we present a gamut mapping algorithm that is based on spectral properties of image Laplacians as image structure descriptors. Using the fact that structurally similar images have similar Laplacian eigenvectors and employing the relation between joint diagonalizability and commutativity of matrices, we minimize the Laplacians commutator w.r.t. the parameters of a color transformation to achieve optimal structure preservation while complying with the target gamut. Our method is computationally efficient, favorably compares to state-of-the-art approaches in terms of quality, allows mapping to devices with any number of primaries, and supports gamma correction, accounting for brightness response of computer displays.