Eurographics Symposium on Geometry Processing 2018 Volume 37 (2018), Number 5 T. Ju and A. Vaxman (Guest Editors) Topological Function Optimization for Continuous Shape Matching Adrien Poulenard1, Primoz Skraba2, Maks Ovsjanikov1 1LIX, Ecole Polytechnique, CNRS, 2Jozef Stefan Institute Abstract We present a novel approach for optimizing real-valued functions based on a wide range of topological criteria. In particular, we show how to modify a given function in order to remove topological noise and to exhibit prescribed topological features. Our method is based on using the previously-proposed persistence diagrams associated with real-valued functions, and on the analysis of the derivatives of these diagrams with respect to changes in the function values. This analysis allows us to use continuous optimization techniques to modify a given function, while optimizing an energy based purely on the values in the persistence diagrams. We also present a procedure for aligning persistence diagrams of functions on different domains, without requiring a mapping between them. Finally, we demonstrate the utility of these constructions in the context of the functional map framework, by first giving a characterization of functional maps that are associated with continuous point-to- point correspondences, directly in the functional domain, and then by presenting an optimization scheme that helps to promote the continuity of functional maps, when expressed in the reduced basis, without imposing any restrictions on metric distortion. We demonstrate that our approach is efficient and can lead to improvement in the accuracy of maps computed in practice. 1. Introduction preservation of the topological structure of the functions before and after the mapping. A core problem in geometry processing consists in quantify- ing similarity between shapes and their parts, as well as detect- In this paper, we show how such problems can be solved by ef- ing detailed region or point-based correspondences [VKZHCO11, ficiently optimizing the topological structure of real-valued func- TCL∗13, BCBB16]. A common approach for both shape compari- tions defined on the shapes, either independently (to promote cer- son and correspondence consists in computing real-valued (for ex- tain structural properties), or jointly (to enforce similarity between ample, descriptor) functions defined on the shapes and comparing such properties), without resorting to combinatorial search or point- the shapes and their parts by comparing the values of such func- to-point maps. The key to our approach is the manipulation of per- tions. This includes both computing correspondences by matching sistence diagrams [CZCG05, Car09]. These diagrams have been in descriptor space, and also, more recently, by computing linear shown to summarize the properties of very general classes of topo- transformations between spaces of real-valued functions using the logical spaces, including, most relevant to us, real-valued func- so-called functional map framework [OBCS∗12, OCB∗17]. tions defined on the surfaces, and also enjoy several key prop- erties such as being stable under a broad range of perturbations Many existing techniques for comparison of functions on the [CSEH07, CCSG∗09]. Existing methods, however, concentrate on shapes directly rely on comparing function values, without analyz- either efficiently constructing persistence diagrams from a given ing the global structure of the functions involved. For example, a signal [CZCG05, MMS11, CK13] or using them as a tool for, e.g. descriptor function computed on one shape can have several promi- shape or image comparison [CZCG05,LOC14] or shape segmenta- nent maxima, whereas on another shape, it can be uniform or with tion [SOCG10] among many others. low variance. Intuitively, pairs of functions with dissimilar struc- tural properties can lead to large errors in the correspondence com- Our main insight is that it is possible to formulate optimization putation. This problem is especially prominent in the context of objectives on the persistence diagrams of real-valued functions, re- functional maps which are linear transformations between spaces gardless of their underlying spatial domain, and to optimize a given of real-valued functions defined on different shapes. In this case, function to improve such objectives, via continuous non-linear op- one is often interested in formulating an objective which would timization. For this, we first show, how the derivative of a persis- promote mapping indicator functions of connected regions to other tence diagram of a function can be computed with respect to the such indicator functions without knowing in advance which regions change in the function values, and then how this computation can should match. At a high level, such an objective should promote the be used to efficiently optimize various energies defined on persis- c 2018 The Author(s) Computer Graphics Forum c 2018 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. A. Poulenard, P. Skraba, M. Ovsjanikov / Topological Function Optimization tence diagrams. Crucially, these computations can be performed in dences and then proposing a practical optimization scheme that any functional basis, which can significantly improve stability and helps to improve the accuracy of functional map computations. computation speed. Moreover, our approach allows us to jointly The functional map representation and the associated correspo- optimize the persistence diagrams of multiple functions, without dence computation pipeline was first introduced in [OBCS∗12] assuming that they are defined over the same domain. and has since then been extended significantly in, e.g., [KBBV15, HO17, RCB 17, EBC17] among many others (see [OCB 17] for We apply these insights to first provide a characterization of ∗ ∗ an overview). The key practical advantage of this representa- functional maps associated with continuous point-to-point maps, tion is that it allows to encode correspondences between shapes based on the preservation of persistence diagrams. Unlike previ- as small-sized matrices that represent linear transformations be- ous results, this characterization does not assume that the map is tween function spaces in some reduced basis. Moreover, comput- isometric or area-preserving and holds for any continuous point- ing functional maps can be done efficiently by leveraging tools to-point map. We then propose an optimization scheme that helps from numerical linear algebra and manifold optimization, as shown to promote continuity of functional maps, even when they are ex- in, e.g., [KBBV15, LRBB17, LRB 16]. One challenge with this pressed in a reduced basis. ∗ approach, however, is that the space of linear functional trans- formations is much larger than that of point-to-point correspon- 2. Related Work dences, which means that in many cases regularization is neces- Persistent (Co)homology Topology is the study of connectivity sary to compute accurate maps. This has prompted work on char- and continuity, and persistent (co)homology is a natural language acterizing how various properties of pointwise maps are mani- for describing it in an applied setting. Persistence has been widely fested in the functional domain. For example, the original work studied [EH10] and has become a central tool for the rapidly devel- showed that area-preserving correspondences lead to orthonormal oping area of topological data analysis. We provide an overview functional maps [OBCS∗12] (Theorem 5.1). Other characteriza- of persistence in Section4. Most relevant to our work is the tions have been shown for conformal maps [ROA∗13, HO17], numerous applications it has found in geometry processing e.g. isometries [OBCS∗12, KBBV15] and for partial correspondences [CZCG05, DLL∗10, SOCG10]. [RCB∗17, LRBB17]. More recently, some works have exploited the relation of point-to-point maps to functional maps that pre- In this paper, we are interested in optimizing functions to achieve serve pointwise products of functions [NO17,NMR∗18]. One large certain prescribed topological criteria. Using persistence for mod- missing piece, however, is to characterize continuous point-to-point ifying functions has been studied as topological simplification, maps purely in the functional domain, and without making assump- which also served as one of the motivations of the original work tions on the metric preservation. In this work, we fill this gap by on persistence [ELZ00]. The simplification problem has been pri- precisely characterizing functional maps that arise from continu- marily stated as a denoising problem [AGH∗09, BLW12], chang- ous point-to-point maps and propose an optimization scheme that ing the function so that “persistent features" are preserved. Un- allows to promote this continuity. like these works, we approach this problem via continuous opti- mization, which allows us to explicitly modify the function, ex- We also note that another commonly used relaxation for match- pressed in an arbitrary basis to optimize topological criteria. Our ing problems is based on the formalism of optimal transport, which approach is inspired by Morse theory, which is deeply tied to per- has recently been used for finding bijective and continous corre- sistence [EHZ01, CCL03]. spondences [SPKS16,MCSK∗17,VLB∗17]. These techniques have benefited from the computational advances in solving large-scale Most related to our work is [GHO16]. The main application of transport problems, especially using the Sinkhorn method