SPOT: Sliced Partial Optimal Transport

International audienceOptimal transport research has surged in the last decade with wide applications in computer graphics. In most cases, however, it has focused on the special case of the so-called ``balanced'' optimal transport problem, that is, the problem of optimally matching positive measures of equal total mass. While this approach is suitable for handling probability distributions as their total mass is always equal to one, it precludes other applications manipulating disparate measures.Our paper proposes a fast approach to the optimal transport of constant distributions supported on point sets of different cardinality via one-dimensional slices. This leads to one-dimensional partial assignment problems akin to alignment problems encountered in genomics or text comparison. Contrary to one-dimensional balanced optimal transport that leads to a trivial linear-time algorithm, such partial optimal transport, even in 1-d, has not seen any closed-form solution nor very efficient algorithms to date.We provide the first efficient 1-d partial optimal transport solver. Along with a quasilinear time problem decomposition algorithm, it solves 1-d assignment problems consisting of up to millions of Dirac distributions within fractions of a second in parallel.We handle higher dimensional problems via a slicing approach, and further extend the popular iterative closest point algorithm using optimal transport -- an algorithm we call Fast Iterative Sliced Transport. We illustrate our method on computer graphics applications such a color transfer and point cloud registration

Efficient Decomposition of Image and Mesh Graphs by Lifted Multicuts

Formulations of the Image Decomposition Problem as a Multicut Problem (MP) w.r.t. a superpixel graph have received considerable attention. In contrast, instances of the MP w.r.t. a pixel grid graph have received little attention, firstly, because the MP is NP-hard and instances w.r.t. a pixel grid graph are hard to solve in practice, and, secondly, due to the lack of long-range terms in the objective function of the MP. We propose a generalization of the MP with long-range terms (LMP). We design and implement two efficient algorithms (primal feasible heuristics) for the MP and LMP which allow us to study instances of both problems w.r.t. the pixel grid graphs of the images in the BSDS-500 benchmark. The decompositions we obtain do not differ significantly from the state of the art, suggesting that the LMP is a competitive formulation of the Image Decomposition Problem. To demonstrate the generality of the LMP, we apply it also to the Mesh Decomposition Problem posed by the Princeton benchmark, obtaining state-of-the-art decompositions

Example-based video color grading

In most professional cinema productions, the color palette of the movie is painstakingly adjusted by a team of skilled colorists -- through a process referred to as color grading -- to achieve a certain visual look. The time and expertise required to grade a video makes it difficult for amateurs to manipulate the colors of their own video clips. In this work, we present a method that allows a user to transfer the color palette of a model video clip to their own video sequence. We estimate a per-frame color transform that maps the color distributions in the input video sequence to that of the model video clip. Applying this transformation naively leads to artifacts such as bleeding and flickering. Instead, we propose a novel differential-geometry-based scheme that interpolates these transformations in a manner that minimizes their curvature, similarly to curvature flows. In addition, we automatically determine a set of keyframes that best represent this interpolated transformation curve, and can be used subsequently, to manually refine the color grade. We show how our method can successfully transfer color palettes between videos for a range of visual styles and a number of input video clips.Engineering and Applied Science

Sliced Wasserstein Barycenter of Multiple Densities

The optimal mass transportation problem provides a framework for interpolating between different probability density functions (pdfs), warping one function toward another. Interpolating between two arbitrary pdfs can be challenging, but interpolating between more than two pdfs just remains untractable. We propose an approximation of such interpolations based on 1D projections, that is efficiently solved via Radon transforms. We observe the expected translational behavior of this interpolation on smooth 2D functions, and prove that it corresponds to the exact interpolant in a few particular cases.Engineering and Applied Science

Ground Metric Learning on Graphs

Optimal transport (OT) distances between probability distributions are parameterized by the ground metric they use between observations. Their relevance for real-life applications strongly hinges on whether that ground metric parameter is suitably chosen. Selecting it adaptively and algorithmically from prior knowledge, the so-called ground metric learning GML) problem, has therefore appeared in various settings. We consider it in this paper when the learned metric is constrained to be a geodesic distance on a graph that supports the measures of interest. This imposes a rich structure for candidate metrics, but also enables far more efficient learning procedures when compared to a direct optimization over the space of all metric matrices. We use this setting to tackle an inverse problem stemming from the observation of a density evolving with time: we seek a graph ground metric such that the OT interpolation between the starting and ending densities that result from that ground metric agrees with the observed evolution. This OT dynamic framework is relevant to model natural phenomena exhibiting displacements of mass, such as for instance the evolution of the color palette induced by the modification of lighting and materials.Comment: Fixed sign of gradien

Relighting Photographs of Tree Canopies

International audienceWe present an image-based approach to relighting photographs of tree canopies. Our goal is to minimize capture overhead; thus the only input required is a set of photographs of the tree taken at a single time of day, while allowing relighting at any other time. We first analyze lighting in a tree canopy both theoretically and using simulations. From this analysis, we observe that tree canopy lighting is similar to volumetric illumination. We assume a single-scattering volumetric lighting model for tree canopies, and diffuse leaf reflectance; we validate our assumptions with synthetic renderings. We create a volumetric representation of the tree from 10-12 images taken at a single time of day and and use a single-scattering participating media lighting model. An analytical sun and sky illumination model provides consistent representation of lighting for the captured input and unknown target times. We relight the input image by applying a ratio of the target and input time lighting representations. We compute this representation efficiently by simultaneously coding transmittance from the sky and to the eye in spherical harmonics. We validate our method by relighting images of synthetic trees and comparing to path-traced solutions. We also present results for photographs where sparse, validating with time-lapse ground truth sequences

Wasserstein Barycentric Coordinates: Histogram Regression Using Optimal Transport

International audienceThis article defines a new way to perform intuitive and geometrically faithful regressions on histogram-valued data. It leverages the theory of optimal transport, and in particular the definition of Wasserstein barycenters, to introduce for the first time the notion of barycentric coordinates for histograms. These coordinates take into account the underlying geometry of the ground space on which the histograms are defined, and are thus particularly meaningful for applications in graphics to shapes, color or material modification. Beside this abstract construction, we propose a fast numerical optimization scheme to solve this backward problem (finding the barycentric coordinates of a given histogram) with a low computational overhead with respect to the forward problem (computing the barycenter). This scheme relies on a backward algorithmic differentiation of the Sinkhorn algorithm which is used to optimize the entropic regularization of Wasserstein barycenters. We showcase an illustrative set of applications of these Wasserstein coordinates to various problems in computer graphics: shape approximation, BRDF acquisition and color editing

Consistent Video Filtering for Camera Arrays

International audienceVisual formats have advanced beyond single-view images and videos: 3D movies are commonplace, researchers have developed multi-view navigation systems, and VR is helping to push light field cameras to mass market. However, editing tools for these media are still nascent, and even simple filtering operations like color correction or stylization are problematic: naively applying image filters per frame or per view rarely produces satisfying results due to time and space inconsistencies. Our method preserves and stabilizes filter effects while being agnostic to the inner working of the filter. It captures filter effects in the gradient domain, then uses \emph{input} frame gradients as a reference to impose temporal and spatial consistency. Our least-squares formulation adds minimal overhead compared to naive data processing. Further, when filter cost is high, we introduce a filter transfer strategy that reduces the number of per-frame filtering computations by an order of magnitude, with only a small reduction in visual quality. We demonstrate our algorithm on several camera array formats including stereo videos, light fields, and wide baselines

Proxy-Guided Texture Synthesis for Rendering Natural Scenes

International audienceLandscapes and other natural scenes are easy to photograph but difficult to model and render. We present a proxy-guided pipeline which allows for simple 3D proxy geometry to be rendered with the rich visual detail found in a suitably pre-annotated example image. This greatly simplifies the geometric modeling and texture mapping of such scenes. Our method renders at near-interactive rates and is designed by carefully adapting guidancebased texture synthesis to our goals. A guidance-map synthesis step is used to obtain silhouettes and borders that have the same rich detail as the source photo, using a Chamfer distance metric as a principled way of dealing with discrete texture labels. We adapt an efficient parallel approach to the challenging guided synthesis step we require, providing a fast and scalable solution. We provide a solution for local temporal coherence, by introducing a reprojection algorithm, which reuses earlier synthesis results when feasible, as measured by a distortion metric. Our method allows for the consistent integration of standard CG elements with the texture-synthesized elements. We demonstrate near-interactive camera motion and landscape editing on a number of examples

CGIntrinsics: Better Intrinsic Image Decomposition through Physically-Based Rendering

Intrinsic image decomposition is a challenging, long-standing computer vision problem for which ground truth data is very difficult to acquire. We explore the use of synthetic data for training CNN-based intrinsic image decomposition models, then applying these learned models to real-world images. To that end, we present \ICG, a new, large-scale dataset of physically-based rendered images of scenes with full ground truth decompositions. The rendering process we use is carefully designed to yield high-quality, realistic images, which we find to be crucial for this problem domain. We also propose a new end-to-end training method that learns better decompositions by leveraging \ICG, and optionally IIW and SAW, two recent datasets of sparse annotations on real-world images. Surprisingly, we find that a decomposition network trained solely on our synthetic data outperforms the state-of-the-art on both IIW and SAW, and performance improves even further when IIW and SAW data is added during training. Our work demonstrates the suprising effectiveness of carefully-rendered synthetic data for the intrinsic images task.Comment: Paper for 'CGIntrinsics: Better Intrinsic Image Decomposition through Physically-Based Rendering' published in ECCV, 201