Shape deformation for morphing or editing purposes is a central challenge in Computer Graphics. While numerous methods exist, few allow for the explicit evaluation of the deformation at arbitrary times and locations directly, without resorting to intricate advection or interpolation schemes. In this paper, we propose a method that provides an explicit expression of the deformation parameterized as a flow, for continuously deforming shapes defined implicitly. Implicit surfaces are indeed particularly well suited for deformation tasks, since they inherently account for both the surface and the enclosed volume. Our approach leverages invertible neural networks to ensure theoretically that the deformation is a valid flow, while also providing differential quantities useful for geometric regularization. We demonstrate applications of this flow to shape morphing with and without landmarks, shape editing, and pairwise-to-any morphing where we compute pairwise morphings to a canonical shape allowing to deduce transformations between any pair through flow composition. The code for our method is available at https://github.com/camillebnm/explicit_flows_for_implicit_surfaces.

Abstract Shape interpolation is a long standing challenge of geometry processing. As it is ill‐posed, shape interpolation methods always work under some hypothesis such as semantic part matching or least displacement. Among such constraints, volume preservation is one of the traditional animation principles. In this paper we propose a method to interpolate between shapes in arbitrary poses favoring volume and topology preservation. To do so, we rely on a level set representation of the shape and its advection by a velocity field through the level set equation, both shape representation and velocity fields being parameterized as neural networks. While divergence free velocity fields ensure volume and topology preservation, they are incompatible with the Eikonal constraint of signed distance functions. This leads us to introduce the notion of adaptive divergence velocity field, a construction compatible with the Eikonal equation with theoretical guarantee on the shape volume preservation. In the non constant volume setting, our method is still helpful to provide a natural morphing, by combining it with a parameterization of the volume change over time. We show experimentally that our method exhibits better volume preservation than other recent approaches, limits topological changes and preserves the structures of shapes better without landmark correspondences.

We propose a deep learning approach for inpainting holes in digital models of fabric surfaces. Leveraging the developable nature of fabric surfaces, we flatten the area surrounding the holes with minor distortion and regularly sample it to obtain a discrete 2D map of the 3D embedding, with an indicator mask outlining holes locations. This enables the use of a standard 2D convolutional neural network to inpaint holes given the 3D positioning of the surface. The provided neural architecture includes an attention mechanism to capture long-range relationships on the surface. Finally, we provide ScarfFolds, a database of folded fabrics patches with varying complexity, which is used to train our convolutional network in a supervised manner. We successfully tested our approach on various examples and illustrated that previous 3D deep learning approaches suffer from several issues when applied to fabrics. Also, our method allows the users to interact with the construction of the inpainted surface. The editing is interactive and supports many tools like vertex grabbing, drape twisting or pinching.

When scanning real shapes, occlusion issues may lead to holes in the reconstructed surface which must be solved using an inpainting technique. When dealing with fabrics with folds, reconstruction gets even more challenging because these occlusion problems become almost inevitable and strong assumptions are implied on the physical model of the inpainted surface. We propose a framework to fill holes in triangle mesh surfaces representing fabrics. The method leverages the developable nature of fabrics to recover the intrinsic geometry of the missing patch in 2D. Our inpainting strategy is then based on a variational method to smoothly incorporate the patch into the surface by minimizing an isometric energy. The proposed approach allows us to produce folds and creases which are difficult to obtain with general purpose hole filling techniques. Moreover, our approach remains relevant in the case where the model is not provided by the digitization of a real fabric as for the acquisition from ancient statues with draperies.

Abstract 3D acquisition of archaeological artefacts has become an essential part of cultural heritage research for preservation or restoration purpose. Statues, in particular, have been at the center of many projects. In this paper, we introduce a way to improve the understanding of acquired statues representing real or imaginary creatures by registering a simple and pliable articulated model to the raw point set data. Our approach performs a Forward And bacKward Iterative Registration (FAKIR) which proceeds joint by joint, needing only a few iterations to converge. We are thus able to detect the pose and elementary anatomy of sculptures, with possibly non realistic body proportions. By adapting our simple skeleton, our method can work on animals and imaginary creatures.

Abstract Detail enhancement is a well‐studied area of 3D rendering and image processing, which has few equivalents for 3D shape processing. To enhance details, one needs an efficient analysis tool to express the local surface dynamics. We introduce Wavejets, a new function basis for locally decomposing a shape expressed over the local tangent plane, by considering both angular oscillations of the surface around each point and a radial polynomial. We link the Wavejets coefficients to surface derivatives and give theoretical guarantees for their precision and stability with respect to an approximate tangent plane. The coefficients can be used for shape details amplification, to enhance, invert or distort them, by operating either on the surface point positions or on the normals. From a practical point of view, we derive an efficient way of estimating Wavejets on point sets and demonstrate experimentally the amplification results with respect to noise or basis truncation.

Abstract Three‐dimensional scanners provide a virtual representation of object surfaces at some given precision that depends on many factors such as the object material, the quality of the laser ray or the resolution of the camera. This precision may even vary over the surface, depending, for example, on the distance to the scanner which results in uneven and unstructured point sets, with an uncertainty on the coordinates. To enhance the quality of the scanner output, one usually resorts to local surface interpolation between measured points. However, object surfaces often exhibit interesting statistical features such as repetitive geometric textures. Building on this property, we propose a new approach for surface super‐resolution that detects repetitive patterns or self‐similarities and exploits them to improve the scan resolution by aggregating scattered measures. In contrast with other surface super‐resolution methods, our algorithm has two important advantages. First, when handling multiple scans, it does not rely on surface registration. Second, it is able to produce super‐resolution from even a single scan. These features are made possible by a new local shape description able to capture differential properties of order above 2. By comparing those descriptors, similarities are detected and used to generate a high‐resolution surface. Our results show a clear resolution gain over state‐of‐the‐art interpolation methods.

We propose a new shape analysis approach based on the non-local analysis of local shape variations. Our method relies on a novel description of shape variations, called Local Probing Field (LPF), which describes how a local probing operator transforms a pattern onto the shape. By carefully optimizing the position and orientation of each descriptor, we are able to capture shape similarities and gather them into a geometrically relevant dictionary over which the shape decomposes sparsely. This new representation permits to handle shapes with mixed intrinsic dimensionality (e.g., shapes containing both surfaces and curves) and to encode various shape features such as boundaries. Our shape representation has several potential applications; here we demonstrate its efficiency for shape resampling and point set denoising for both synthetic and real data.

LiDAR (Light Detection And Ranging) acquisition is a widespread method for measuring urban scenes, be it a small town neighborhood or an entire city. It is even more interesting when this acquisition is coupled with a collection of pictures registered with the data, permitting to recover the color information of the points. Yet, this added color can be perturbed by shadows that are very dependent on the sun direction and weather conditions during the acquisition. In this paper, we focus on the problem of automatically detecting and correcting the shadows from the LiDAR data by exploiting both the images and the point set laser reflectance. Building on the observation that shadow boundaries are characterized by both a significant color change and a stable laser reflectance, we propose to first detect shadow boundaries in the point set and then segment ground shadows using graph cuts in the image. Finally using a simplified illumination model we correct the shadows directly on the colored point sets. This joint exploitation of both the laser point set and the images renders our approach robust and efficient, avoiding user interaction.

Abstract Most surfaces, be it from a fine‐art artifact or a mechanical object, are characterized by a strong self‐similarity. This property finds its source in the natural structures of objects but also in the fabrication processes: regularity of the sculpting technique, or machine tool. In this paper, we propose to exploit the self‐similarity of the underlying shapes for compressing point cloud surfaces which can contain millions of points at a very high precision. Our approach locally resamples the point cloud in order to highlight the self‐similarity of the shape, while remaining consistent with the original shape and the scanner precision. It then uses this self‐similarity to create an ad hoc dictionary on which the local neighborhoods will be sparsely represented, thus allowing for a light‐weight representation of the total surface. We demonstrate the validity of our approach on several point clouds from fine‐arts and mechanical objects, as well as a urban scene. In addition, we show that our approach also achieves a filtering of noise whose magnitude is smaller than the scanner precision.

Abstract In this paper, we propose a novel progressive lossless mesh compression algorithm based on Incremental Parametric Refinement, where the connectivity is uncontrolled in a first step, yielding visually pleasing meshes at each resolution level while saving connectivity information compared to previous approaches. The algorithm starts with a coarse version of the original mesh, which is further refined by means of a novel refinement scheme. The mesh refinement is driven by a geometric criterion, in spirit with surface reconstruction algorithms, aiming at generating uniform meshes. The vertices coordinates are also quantized and transmitted in a progressive way, following a geometric criterion, efficiently allocating the bit budget. With this assumption, the generated intermediate meshes tend to exhibit a uniform sampling. The potential discrepancy between the resulting connectivity and the original one is corrected at the end of the algorithm. We provide a proof‐of‐concept implementation, yielding very competitive results compared to previous works in terms of rate/distortion trade‐off.

During a highly productive period running from 1995 to about 2002, the research in lossless compression of surface meshes mainly consisted in a hard battle for the best bitrates. However, for a few years, compression rates seem stabilized around 1.5 bit per vertex for the connectivity coding of usual triangular meshes, and more and more work is dedicated to remeshing, lossy compression, or gigantic mesh compression, where memory access and CPU optimizations are the new priority. However, the size of 3D models keeps growing, and many application fields keep requiring lossless compression. In this paper, we present a new contribution for single-rate lossless connectivity compression, which first brings improvement over current state of the art bitrates, and second, does not constraint the coding of the vertex positions, offering therefore a good complementarity with the best performing geometric compression methods. The initial observation having motivated this work is that very often, most of the connectivity part of a mesh can be automatically deduced from its geometric part using reconstruction algorithms. This has already been used within the limited framework of projectable objects (essentially, terrain models and GIS), but finds here its first generalization to arbitrary triangular meshes, without any limitation regarding the topological genus, the number of connected components, the manifoldness or the regularity. This can be obtained by constraining and guiding a Delaunay-based reconstruction algorithm so that it outputs the initial mesh to be coded. The resulting rates seem extremely competitive when the meshes are fully included in Delaunay, and are still good compared to the state-of-the-art in the case of scanned models.

Reconstruction algorithms make it possible to retrieve a surface from the Delaunay tetrahedralisation (DT) of a point sampling, whose density reflects the surface local geometry and thickness. Most of these algorithms are static and some work remains to be done to handle deforming surfaces. In such case, we defend the idea that each point of the sampling should move with the surface using the information given by the motion to allow fast reconstruction. In this article, we tackle the problem of producing a good evolving sampling of a deforming surface S, and maintaining its DT along the motion. The surface is known only through a projection operator (O1): i3→ S , and a normal operator (O2) that returns the oriented normal at a point on the surface. On that basis, we offer some perspectives on how reconstruction algorithms can be extended to the tracking of deforming surfaces.

In this paper, we present a new and efficient spherical harmonics decomposition for spherical functions defining 3D triangulated objects. Such spherical functions are intrinsically associated to star-shaped objects. However, our results can be extended to any triangular object after segmentation into star-shaped surface patches and recomposition of the results in the implicit framework. There is thus no restriction about the genus number of the object. We demonstrate that the evaluation of the spherical harmonics coefficients can be performed by a Monte Carlo integration over the edges, which makes the computation more accurate and faster than previous techniques, and provides a better control over the precision error in contrast to the voxel-based methods. We present several applications of our research, including fast spectral surface reconstruction from point clouds, local surface smoothing and interactive geometric texture transfer.

We present a streaming algorithm for reconstructing closed surfaces from large non-uniform point sets based on a geometric convection technique. Assuming that the sample points are organized into slices stacked along one coordinate axis, a triangle mesh can be efficiently reconstructed in a streamable layout with a controlled memory footprint. Our algorithm associates a streaming 3D Delaunay triangulation data-structure with a multilayer version of the geometric convection algorithm. Our method can process millions of sample points at the rate of 50k points per minute with 350 MB of main memory.

During a highly productive period running from 1995 to about 2002, the research in lossless compression of 3D meshes mainly consisted in a hard battle for the best bitrates. But for a few years, compression rates seem stabilized around 1.5 bit per vertex for the connectivity coding of usual meshes, and more and more work is dedicated to remeshing, lossy compression, or gigantic mesh compression, where memory and CPU optimizations are the new priority. However, the size of 3D models keeps growing, and many application fields keep requiring lossless compression. In this paper, we present a new contribution for single-rate lossless connectivity compression, which first brings improvement over current state of the art bitrates, and secondly, does not constraint the coding of the vertex positions, offering therefore a good complementarity with the best performing geometric compression methods. The initial observation having motivated this work is that very often, most of the connectivity part of a mesh can be automatically deduced from its geometric part using reconstruction algorithms. This has already been used within the limited framework of projectable objects (essentially terrain models and GIS), but finds here its first generalization to arbitrary triangular meshes, without any limitation regarding the topological genus, the number of connected components, the manifoldness or the regularity. This can be obtained by constraining and guiding a Delaunay-based reconstruction algorithm so that it outputs the initial mesh to be coded. The resulting rates seem extremely competitive when the meshes are fully included in Delaunay, and are still good compared to the state of the art in the general case.

In this paper, we introduce a flexible framework for the reconstruction of a surface from an unorganized point set, extending the geometric convection approach introduced by Chaine. Given a dense input point cloud, we first extract a triangulated surface that interpolates a subset of the initial data. We compute this surface in an output sensitive manner by decimating the input point set on-the-fly during the reconstruction process. Our simplification procedure relies on a simple criterion that locally detects and reduces oversampling. If needed, we then operate in a dynamic fashion for local refinement or further simplification of the reconstructed surface. Our method allows to locally update the reconstructed surface by inserting or removing sample points without restarting the convection process from scratch. This iterative correction process can be controlled interactively by the user or automatized given some specific local sampling constraints.