Thesis Chapters by Leonardo Sacht
IMPA doctoral disseration, 2014
We first provide a review on the sampling and reconstruction problem. This review focuses mainly ... more We first provide a review on the sampling and reconstruction problem. This review focuses mainly on theoretical and historic aspects that will be the fundamental to present our new framework. We then obtain new quasi-interpolators for continuous reconstruction of sampled images by minimizing a new objective function that takes into account the approximation error over the full Nyquist interval. To achieve this goal, we optimize with respect to all possible degrees of freedom in the approximation scheme. We consider three study cases offering different trade-offs between quality and computational cost: a linear, a quadratic, and a cubic scheme. Experiments with compounded rotations and translations confirm that our new quasi- interpolators perform better than the state-of-the-art for a similar computational cost. We then turn our attention to a different problem on the field geometry processing: the one of generating consistent tetrahedral discretizations inside self-intersecting triangle meshes. With the goal of proposing an algorithm for this problem, we first introduce the main concepts of tetrahedral meshes, geometric flows, deformation energies and parametrization. We then observe that most steps in the geometry processing pipeline for surfaces, like deformation, smoothing, subdivision and decimation, may create self-intersections. Volumetric processing of solid shapes then becomes difficult, because obtaining a correct volumetric discretization is impossible: existing tet-meshing methods require watertight input. We propose an algorithm that produces a tetrahedral mesh that overlaps itself consistently with the self-intersections in the input surface. This enables volumetric processing on self-intersecting models. We leverage conformalized mean-curvature flow, which removes self-intersections, and define an intrinsically similar reverse flow, which prevents them. We tetrahedralize the resulting surface and map the mesh inside the original surface. We demonstrate the effectiveness of our method with applications to automatic skinning weight computation, physically based simulation and geodesic distance computation.
IMPA Master's Thesis, 2010
Common cameras usually capture a very narrow field of view (FOV), around ninety degrees. The reas... more Common cameras usually capture a very narrow field of view (FOV), around ninety degrees. The reason for this fact is that when the field of view becomes wider, the projection that these cameras use starts introducing unnatural and nontrivial distortions. This thesis studies these distortions in order to obtain panoramic images, i.e., images of wide fields of view.
After modeling the FOV as a unit sphere, the problem becomes finding a projection from a subset of the unit sphere to the image plane, with desirable properties. We provide an in-depth discussion of Carroll et. al ([1]), where preservation of straight lines and object shapes are stated as the main desirable properties and an optimization solution is proposed. Next, we show panoramic images obtained by this method and conclude that it works well in a variety of scenes.
This thesis also provides a novel study about panoramic videos, i.e., videos where each frame is constructed from a wide FOV. We introduce a mathematical model for this pro- blem, discuss desirable temporal coherence properties, formulate equations that represent this properties, propose an optimization solution for a particular case and point future directions.
Papers by Leonardo Sacht
Computers & Graphics, 2022
We propose a new image processing problem that consists in approximating an input image with a se... more We propose a new image processing problem that consists in approximating an input image with a set of plastic bottle caps. This problem is motivated by the appreciation caused by low resolution imaging, combined with the goal of finding a new destination for plastic caps that would be discarded and possibly damage the environment. Our solution consists in formulating an energy that measures how well a bottle cap art reproduces structures and features from the reference image and maximizing it with a simulated annealing strategy. Examples in the paper show that our method produces high quality results in this very low spatial and color resolution setting.
Computer & Graphics, 2020
We propose a new method to correct distortions in equirectangular images, i.e., images that use t... more We propose a new method to correct distortions in equirectangular images, i.e., images that use the longitude/latitude representation to describe the full spherical field of view around a given viewpoint. We show that Möbius transformations are the correct mathematical tool to deal with the conflicting
distortions in this setting: they are conformal, are able to rectify lines, perform rotations, translations and scales on the sphere and are bijective.
Multiple transformations are specified through points, lines and cages on the sphere. We associate three points that uniquely define a Möbius transformation to each one of these geometric handles. Linear blend skinning with bounded biharmonic weights combine the different transformations into a single full spherical modified image.
We present a collection of results in challenging settings and show that our method is more flexible and produces higher quality results when compared to previous methods.
SIBGRAPI Workshop of Works in Progress, 2019
We present a partial solution for the problem of reducing overhanging parts of a surface to 3D pr... more We present a partial solution for the problem of reducing overhanging parts of a surface to 3D print it with minimal number of supports. We first present a summary about how a 3D printer works and why the overhanging problem happens. This review focuses on a specific type of 3D printer that uses polymer melted to print solids on cross-sectional layers. We then do a fast review of three-dimensional surface representation in a computer and its discrete representation. Then we present our minimization problem and show some test results, using libigl library and gptoolbox functions, to observe the solution of problem.
International Journal of Image and Graphics, 2018
In this work, we propose a framework that performs a number of popular image-processing operation... more In this work, we propose a framework that performs a number of popular image-processing operations in the continuous domain. This is in contrast to the standard practice of defining them as operations over discrete sequences of sampled values. The guiding principle is that, in order to prevent aliasing, non-linear image-processing operations should ideally be performed prior to prefiltering and sampling. This is of course impractical, as we may not have access to the continuous input. Even so, we show that it is best to apply image-processing operations over the continuous reconstruction of the input. This transformed continuous representation is then prefiltered and sampled to produce the output. The use of high-quality reconstruction strategies brings this alternative much closer to the ideal than directly operating over discrete values. We illustrate the advantages of our framework with several popular effects. In each case, we demonstrate the quality difference between continuous image-processing, their discrete counterparts and previous anti-aliasing alternatives. Finally, our GPU implementation shows that current graphics hardware has enough computational power to perform continuous image processing in real-time.
SIBGRAPI Workshop of Undergraduate Works, 2017
Input data for geometry processing is commonly problematic: the discretization of the volume of a... more Input data for geometry processing is commonly problematic: the discretization of the volume of a surface may contain self-intersections or noise. These problems are related to curvature. In this work we review some concepts in differential geometry, geometry processing and present the Mean Curvature Flow to solve those problems. We review some modifications in the discrete case to improve the discrete flow. Then we analyse an implementation of the Conformalized Mean Curvature Flow.
SIBGRAPI Workshop of Undergraduate Works, 2017
In this paper we discuss the application of specific constraints to quasi-interpolators, which ar... more In this paper we discuss the application of specific constraints to quasi-interpolators, which are a special class of functions used for the continuous reconstruction of an input image when sampled at a given spacing rate. In addition, this research intends to explore some mathematical concepts related to image processing, which are necessary for the comprehension of all the processes along the stages of the traditional sampling framework. This article also focuses on minimizing the residual error of linear schemes, obtained from the optimized quasiinterpolators.
Computers & Graphics, 2017
This work presents a new method to transform omnidirectional images based on a combination of Moe... more This work presents a new method to transform omnidirectional images based on a combination of Moebius transformations in the complex plane and weighting functions that restrict the action of these mappings to regions of interest. The transformations are calculated based on the specification of the image of three points and the weighting functions are designed to achieve specific goals such as local zoom or straight line rectification. Since no optimization or numerical methods are involved, our implementation of the proposed method can be upgraded to reach real-time performance. We provide a user interface and present many results that illustrate the potential of the proposed technique.
Journal of Real-Time Image Processing, 2015
Wide-angle images gained a huge popularity in the last years due to the development of computatio... more Wide-angle images gained a huge popularity in the last years due to the development of computational photography and imaging technological advances. They present the information of a scene in a way which is more natural for the human eye but, on the other hand, they introduce artifacts such as bent lines. These artifacts become more and more unnatural as the field of view increases. In this work, we present a technique aimed to improve the perceptual quality of panorama visualization. The main ingredients of our approach are, on one hand, considering the viewing sphere as a Riemann sphere, what makes natural the application of Möbius (complex) transformations to the input image, and, on the other hand, a projection scheme which changes in function of the field of view used. We also introduce an implementation of our method, compare it against images produced with other methods and show that the transformations can be done in real-time, which makes our technique very appealing for new settings, as well as for existing interactive panorama applications.
GRAPP, 2013
We present a method for manipulation and visualization of wide-angle images using transformations... more We present a method for manipulation and visualization of wide-angle images using transformations defined on the complex plane C. We map the unit sphere S 2 to C using the stereographic projection, multiply the complex plane by a given complex number, and map the result back to the sphere using the inverse of the stereographic projection. Since all these transformations preserve angle, we obtain a result containing only distortions due to the latitude/longitude representation of the sphere, which were already present in the input image. We then explore the possibility given by our technique of mapping wide fields of view to narrower ones. This makes possible to apply perspective projection to wider fields of view, leading to a natural generalization of the perspective projection in the context of panoramic images. Our results are generated in real-time and compare competitively with state-of-the-art methods used to project the viewing sphere to the image plane.
SIBGRAPI Workshop of Thesis and Dissertations, 2010
This paper studies the problem of obtaining panoramic images, i.e., images of wide fields of view... more This paper studies the problem of obtaining panoramic images, i.e., images of wide fields of view. We start by analyzing previous approaches, in order to understand the difficulties of this problem. Then we discuss in detail the work by Carroll et al., entitled "Optimizing content-preserving projections for wide-angle images" and show results of this method that prove that it produces good results in a variety of scenes. We discuss some aspects that were not emphasized in that work, such as the optimization method used to produce the final result. In addition, we propose some important extensions: an interface with some additional features to the one proposed by them; and Computer Vision methods to detect faces and straight lines in equirectangular images, which are very important features for the method. We also provide a novel study about panoramic videos, i.e., videos where each frame is constructed from a wide FOV. We introduce a mathematical model for this problem, discuss desirable temporal coherence properties, formulate equations that represent these properties, propose an optimization solution for a particular case and point future directions.
Workshop de Visão Computatcional, 2010
In this paper, we address the problem of detecting faces and straight lines in a scene represente... more In this paper, we address the problem of detecting faces and straight lines in a scene represented by an equirectangular image, i.e., a spherical image of the 360 degree longitude/180 degree latitude field of view. The main difficulty of these problems is that straight lines in the scene usually appear curved in the equirectangular images and faces may appear too distorted. Thus, standard detectors for lines and faces can not be used directly in this context. The solutions we present in this paper use projections of the equirectangular image where these features are well preserved. To detect faces, we use the Mercator projection, which preserves shape and orientation of the faces. To detect lines, we use perspective projections, which preserve straight lines. To deal with the whole equirectangular image, six different projections are used, one for each face of the bounding cube. After some processing steps are applied to the Mercator and perspective projections, the standard detectors are applied and the detected features are mapped back to the equirectangular image.
arXiv, 2010
The main goal of the paper is to introduce methods which compute Bézier curves faster than Castel... more The main goal of the paper is to introduce methods which compute Bézier curves faster than Casteljau's method does. These methods are based on the spectral factorization of a n × n Bernstein matrix, B e n (s) = P n G n (s)P −1 n , where P n is the n × n lower triangular Pascal matrix. So we first calculate the exact optimum positive value t in order to transform P n in a scaled Toeplitz matrix, which is a problem that was partially solved by X. Wang and J. Zhou (2006). Then fast Pascal matrixvector multiplications and strategies of polynomial evaluation are put together to compute Bézier curves. Nevertheless, when n increases, more precise Pascal matrixvector multiplications allied to affine transformations of the vectors of coordinates of the control points of the curve are then necessary to stabilize all the computation.
Drafts by Leonardo Sacht
SIGGRAPH Asia Technical Sketches, 2011
This work presents a method for obtaining perceptually natural panoramic videos, that is, videos ... more This work presents a method for obtaining perceptually natural panoramic videos, that is, videos composed of wide-angle frames where straight lines and object shapes are preserved. The shape preservation for moving objects has a special treatment that ensures temporal coherence. Energies describing these properties are obtained for the case of a fixed omni-directional camera, and an optimization procedure is presented. Our optimization works per-frame, which makes the method scalable for arbitrarily long scenes.
CNMAC abstracts, 2010
Common cameras capture a limited field of view, usually of up to ninety degrees. The reason for t... more Common cameras capture a limited field of view, usually of up to ninety degrees. The reason for this fact is that when the field of view (FOV) becomes wider, the projection used by these cameras introduces strong distortions in the objects. Panoramic images, which are images of wide FOVs, can better represent an entire scene. Our work is based in Carroll et al. [1], who suggested all the distortions we consider. The contributions of this abstract are the details about the optimization methods, which were briefly mentioned in [1].
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Thesis Chapters by Leonardo Sacht
After modeling the FOV as a unit sphere, the problem becomes finding a projection from a subset of the unit sphere to the image plane, with desirable properties. We provide an in-depth discussion of Carroll et. al ([1]), where preservation of straight lines and object shapes are stated as the main desirable properties and an optimization solution is proposed. Next, we show panoramic images obtained by this method and conclude that it works well in a variety of scenes.
This thesis also provides a novel study about panoramic videos, i.e., videos where each frame is constructed from a wide FOV. We introduce a mathematical model for this pro- blem, discuss desirable temporal coherence properties, formulate equations that represent this properties, propose an optimization solution for a particular case and point future directions.
Papers by Leonardo Sacht
distortions in this setting: they are conformal, are able to rectify lines, perform rotations, translations and scales on the sphere and are bijective.
Multiple transformations are specified through points, lines and cages on the sphere. We associate three points that uniquely define a Möbius transformation to each one of these geometric handles. Linear blend skinning with bounded biharmonic weights combine the different transformations into a single full spherical modified image.
We present a collection of results in challenging settings and show that our method is more flexible and produces higher quality results when compared to previous methods.
Drafts by Leonardo Sacht
After modeling the FOV as a unit sphere, the problem becomes finding a projection from a subset of the unit sphere to the image plane, with desirable properties. We provide an in-depth discussion of Carroll et. al ([1]), where preservation of straight lines and object shapes are stated as the main desirable properties and an optimization solution is proposed. Next, we show panoramic images obtained by this method and conclude that it works well in a variety of scenes.
This thesis also provides a novel study about panoramic videos, i.e., videos where each frame is constructed from a wide FOV. We introduce a mathematical model for this pro- blem, discuss desirable temporal coherence properties, formulate equations that represent this properties, propose an optimization solution for a particular case and point future directions.
distortions in this setting: they are conformal, are able to rectify lines, perform rotations, translations and scales on the sphere and are bijective.
Multiple transformations are specified through points, lines and cages on the sphere. We associate three points that uniquely define a Möbius transformation to each one of these geometric handles. Linear blend skinning with bounded biharmonic weights combine the different transformations into a single full spherical modified image.
We present a collection of results in challenging settings and show that our method is more flexible and produces higher quality results when compared to previous methods.