Results 1 - 10
of
44
A Riemannian Framework for Tensor Computing
- INTERNATIONAL JOURNAL OF COMPUTER VISION
, 2006
"... Positive definite symmetric matrices (so-called tensors in this article) are nowadays a common source of geometric information. In this paper, we propose to provide the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of ..."
Abstract
-
Cited by 286 (27 self)
- Add to MetaCart
Positive definite symmetric matrices (so-called tensors in this article) are nowadays a common source of geometric information. In this paper, we propose to provide the tensor space with an affine-invariant Riemannian metric. We demonstrate that it leads to strong theoretical properties: the cone of positive definite symmetric matrices is replaced by a regular manifold of constant curvature without boundaries (null eigenvalues are at the infinity), the geodesic between two tensors and the mean of a set of tensors are uniquely defined, etc. We have
Fast Anisotropic Smoothing of Multi-Valued Images using Curvature-Preserving PDE’s
- Research Report “Les Cahiers du GREYC”, No 05/01. Equipe IMAGE/GREYC (CNRS UMR 6072), Février
, 2005
"... We are interested in PDE’s (Partial Differential Equations) in order to smooth multi-valued images in an anisotropic manner. Starting from a review of existing anisotropic regularization schemes based on diffusion PDE’s, we point out the pros and cons of the different equations proposed in the liter ..."
Abstract
-
Cited by 65 (3 self)
- Add to MetaCart
We are interested in PDE’s (Partial Differential Equations) in order to smooth multi-valued images in an anisotropic manner. Starting from a review of existing anisotropic regularization schemes based on diffusion PDE’s, we point out the pros and cons of the different equations proposed in the literature. Then, we introduce a new tensor-driven PDE, regularizing images while taking the curvatures of specific integral curves into account. We show that this constraint is particularly well suited for the preservation of thin structures in an image restoration process. A direct link is made between our proposed equation and a continuous formulation of the LIC’s (Line Integral Convolutions by Cabral and Leedom [11]). It leads to the design of a very fast and stable algorithm that implements our regularization method, by successive integrations of pixel values along curved integral lines. Besides, the scheme numerically performs with a sub-pixel accuracy and preserves then thin image structures better than classical finite-differences discretizations. Finally, we illustrate the efficiency of our generic curvature-preserving approach- in terms of speed and visual quality- with different comparisons and various applications requiring image smoothing: color images denoising, inpainting and image resizing by nonlinear interpolation.
Capture of Hair Geometry from Multiple Images
, 2004
"... Hair is a major feature of digital characters. Unfortunately, it has a complex geometry which challenges standard modeling tools. Some dedicated techniques exist, but creating a realistic hairstyle still takes hours. Complementary to user-driven methods, we here propose an image-based approach to ca ..."
Abstract
-
Cited by 54 (4 self)
- Add to MetaCart
Hair is a major feature of digital characters. Unfortunately, it has a complex geometry which challenges standard modeling tools. Some dedicated techniques exist, but creating a realistic hairstyle still takes hours. Complementary to user-driven methods, we here propose an image-based approach to capture the geometry of hair. The novelty of this work is that we draw information from the scattering properties of the hair that are normally considered a hindrance. To do so, we analyze image sequences from a fixed camera with a moving light source. We first introduce a novel method to compute the image orientation of the hairs from their anisotropic behavior. This method is proven to subsume and extend existing work while improving accuracy. This image orientation is then raised into a 3D orientation by analyzing the light reflected by the hair fibers. This part relies on minimal assumptions that have been proven correct in previous work. Finally, we show how to use several such image sequences to reconstruct the complete hair geometry of a real person. Results are shown to illustrate the fidelity of the captured geometry to the original hair. This technique paves the way for a new approach to digital hair generation.
Channel smoothing: Efficient robust smoothing of low-level signal features
- IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2006
"... In this paper, we present a new and efficient method to implement robust smoothing of low-level signal features: B-spline channel smoothing. This method consists of three steps: encoding of the signal features into channels, averaging of the channels, and decoding of the channels. We show that line ..."
Abstract
-
Cited by 36 (22 self)
- Add to MetaCart
(Show Context)
In this paper, we present a new and efficient method to implement robust smoothing of low-level signal features: B-spline channel smoothing. This method consists of three steps: encoding of the signal features into channels, averaging of the channels, and decoding of the channels. We show that linear smoothing of channels is equivalent to robust smoothing of the signal features if we make use of quadratic B-splines to generate the channels. The linear decoding from B-spline channels allows the derivation of a robust error norm, which is very similar to Tukey’s biweight error norm. We compare channel smoothing with three other robust smoothing techniques: nonlinear diffusion, bilateral filtering, and mean-shift filtering, both theoretically and on a 2D orientation-data smoothing task. Channel smoothing is found to be superior in four respects: It has a lower computational complexity, it is easy to implement, it chooses the global minimum error instead of the nearest local minimum, and it can also be used on nonlinear spaces, such as orientation space.
Measuring brain variability by extrapolating sparse tensor fields measured on sulcal lines
- Neuroimage
, 2007
"... Abstract. Modeling and understanding the variability of brain structures is a fundamental problem in the neurosciences. Improved mathematical representations of structural brain variation are needed to help detect and understand genetic or disease related sources of abnormality, as well as to improv ..."
Abstract
-
Cited by 32 (15 self)
- Add to MetaCart
Abstract. Modeling and understanding the variability of brain structures is a fundamental problem in the neurosciences. Improved mathematical representations of structural brain variation are needed to help detect and understand genetic or disease related sources of abnormality, as well as to improve statistical power when integrating functional brain mapping data across subjects. In this paper, we develop a new mathematical model of normal brain variation based on a large set of cortical sulcal landmarks (72 per brain) delineated in each of 98 healthy human subjects scanned with 3D MRI (age: 51.8 +/- 6.2 years). We propose an original method to compute an average representation of the sulcal curves, which constitutes the mean anatomy. After a ne alignment of the individual data across subjects, the second order moment distribution of the sulcal position is modeled as a sparse eld of covariance tensors (symmetric, positive de nite matrices). To extrapolate this information to the full brain, one has to overcome the limitations of the standard Euclidean matrix calculus. We propose an a ne-invariant Riemannian framework to perform computations with tensors. In particular, we generalize radial basis function (RBF) interpolation and harmonic di usion partial di erential equations (PDEs) to tensor elds. As a result, we obtain a dense 3D variability map which agrees well with prior results on smaller subject samples. Moreover, "leave
Flow-based image abstraction
- IEEE Transactions on Visualization and Computer Graphics
, 2009
"... Abstract—We present a nonphotorealistic rendering technique that automatically delivers a stylized abstraction of a photograph. Our approach is based on shape/color filtering guided by a vector field that describes the flow of salient features in the image. This flow-based filtering significantly im ..."
Abstract
-
Cited by 31 (4 self)
- Add to MetaCart
(Show Context)
Abstract—We present a nonphotorealistic rendering technique that automatically delivers a stylized abstraction of a photograph. Our approach is based on shape/color filtering guided by a vector field that describes the flow of salient features in the image. This flow-based filtering significantly improves the abstraction performance in terms of feature enhancement and stylization. Our method is simple, fast, and easy to implement. Experimental results demonstrate the effectiveness of our method in producing stylistic and feature-enhancing illustrations from photographs. Index Terms—Nonphotorealistic rendering, image abstraction, flow-based filtering, line drawing, bilateral filter. Ç 1
B.: Level-set methods for tensor-valued images
- Proc. Second IEEE Workshop on Geometric and Level Set Methods in Computer Vision
, 2003
"... Tensor-valued data are becoming more and more important as input for todays image analysis problems. This has been caused by a number of applications including diffusion ten-sor (DT-) MRI and physical measurements of anisotropic be-haviour such as stress-strain relationships, interia and per-mittivi ..."
Abstract
-
Cited by 30 (5 self)
- Add to MetaCart
(Show Context)
Tensor-valued data are becoming more and more important as input for todays image analysis problems. This has been caused by a number of applications including diffusion ten-sor (DT-) MRI and physical measurements of anisotropic be-haviour such as stress-strain relationships, interia and per-mittivity tensors. Consequently, there arises the need to fil-ter and segment such tensor fields. In this paper we extend three important level set methods to tensor-valued data. To this end we first generalise Di Zenzo’s concept of a struc-ture tensor for vector-valued images to tensor-valued data. This allows us to derive formulations of mean curvature mo-tion and self-snakes in the case of tensor-valued images. We prove that these processes maintain positive semidefiniteness if the initial matrix data are positive semidefinite. Finally we present a geodesic active contour model for segmenting ten-sor fields. Since it incorporates information from all chan-nels, it gives a contour representation that is highly robust under noise. 1
Unsupervised Learning of Object Deformation Models
"... The aim of this work is to learn generative models of object deformations in an unsupervised manner. Initially, we introduce an Expectation Maximization approach to estimate a linear basis for deformations by maximizing the likelihood of the training set under an Active Appearance Model (AAM). This ..."
Abstract
-
Cited by 23 (5 self)
- Add to MetaCart
(Show Context)
The aim of this work is to learn generative models of object deformations in an unsupervised manner. Initially, we introduce an Expectation Maximization approach to estimate a linear basis for deformations by maximizing the likelihood of the training set under an Active Appearance Model (AAM). This approach is shown to successfully capture the global shape variations of objects like faces, cars and hands. However the AAM representation cannot deal with articulated objects, like cows and horses. We therefore extend our approach to a representation that allows for multiple parts with the relationships between them modeled by a Markov Random Field (MRF). Finally, we propose an algorithm for efficiently performing inference on part-based MRF object models by speeding up the estimation of observation potentials. We use manually collected landmarks to compare the alternative models and quantify learning performance. 1.
Tensor Field Interpolation with PDEs
, 2005
"... We present a unified framework for interpolation and regularisation of scalar- and tensor-valued images. This framework is based on elliptic partial differential equations (PDEs) and allows rotationally invariant models. Since it does not require a regular grid, it can also be used for tensor-valued ..."
Abstract
-
Cited by 16 (8 self)
- Add to MetaCart
We present a unified framework for interpolation and regularisation of scalar- and tensor-valued images. This framework is based on elliptic partial differential equations (PDEs) and allows rotationally invariant models. Since it does not require a regular grid, it can also be used for tensor-valued scattered data interpolation and for tensor field inpainting. By choosing suitable differential operators, interpolation methods using radial basis functions are covered. Our experiments show that a novel interpolation technique based on anisotropic diffusion with a diffusion tensor should be favoured: It outperforms interpolants with radial basis functions, it allows discontinuity-preserving interpolation with no additional oscillations, and it respects positive semidefiniteness of the input tensor data.
A Riemannian framework for the processing of tensor-valued images
- In Ole Fogh Olsen, Luc Florak, and Arjan Kuijper, editors, Deep Structure, Singularities, and Computer Vision (DSSCV), number 3753 in LNCS
, 2005
"... Abstract. In this paper, we present a novel framework to carry out computations on tensors, i.e. symmetric positive definite matrices. We endow the space of tensors with an affine-invariant Riemannian metric, which leads to strong theoretical properties: The space of positive definite symmetric matr ..."
Abstract
-
Cited by 12 (5 self)
- Add to MetaCart
(Show Context)
Abstract. In this paper, we present a novel framework to carry out computations on tensors, i.e. symmetric positive definite matrices. We endow the space of tensors with an affine-invariant Riemannian metric, which leads to strong theoretical properties: The space of positive definite symmetric matrices is replaced by a regular and geodesically complete manifold without boundaries. Thus, tensors with non-positive eigenvalues are at an infinite distance of any positive definite matrix. Moreover, the tools of differential geometry apply and we generalize to tensors numerous algorithms that were reserved to vector spaces. The application of this framework to the processing of diffusion tensor images shows very promising results. We apply this framework to the processing of structure tensor images and show that it could help to extract low-level features thanks to the affine-invariance of our metric. However, the same affine-invariance causes the whole framework to be noise sensitive and we believe that the choice of a more adapted metric could significantly improve the robustness of the result. 1
