8 research outputs found
Nonlinear Diffusion on the 2D Euclidean Motion Group
Linear and nonlinear diffusion equations are usually considered on an image, which is in fact a function on the translation group. In this paper we study diffusion on orientation scores, i.e. on functions on the Euclidean motion group SE(2). An orientation score is obtained from an image by a linear invertible transformation. The goal is to enhance elongated structures by applying nonlinear left-invariant diffusion on the orientation score of the image. For this purpose we describe how we can use Gaussian derivatives to obtain regularized left-invariant derivatives that obey the non-commutative structure of the Lie algebra of SE(2). The Hessian constructed with these derivatives is used to estimate local curvature and orientation strength and the diffusion is made nonlinearly dependent on these measures. We propose an explicit finite difference scheme to apply the nonlinear diffusion on orientation scores. The experiments show that preservation of crossing structures is the main advantage compared to approaches such as coherence enhancing diffusion
A geometric model of multi-scale orientation preference maps via Gabor functions
In this paper we present a new model for the generation of orientation
preference maps in the primary visual cortex (V1), considering both orientation
and scale features. First we undertake to model the functional architecture of
V1 by interpreting it as a principal fiber bundle over the 2-dimensional
retinal plane by introducing intrinsic variables orientation and scale. The
intrinsic variables constitute a fiber on each point of the retinal plane and
the set of receptive profiles of simple cells is located on the fiber. Each
receptive profile on the fiber is mathematically interpreted as a rotated Gabor
function derived from an uncertainty principle. The visual stimulus is lifted
in a 4-dimensional space, characterized by coordinate variables, position,
orientation and scale, through a linear filtering of the stimulus with Gabor
functions. Orientation preference maps are then obtained by mapping the
orientation value found from the lifting of a noise stimulus onto the
2-dimensional retinal plane. This corresponds to a Bargmann transform in the
reducible representation of the group. A
comparison will be provided with a previous model based on the Bargman
transform in the irreducible representation of the group,
outlining that the new model is more physiologically motivated. Then we present
simulation results related to the construction of the orientation preference
map by using Gabor filters with different scales and compare those results to
the relevant neurophysiological findings in the literature
Cortical Functional architectures as contact and sub-Riemannian geometry
In a joint paper, Jean Petitot together with the authors of the present paper
described the functional geometry of the visual cortex as the symplectization
of a contact form to describe the family of cells sensitive to position,
orientation and scale. In the present paper, as a "homage" to the enormous
contribution of Jean Petitot to neurogeometry, we will extend this approach to
more complex functional architectures built as a sequence of contactization or
a symplectization process, able to extend the dimension of the space. We will
also outline a few examples where a sub-Riemannian lifting is needed
Left-invariant evolutions of wavelet transforms on the Similitude Group
Enhancement of multiple-scale elongated structures in noisy image data is
relevant for many biomedical applications but commonly used PDE-based
enhancement techniques often fail at crossings in an image. To get an overview
of how an image is composed of local multiple-scale elongated structures we
construct a multiple scale orientation score, which is a continuous wavelet
transform on the similitude group, SIM(2). Our unitary transform maps the space
of images onto a reproducing kernel space defined on SIM(2), allowing us to
robustly relate Euclidean (and scaling) invariant operators on images to
left-invariant operators on the corresponding continuous wavelet transform.
Rather than often used wavelet (soft-)thresholding techniques, we employ the
group structure in the wavelet domain to arrive at left-invariant evolutions
and flows (diffusion), for contextual crossing preserving enhancement of
multiple scale elongated structures in noisy images. We present experiments
that display benefits of our work compared to recent PDE techniques acting
directly on the images and to our previous work on left-invariant diffusions on
orientation scores defined on Euclidean motion group.Comment: 40 page
A sub-Riemannian model of the visual cortex with frequency and phase
In this paper we present a novel model of the primary visual cortex (V1) based on orientation, frequency and phase selective behavior of the V1 simple cells. We start from the first level mechanisms of visual perception: receptive profiles. The model interprets V1 as a fiber bundle over the 2-dimensional retinal plane by introducing orientation, frequency and phase as intrinsic variables. Each receptive profile on the fiber is mathematically interpreted as a rotated, frequency modulated and phase shifted Gabor function. We start from the Gabor function and show that it induces in a natural way the model geometry and the associated horizontal connectivity modeling the neural connectivity patterns in V1. We provide an image enhancement algorithm employing the model framework. The algorithm is capable of exploiting not only orientation but also frequency and phase information existing intrinsically in a 2-dimensional input image. We provide the experimental results corresponding to the enhancement algorithm
A sub-Riemannian model of the visual cortex with frequency and phase
International audienceIn this paper we present a novel model of the primary visual cortex (V1) based on orientation, frequency and phase selective behavior of the V1 simple cells. We start from the first level mechanisms of visual perception: receptive profiles. The model interprets V1 as a fiber bundle over the 2-dimensional retinal plane by introducing orientation, frequency and phase as intrinsic variables. Each receptive profile on the fiber is mathematically interpreted as a rotated, frequency modulated and phase shifted Gabor function. We start from the Gabor function and show that it induces in a natural way the model geometry and the associated horizontal connectivity modeling the neural connectivity patterns in V1. We provide an image enhancement algorithm employing the model framework. The algorithm is capable of exploiting not only orientation but also frequency and phase information existing intrinsically in a 2-dimensional input image. We provide the experimental results corresponding to the enhancement algorithm
Local Geometric Transformations in Image Analysis
The characterization of images by geometric features facilitates the precise analysis of the structures found in biological micrographs such as cells, proteins, or tissues. In this thesis, we study image representations that are adapted to local geometric transformations such as rotation, translation, and scaling, with a special emphasis on wavelet representations. In the first part of the thesis, our main interest is in the analysis of directional patterns and the estimation of their location and orientation. We explore steerable representations that correspond to the notion of rotation. Contrarily to classical pattern matching techniques, they have no need for an a priori discretization of the angle and for matching the filter to the image at each discretized direction. Instead, it is sufficient to apply the filtering only once. Then, the rotated filter for any arbitrary angle can be determined by a systematic and linear transformation of the initial filter. We derive the Cramér-Rao bounds for steerable filters. They allow us to select the best harmonics for the design of steerable detectors and to identify their optimal radial profile. We propose several ways to construct optimal representations and to build powerful and effective detector schemes; in particular, junctions of coinciding branches with local orientations. The basic idea of local transformability and the general principles that we utilize to design steerable wavelets can be applied to other geometric transformations. Accordingly, in the second part, we extend our framework to other transformation groups, with a particular interest in scaling. To construct representations in tune with a notion of local scale, we identify the possible solutions for scalable functions and give specific criteria for their applicability to wavelet schemes. Finally, we propose discrete wavelet frames that approximate a continuous wavelet transform. Based on these results, we present a novel wavelet-based image-analysis software that provides a fast and automatic detection of circular patterns, combined with a precise estimation of their size
Nonlinear diffusion on the 2D Euclidean motion group
Linear and nonlinear diffusion equations are usually considered on an image, which is in fact a function on the translation group. In this paper we study diffusion on orientation scores, i.e. on functions on the Euclidean motion group SE(2). An orientation score is obtained from an image by a linear invertible transformation. The goal is to enhance elongated structures by applying nonlinear left-invariant diffusion on the orientation score of the image. For this purpose we describe how we can use Gaussian derivatives to obtain regularized left-invariant derivatives that obey the non-commutative structure of the Lie algebra of SE(2). The Hessian constructed with these derivatives is used to estimate local curvature and orientation strength and the diffusion is made nonlinearly dependent on these measures. We propose an explicit finite difference scheme to apply the nonlinear diffusion on orientation scores. The experiments show that preservation of crossing structures is the main advantage compared to approaches such as coherence enhancing diffusion