Akihiko Torii

Variational analysis of spherical images

This paper focuses on variational image analysis on a sphere. Since a sphere is a closed Riemannian manifold with the positive constant curvature and no holes, a sphere has similar geometrical properties with a plane, whose curvature is zero. Images observed through a catadioptric system with a conic-mirror and a dioptric system with fish-eye lens are transformed to images on the sphere. Therefore, in robot vision, image analysis on the sphere is an essential requirement to the application of the omni-directional imaging system with conic-mirror and fish-eye lens for navigation and control. We introduce algorithms for optical flow computation for images on a sphere.

Optical flow computation for compound eyes: variational analysis of omni-directional views

This paper focuses on variational optical flow computation for spherical images. It is said that some insects recognise the world through optical flow observed by their compound eyes, which observe spherical views. Furthermore, images observed through a catadioptric system with a conic mirror and a fish-eye-lens camera are transformed to images on the sphere. Spherical motion field on the spherical retina has some advantages for the ego-motion estimation of autonomous mobile observer. We provide a framework for motion field analysis on the spherical retina using variational method for image analysis.

Panoramic image transform of omnidirectional images using discrete geometry techniques

This work proposes an omnidirectional-to-panoramic image transform with high accuracy using PDE-based resampling models. For the application of computer-vision techniques to omnidirectional images, the transformation of omnidirectional images to uniform-resolution quadric-surface images is needed in the two reasons. First, an omnidirectional image does not have a uniform resolution. Second, the development of the computer-vision-based techniques on the quadric surface is mathematically accurate compared with the development of the techniques on the omnidirectional image directly. Therefore, our aim is to generate uniform-resolution panoramic images on cylindrical surface from nonuniform-resolution omnidirectional images. The uniform-resolution panoramic images allow us to reconstruct 3D objects and scenes from omnidirectional images robustly. Our panoramic transformation selects the uniform resolution pixels on omnidirectional images employing the geometrical configuration of cameras in the estimation and resampling process. Therefore, our method is mathematically accurate comparing to the traditional panoramic transformation using point-to-point correspondences with the geometries of cameras and the cubic convolution.

Circle-Marker Detection Method for Omnidirectional Images and its Application to Robot Positioning

Introduction In this paper, we develop an algorithm for the detection of circles from an image captured by a monocular omnidirectional camera system. We assume that an image captured by an omnidirectional camera system is normalised to a spherical image, the image on the unit sphere. Using this geometrical property of the omnidirectional images, we introduce a method for marker-based positioning and navigation of autonomous mobile robots which mounts a monocular omnidirectional camera system. We first clarify the geometric properties of the spherical image of a circle marker placed on the ground plane, and we show that, for the detection of a plane from circle markers, we are required to capture at least two coplanar circle markers. We prove that the image of a circle on the spherical image is a fourth-order algebraic curve, which is the intersection of a sphere and an oblique elliptic cone. For the detection of marker images on the spherical images, we introduce a method for transforming the detection of this fourth-order algebraic curve to the detection of a spatial conic, which is quadric. Second, we develop a voting method for the extraction of images of planar circle markers on a spherical image, using the spatial-quadric detection strategy. Finally, using the assumption for the geometrical configuration of the camera system and the circle markers on the plane on which the robot moves, we derive a positioning algorithm. This positioning method for the robot mounting a monocular omnidirectional camera system allows us to navigate a robot using our circle-detection algorithm. We show some numerical examples both for synthetic and real images.

Mathematics of a Multiple Omni-Directional System

We formulate multiple-view geometry for omni-directional and panorama-camera systems. The mathematical formu-lations enable us to derive the geometrical and algebraic constraints for multiple panorama-camera configurations. The constraints permit us to reconstruct three-dimensional objects for a large feasible region.

Inverse quantization for resolution conversion

In this paper, we introduce a resolution-conversion method for two- and three-dimensional discrete objects. We first derive a method for boundary extraction, second, introduce a method for the estimation of a smooth boundary, and third, construct an algorithm for resolution conversion.

The randomized hough transform for spherical images

We propose the algorithm for detecting great circles on images on a sphere using the Hough transform. Since our Hough transform on images on a sphere is derived on the basis of the duality, the Hough transform employs a dual sphere as the accumulator of the voting procedure. Furthermore, we propose a robust algorithm based on three-point Hough transform and the segmentation of points on the dual sphere using the metric defined on a sphere.

Tomography on Finite Graphs

Abstract This paper focuses on tomography of functions on finite graphs. Graphs are discrete topological objects, which provide mathematical tools for the analysis of circuits and networks. The main problem in tomography on a graph deals with the reconstruction of a function defined on vertices of this graph and weights of edges from summations of these values along paths and cycles on this graph. We introduce some definitions of the problems, examples and open problems for tomography on finite graphs.

Resolution conversion of gray-level images by discrete geometry

We propose a superresolution process for gray-level images based on a resolution-conversion method for discrete terrain in a space. With our resolution-conversion method, sampling a terrain and expressing it as a discrete surface allows us to estimate an original surface from a low-resolution one applying the resolution-conversion method.

PDE Based Method for Superresolution of Gray-Level Images

We propose a superresolution method for gray-level images. The method is based on resolution conversion of discrete terrains in a space by regarding gray-level images as discrete terrains. The deformation process using a discrete diffusion enables us to estimate the smooth boundary surface from a low-resolution data of discrete terrain. Furthermore, the resampling process of the estimated smooth boundary surface produces a high-resolution data of discrete terrain.