AN INTEGRAL PROJECTION APPROACH TO 3D RIGID BODY TRANSFORMATIONS Stefan Lehmann, I. Vaughan L. Clarkson, and Peter J. Kootsookos Intelligent Real-Time Imaging and Sensing Group School of Information Technology and Electrical Engineering The University of Queensland 4072 AUSTRALIA (lehmann,v.clarkson,kootsoop)@itee.uq.edu.au ABSTRACT The analysis of 3D rigid body transformations based on camera images is of great significance in many research ar- eas. Classical methods to recover 3D object information often use a set of cameras that cover a static scene from different view angles. Key areas in this context include Structure from Motion, Motion from Structure and Cam- era Calibration. This paper introduces a new approach to analyse 3D rigid body transformations which we call inte- gral projection. In this model, we are able to use frequency- domain information to estimate parameters of the transfor- mation. Simulations are presented which demonstrate our initial successes. 1. INTRODUCTION One of the most challenging research areas in computer vi- sion is to gain 3D information about an object from cam- era images of this object. In the case of a static object, at least two cameras with different view angles are required to compute depth maps that enable back-projection of the 2D pixels, i.e., the transformation of a 2D pixel into the cor- responding 3D voxel of the object [1]. On the other hand, for dynamic scenes, the use of one camera only allows the acquisition of 3D information, provided that at least two frames of an image sequence from that camera are being processed. Luong and Faugeras, amongst many others, have shown how 3D structure and motion information can be es- timated based on point correspondences [2]. Gaining structural information about the object or scene based on rigid body transformations of a 3D object or scene is known as Structure from Motion. A variety of approaches have been proposed in this context [3, 4, 5]. An overview of different methods can be found in [6]. On the other hand, the aim of Motion from Structure is to extract 3D motion parameters from camera images. In the dynamic case, a sequence of images from one camera can be analysed to determine the 3D motion parameters of a moving object. In the static case, the images of a fixed scene from multiple cameras can be used for camera calibration. Finding the depth information of the image pixels relies on an accurate determination of rigid body transformations, i.e., the relative rotation and translation of the 3D object with respect to the camera. The camera model determines the relationship between the 3D voxels of the object and the corresponding 2D pixels in the image plane. In computer vision, the most commonly used projection models are ei- ther parallel or perspective projections. Transformations in the space domain that are composed of rotations and translations correspond to pure rotations of the spectral magnitudes in the frequency domain [7]. A spa- tial translation yields a phase shift in the frequency domain. However, common projection techniques such as parallel projection are non-linear transformations between the 3D voxel model and the 2D image model. As a consequence, the transformation between the two Fourier spectra of the images that result from the projections of the original and the transformed object respectively is not straightforward. We propose an integral projection model that is a linear operation and establishes a straightforward relationship be- tween the two image spectra. There is a correspondence between parallel and integral projection which will be dis- cussed in Section 2.1. Section 2 introduces our model and shows its relevance for rigid body transformations. An algorithm for estimating transformation parameters is presented in Section 3. Initial experimental results are shown in Section 4. We conclude with an outlook on future research. 2. INTEGRAL PROJECTION 2.1. Concept and relationship to parallel projection We will illustrate the integral projection scheme by project- ing a simple 2D object into a 1D projection function. The integral projection model determines the 1D projection val- ues by integrating the 2D object along lines that run parallel to the view axis. This model comprises two simplifications: use of integration to perform the projection and use of par- allel, rather than fan or perspective, projection.