Better Global Polynomial Approximation for Image Rectification

📝 Abstract

When using images to locate objects, there is the problem of correcting for distortion and misalignment in the images. An elegant way of solving this problem is to generate an error correcting function that maps points in an image to their corrected locations. We generate such a function by fitting a polynomial to a set of sample points. The objective is to identify a polynomial that passes “sufficiently close” to these points with “good” approximation of intermediate points. In the past, it has been difficult to achieve good global polynomial approximation using only sample points. We report on the development of a global polynomial approximation algorithm for solving this problem. Key Words: Polynomial approximation, interpolation, image rectification.

💡 Analysis

When using images to locate objects, there is the problem of correcting for distortion and misalignment in the images. An elegant way of solving this problem is to generate an error correcting function that maps points in an image to their corrected locations. We generate such a function by fitting a polynomial to a set of sample points. The objective is to identify a polynomial that passes “sufficiently close” to these points with “good” approximation of intermediate points. In the past, it has been difficult to achieve good global polynomial approximation using only sample points. We report on the development of a global polynomial approximation algorithm for solving this problem. Key Words: Polynomial approximation, interpolation, image rectification.

📄 Content

arXiv:0904.3944v1 [cs.CV] 24 Apr 2009 BETTER GLOBAL POLYNOMIAL APPROXIMATION FOR IMAGE RECTIFICATION∗ Christopher O. Ward Department of Mathematics and Computer Science University of West Indies (St. Augustine), Trinidad & Tobago West Indies christopher.ward@sta.uwi.edu June 24, 2018 Abstract When using images to locate objects, there is the problem of correcting for dis- tortion and misalignment in the images. An elegant way of solving this problem is to generate an error correcting function that maps points in an image to their corrected locations. We generate such a function by fitting a polynomial to a set of sample points. The objective is to identify a polynomial that passes “sufficiently close” to these points with “good” approximation of intermediate points. In the past, it has been difficult to achieve good global polynomial approximation using only sample points. We report on the development of a global polynomial approx- imation algorithm for solving this problem. Key Words: Polynomial approximation, interpolation, image rectification. 1 Introduction The problem that is addressed here occurred in the context of the development of a simple, low-cost robotic exhibit for demonstrating the concept of intelligent robotics to the general public. Intelligent robotics deals with the use of sensors to enhance a robots performance in an uncertain environment. For the exhibit, we implemented a visually- guided pick-and-place robot. The robot uses an image to determine the location of objects placed arbitrarily on a flat surface and demonstrates success in locating the objects by manipulating them. The exhibit consists of a robotic arm that is fixed in front of a flat work surface. Two pedestals are placed anywhere within reach of the ∗The original Paper entitled Better Global Polynomial Approximation for Image Rectification, was pub- lished in the International Journal of Modelling and Simulation, Vol. 28, No. 3, 2008, pp 299-308. 1 arm. One pedestal is blue and the other is green. A blue ball is placed on the blue pedestal and the robot must pick up the ball and place it on the green pedestal. An ordinary webcam is placed in a frame above the work surface and is used to determine the location of the pedestals so that the arm can be guided accordingly. Fig. 1(a) shows the camera’s view of the workspace. Once the exhibit has been set up, camera, robot and workspace are all fixed relative to one another. (a) unrectified image (b) rectified image Figure 1: Rectification of an image of a test pattern placed over the robot’s workspace taken from a severely misaligned camera. A grid has been superimposed on the images to aid in comparing the horizontal and vertical alignment of image features. Although the original images are colour coded, the images shown are monochrome. If the upper left dot and the upper right dot are ignored, the rest of the dots in (b) sample the area of the workspace that is within reach of the robot. There is therefore the problem of determining the location of the colour-coded pedestals based on their image. This problem is compounded by the fact that the we- bcam produces significant image distortion, and by the fact that the position and ori- entation of the camera relative to the work surface may not be exactly the same every time the exhibit is set up. A similar problem occurs when any automated mechanism is taken apart for maintenance. The mechanism usually must be recalibrated when it is reassembled. In this sense, we are addressing the problem of easy recalibration of the vision component of our robotic exhibit. In processing the image, there is a need to rectify significant image distortion caused by the optical properties of the camera and by errors in positioning the cam- era during set up of the exhibit. We solve this problem by determining a mapping from the pixel locations of image points to the physical locations of corresponding source points. The mapping has to be determined empirically in order to recalibrate the vision system each time the apparatus is set up. The pixel positions of the images of key points in a test pattern are mapped to the known locations of these key points. This partial mapping is then used to approximate a mapping for the entire image. In [1], Brown surveyed and classified several established methods for determining the mapping from pixel position to source location for a camera. Using Brown’s clas- sification, the errors caused by distortion and misalignment are static in the sense that they do not change from image to image taken with the same camera in the same posi- tion. Static distortions can be rectified in a one-time-only setup process via calibration techniques. Our method may be regarded as a calibration technique. Brown classifies the distortion due to camera properties as internal and the mis- 2 alignment as external. Our approach does not require the use of any sort of model of the characteristics of the camera or the geometry of how an image is captu

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