# UCSD CSE 252C - A Model of Perpendicular Texture (8 pages)

Previewing pages*1, 2, 3*of 8 page document

**View the full content.**## A Model of Perpendicular Texture

Previewing pages
*1, 2, 3*
of
actual document.

**View the full content.**View Full Document

## A Model of Perpendicular Texture

0 0 99 views

- Pages:
- 8
- School:
- University of California, San Diego
- Course:
- Cse 252c - Selected Topics in Vision and Learning

**Unformatted text preview:**

cse 252c Fall 2004 Project Report A Model of Perpendicular Texture for Determining Surface Geometry Steven Scher December 2 2004 Steven Scher SteveScher alumni princeton edu Abstract Three dimensional objects distributed over a surface occlude each other in a manner dependent upon the angle sigma between the surface normal and the viewer Leung and Malik note that occlusion allows the top of the objects to be seen but not the bottom For the case where the objects are differently colored as a function of height the color content of the scene will thus depend upon sigma A model is described that predicts the fraction of color expected to be present in each portion of the image for a given value of sigma This model can be compared to the actual color fractions in an image of unknown sigma in order to furnish an error function so that the sigma can be found using an optimization scheme A set of synthetic images of cylinders on a plane are generated and sigma is found to a mean accuracy of two degrees 1 Project Description Solution of the Shape from Texture problem infers the local surface orientation from local feature vectors extracted from the image Much work on this problem focuses on repeating 2 D patterns painted onto a surface For example 2 D texture elements e g ovals found in an image may be warped to match a template texture element e g circles and the warp used to recover the local surface normal A different approach assumes a parameterized distribution of 3 D texture elements over a surface where the texture elements vary in color with height Leung and Malik On Perpendicular Texture 1997 For example a field of flowers is modeled as a field of cylinders whose top is yellow and bottom is green This model is used to find the probability of a pixel being a given color as a function of the slant the angle between the surface tangent and the viewing 1 Figure 1 Far off flowers stems are more likely to be occluded by other flowers direction and of the parameters of the distribution Significantly the effects of perspective are incorporated into the calculation The observed proportion of a color in a region is then used to find the slant of the surface if the parameters of the distribution of elements are known If these parameters are not known they can be found by minimizing a cost function that matches the observed frequencies of the image to those calculated by the parameters This project implemented the technique described above with synthetic images akin to those used in Leung and Malik The camera was pitched down toward the ground at an unknown slant angle Each image was separated into horizontal bands and the fraction of each band occupied by flower stem and background was measured A cost function was formed measuring the dissimilarity between the observed flower fractions in each band and those predicted by a candidate slant angle The unknown slant angle was found by minimizing the cost function and was determined to an accuracy of a few degrees 2 Model The scene is modeled as a set of identical cylinders of height H and radius R randomly distributed according to a Poisson process with intensity the expected number of cylinders centered in a unit area is The color of each cylinder from the bottom to height y differs from the color of the cylinders from y to the top Because the slant angle to the ground nearby differs from the angle to the ground far away the stems of the far off flowers are more likely to be occluded than the nearby stems as in Figure 1 The scene is viewed through a camera placed above the scene with perspective projection An example image is seen in Figure 2 2 Figure 2 Synthetic Image 3 Figure 3 slant angle dependent occlusion determines the likelihood of seeing the flower or the stem 3 Probability of a Given Color Consider a cylinder painted the color of a stem everywhere but painted the color of a flower from y1 to y2 as in Figure 3 The pixel along the line of sight will be flower colored if 1 there is a flower to be seen in Region1 and 2 there is no flower to occlude it in Region2 Since we assumed a Poisson distribution the probability that there is no flower in Region2 is exp Area2 and the probability that there does exist a flower in Region1 is 1 exp Area1 Here Area2 is a rectangle whose width is the cylinders diameter and whose length is proportional to y2 the height of the top of the flower colored region and the tangent of the slant angle Area1 is similarly defined by the geometry This gives PRegion2 empty exp 2R H 1 y2 tan PRegion1 F lowerd 1 exp 2R H y2 y1 tan Thus the probability that a given pixel will be flower colored is dependent upon the slant angle Averaged over a window the probability of seeing the flowers color is approximated by the fraction of pixels having that color Given the parameters of the cylinders and their distribution the observed fraction of flower colored pixels can reveal the slant angle of the ground with respect to the camera With the cameras orientation known the scenes topography is revealed 4 Synthetic Images Synthetic images were created by randomly populating a dense grid with identical cylinders according to a Poisson Process The cylinders have height H 20 and radius R 1 and fill a rectangular area 100 units wide and 1000 units deep There are approximately 1000 cylinders with 01 The bottom Stem and top Flower of each cylinder are separated at y 75 H The scene is viewed from a camera at the edge of the rectangular area The camera has zero roll the horizon is horizontal and is pitched down so that the 4 Pstem Pground Pflower 1 1 1 0 9 0 9 0 9 0 8 0 8 0 8 0 7 0 7 0 7 0 6 0 6 0 6 0 5 0 5 0 5 0 4 0 4 0 4 0 3 0 3 0 3 0 2 0 2 0 2 0 1 0 1 0 1 0 0 50 Slant Angle 0 0 50 Slant Angle 0 0 50 Slant Angle Figure 4 Probabilities based on true 0 provide a reasonable fit to to measured values Pf lower provides a better match thatn Pstem or Pground vector from the camera to the image center makes an angle 0 with the surface normal Vectors from the camera to points above the image center make angles greater than 0 and those to points below the image center make angles less than 0 The angular field of view of the camera was set to 60 degrees The synthetic view is captured and a narrow swatch is extracted covering the full height of the image but only the center 10 of the image width This allowed a much smaller area to be populated with cylinders The image is then divided into ten …

View Full Document