cse 252c Fall 2004 Project Report:A Model of Perpendicular Texture forDetermining Surface GeometrySteven ScherDecember 2, 2004Steven [email protected] objects distributed over a surface occlude each otherin a manner dependent upon the angle (sigma) between the surface nor-mal and the viewer. Leung and Malik note that occlusion allows the topof the objects to be seen, but not the bottom. For the case where theobjects are differently colored as a function of height, the color content ofthe scene will thus depend upon sigma. A model is described that predictsthe 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 actualcolor fractions in an image of unknown sigma, in order to furnish an errorfunction, so that the sigma can be found using an optimization scheme.A set of synthetic images of cylinders on a plane are generated, and sigmais found to a mean accuracy of two degrees.1 Project DescriptionSolution of the Shape from Texture problem infers the local surface orientationfrom local feature vectors extracted from the image. Much work on this problemfocuses on repeating 2-D patterns, painted onto a surface. For example, 2-Dtexture elements (e.g. ovals) found in an image may be warped to match atemplate texture element (e.g. circles), and the warp used to recover the localsurface normal.A different approach assumes a parameterized distribution of 3-D textureelements over a surface, where the texture elements vary in color with height[Leung and Malik, On Perpendicular Texture, 1997]. For example, a field offlowers is modeled as a field of cylinders whose top is yellow, and bottom isgreen. 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 viewing1Figure 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 ofperspective are incorporated into the calculation.The observed proportion of a color in a region is then used to find the slantof 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 costfunction that matches the observed frequencies of the image to those calculatedby the parameters.This project implemented the technique described above with synthetic im-ages akin to those used in Leung and Malik. The camera was pitched downtoward the ground at an unknown slant angle. Each image was separated intohorizontal bands, and the fraction of each band occupied by flower, stem, andbackground was measured. A cost function was formed measuring the dissimi-larity between the observed flower fractions in each band, and those predictedby a candidate slant angle. The unknown slant angle was found by minimizingthe cost function, and was determined to an accuracy of a few degrees.2 ModelThe scene is modeled as a set of identical cylinders of height H and radiusR, randomly distributed according to a Poisson process with intensity λ (theexpected number of cylinders centered in a unit area is λ). The color of eachcylinder from the bottom to height y differs from the color of the cylinders fromy to the top.Because the slant angle to the ground nearby differs from the angle to theground far away, the stems of the far-off flowers are more likely to be occludedthan the nearby stems, as in Figure 1.The scene is viewed through a camera placed above the scene, with perspec-tive projection. An example image is seen in Figure 2.2Figure 2: Synthetic Image3Figure 3: slant angle-dependent occlusion determines the likelihood of seeingthe flower or the stem3 Probability of a Given ColorConsider a cylinder painted the color of a stem everywhere, but painted thecolor of a flower from y1to y2, as in Figure 3. The pixel along the line of sightwill be flower-colored if (1) there is a flower to be seen, in Region1, and (2) thereis no flower to occlude it, in Region2. Since we assumed a Poisson distribution,the probability that there is no flower in Region2is exp(−λ ∗ Area2) and theprobability that there does exist a flower in Region1is 1 − exp(−λ ∗ Area1).Here, Area2is a rectangle whose width is the cylinders diameter, and whoselength is proportional to y2, the height of the top of the flower-colored region,and the tangent of the slant angle σ. Area1is similarly defined by the geometry.This gives:PRegion2empty= exp(−λ ∗ 2R ∗ H(1 − y2) ∗ tan(σ))PRegion1F lowerd= 1 − exp(−λ ∗ 2R ∗ H(y2 − y1) ∗ tan(σ))Thus, the probability that a given pixel will be flower-colored is dependentupon the slant angle. Averaged over a window, the probability of seeing theflowers color is approximated by the fraction of pixels having that color. Giventhe parameters of the cylinders and their distribution, the observed fractionof flower-colored pixels can reveal the slant angle of the ground with respectto the camera. With the cameras orientation known, the scenes topography isrevealed.4 Synthetic ImagesSynthetic images were created by randomly populating a dense grid with iden-tical cylinders, according to a Poisson Process. The cylinders have height H=20and radius R=1, and fill a rectangular area 100 units wide and 1000 units deep.There are approximately 1000 cylinders, with λ=.01. The bottom (Stem) andtop (Flower) of each cylinder are separated at y=.75*H.The scene is viewed from a camera at the edge of the rectangular area. Thecamera has zero roll (the horizon is horizontal) and is pitched down so that the40 5000.10.20.30.40.50.60.70.80.91Slant AnglePstem0 5000.10.20.30.40.50.60.70.80.91Slant AnglePground0 5000.10.20.30.40.50.60.70.80.91Slant AnglePflowerFigure 4: Probabilities based on true σ0provide a reasonable fit to to measuredvalues. Pf lowerprovides a better match thatn Pstemor Pgroundvector from the camera to the image center makes an angle σ0with the surfacenormal. Vectors from the camera to points above the image center make anglesgreater than σ0, and those to points below the image center make angles lessthan σ0. The angular field of view of the camera was set to 60 degrees.The synthetic view is captured, and a narrow swatch is extracted, coveringthe full height of the image, but only the center 10% of the image width. Thisallowed a much smaller area to be populated with cylinders.