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## CS 445 / 645 Introduction to Computer Graphics

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**CS 445 / 645Introduction to Computer Graphics**Lecture 18 Shading**Ray Tracing Assignment**• Deadline extended until Tuesday the 13th**Ray Tracing Assignment**• Barycentric coordinates (Möbius, 1827) • Consider a triangle defined by (A1, A2, A3) • These points are defined relative to world origin • A point within triangle could also be defined as (x, y, z) relative to world origin • A point can be defined as (t1, t2, t3) corresponding to its position with respect to A1, A2, A3.**Barycentric Coordinates**• Solve for (t1, t2, t3) such that • t1 + t2 + t3 = 1 • t1A1 + t2A2 + t3A3 = P mathworld.com**Barycentric Coordinates**• An observation • t1, t2, and t3 are weights such that when they are used to represent the mass at the vertices of the triangle, P is at its center of mass • t1, t2, and t3 are weights that represent the ratio of the area of each of the three subtriangles to the area of the whole**Barycentric coordinates**• The cross product computes a triangle’s area • || (A2 - A1) x (A3-A1) || = (area of A1A2A3)* 2 Where || x || = the area of the triangle x… the cross product**Barycentric Coordinates**• All points of triangle are unique, all points in space can be represented with barycentric coordinates • A1, A2, and A3 form an affine space • If 0 ≤ t1, t2, t3 ≤ 1, the point is in the triangle**Applying Illumination**• We have an illumination model for a point on a surface • Assuming that our surface is defined as a mesh of polygonal facets, which pointsshould we use? • Keep in mind: • It’s a fairly expensive calculation • Several possible answers, each with different implications for the visual quality of the result**Applying Illumination**• With polygonal/triangular models: • Each facet has a constant surface normal • If the light is directional, the diffuse reflectance is constant across the facet. Why?**Flat Shading**• The simplest approach, flat shading, calculates illumination at a single point for each polygon: • If an object really is faceted, is this accurate?**For specular reflectance, direction to eye varies across**the facet Is flat shading realistic for faceted object? • No: • For point sources, the direction to light varies across the facet**Flat Shading**• We can refine it a bit by evaluating the Phong lighting model at each pixel of each polygon, but the result is still clearly faceted:**Vertex Normals**• To get smoother-looking surfaceswe introduce vertex normals at eachvertex • Usually different from facet normal • Used onlyfor shading • Think of as a better approximation of therealsurface that the polygons approximate**Vertex Normals**• Vertex normals may be • Provided with the model • Computed from first principles • Approximated by averaging the normals of the facets that share the vertex**Gouraud Shading**• This is the most common approach • Perform Phong lighting at the vertices • Linearly interpolate the resulting colors over faces • Along edges • Along scanlines c1 + t1(c2-c1) C1 • This is what OpenGL does • Does this eliminate the facets? C3 C2 c1 + t1(c2-c1) + t3(c1 + t2(c3-c1)- c1 + t1(c2-c1)) c1 + t2(c3-c1)**C1**C3 C2 Can’t shade that effect! Gouraud Shading • Artifacts • Often appears dull, chalky • Lacks accurate specular component • If included, will be averaged over entire polygon**Gouraud Shading**• Artifacts • Mach Banding • Artifact at discontinuities in intensity or intensity slope C1 C4 C3 C2 Discontinuity in rateof color changeoccurs here http://www.edcenter.sdsu.edu/slides/GA/visteacher/sld048.htm**Phong Shading**• Phong shading is not the same as Phong lighting, though they are sometimes mixed up • Phong lighting: the empirical model we’ve been discussing to calculate illumination at a point on a surface • Phong shading: linearly interpolating the surface normal across the facet, applying the Phong lighting model at every pixel • Same input as Gouraud shading • Usually very smooth-looking results: • But, considerably more expensive**Phong Shading**• Linearly interpolate the vertex normals • Compute lighting equations at each pixel • Can use specular component N1 N4 Remember: Normals used in diffuse and specular terms Discontinuity in normal’s rate of change is harder to detect N3 N2**Shortcomings of Shading**• Polygonal silhouettes remain Gouraud Phong**i**i i i i i Perspective Distortion Linear interpolationin image space isnot accuratesampling of 3Dspace Imageplane Break up large polygonswith many smaller ones Polygon u u u u u u Z – into the scene Notice that linear interpolation in screen spacedoes not align with linear interpolation in world space Nonlinear color shift fromblue (left) to red (right)**Perspective Distortion**Imageplane Break up large polygonswith many smaller ones Z – into the scene Notice that linear interpolation in screen spacedoes not align with linear interpolation in world space**A**B i i C B A D D C Interpolate betweenCD and AD Interpolate betweenAB and AD Interpolation dependent on polygon orientation Rotate -90oand colorsame point**Problems at Shared Vertices**Vertex B is shared by the two rectangles on the right, but not by the one on the left C H D The first portion of the scanlineis interpolated between DE and ACThe second portion of the scanlineis interpolated between BC and GHA large discontinuity could arise B G F E A**Shading Models (Direct lighting)**• Flat Shading • Compute Phong lighting once for entire polygon • Gouraud Shading • Compute Phong lighting at the vertices and interpolate lighting values across polygon • Phong Shading • Interpolate normals across polygon and perform Phong lighting across polygon