Texture Mapping Outline • Modeling surface details with images. Roger Crawfis • Texture parameterization Ohio State University • Texture evaluation • Anti-aliasing and textures. Texture Mapping Texture Mapping • Why use textures? • Modeling complexity Quote Texture Mapping “I am interested in the effects on an object that speak of • Given an object and an image: human intervention. This is another factor that you must take into consideration. How many times has the object – How does the image map to the vertices or set been painted? Written on? Treated? Bumped into? of points defining the object? Scraped? This is when things get exciting. I am curious about: the wearing away of paint on steps from continual use; scrapes made by a moving dolly along the baseboard of a wall; acrylic paint peeling away from a previous coat of an oil base paint; cigarette burns on tile or wood floors; chewing gum – the black spots on city sidewalks; lover’s names and initials scratched onto park benches…” -Owen Demers [digital] Texturing & Painting, 2002 Jason Bryan Texture Mapping Texture Mapping Jason Bryan Jason Bryan Texture Mapping Texture Mapping Jason Bryan Jason Bryan Texture Mapping Texture Mapping • Given an object and an image: • Given an object with an image mapped to it: – How does the image map to the vertices or set – How do we use the color information from the of points defining the object? texture image to determine a pixel’s color? Texture Mapping Texture Mapping • Problem #1 Fitting a square peg in a round • Problem #2 Mapping from a pixel to a texel hole What is an image? What is a Texture? • How would I rotate an image 45 degrees? • Given the (u,v), want: • How would I translate it 0.5 pixels? – F(u,v) ==> a continuous reconstruction • = { R(u,v), G(u,v), B(u,v) } • = { I(u,v) } • = { index(u,v) } • = { alpha(u,v) } • = { normals(u,v) } • = { surface_height(u,v) } • = ... What is the source of your Texture? Procedural Texture • Procedural Image Periodic and everything else • RGB Image Checkerboard • Intensity image Scale: s= 10 • Opacity table If (u * s) % 2=0 && (v * s)%2=0 texture(u,v) = 0; // black Else texture(u,v) = 1; // white RGB Textures Intensity Modulation Textures • Places an image on the object • Multiply the objects color by that of the texture. Camuto 1998 Opacity Textures Color Index Textures • A binary mask, really redefines the • New Microsoft Extension for 8-bit textures. geometry. • Also some cool new extensions to SGI’s OpenGL to perform table look-ups after the texture samples have been computed. Lao 1998 Bump Mapping Displacement Mapping • This modifies the surface normals. • Modifies the surface position in the • More on this later. direction of the surface normal. Lao 1998 Lao 1998 Reflection Properties Texture and Texel • Each pixel in a texture map is called a Texel • Kd, Ks • Each Texel is associated with a 2D, (u,v), • BDRF’s texture coordinate – Brushed Aluminum • The range of u, v is [0.0,1.0] – Tweed – Non-isotropic or anisotropic surface micro facets. (u,v) tuple Two-Stage Mapping • For any (u,v) in the range of (0-1, 0-1), we 1. Model the mapping: (x,y,z) -> (u,v) can find the corresponding value in the 2. Do the mapping texture using some interpolation Image space scan Image space scan For each scanline, y • Problems: For each pixel, x – Finding the inverse mapping compute u(x,y) and v(x,y) • Use one of the analytical mappings that are copy texture(u,v) to image(x,y) invertable. • Bi-linear or triangle inverse mapping • Samples the warped texture at the – May miss parts of the texture map appropriate image pixels. Image Texture • inverse mapping Texture space scan Texture space scan For each v • Problems: For each u – May not fill image compute x(u,v) and y(u,v) – Forward mapping needed copy texture(u,v) to image(x,y) • Places each texture sample to the mapped image pixel. Texture Image • forward mapping Continuous functions F(u,v) How do we get F(u,v)? • We are given a discrete set of values: • Higher-order interpolation – F[i,j] for i=0,…,N, j=0,…,M – F(u,v) = ∑ i∑j F[i,j] h(u,v) • Nearest neighbor: – h(u,v) is called the reconstruction kernel – F(u,v) = F[ round(N*u), round(M*v) ] • Guassian • Sinc function • Linear Interpolation: • splines – i = floor(N*u), j = floor(M*v) – Like linear interpolation, need to find – interpolate from F[i,j], F[i+1,j], F[i,j+1], neighbors. F[i+1,j+1] • Usually four to sixteen Texture Parameterization Texture Parameterization • Definition: • Mapping to a 3D Plane – The process of assigning texture coordinates or – Simple Affine transformation a texture mapping to an object. • rotate • scale • The mapping can be applied: y • translate – Per-pixel v z – Per-vertex x u Texture Parameterization Texture Parameterization • Mapping to a Cylinder • Mapping to Sphere – Rotate, translate and scale in the uv-plane – Impossible!!!! – u -> theta – Severe distortion at the poles – v -> z – u -> theta – v -> phi – x = r cos(theta), y = r sin(theta) v – x = r sin(theta) cos(phi) – y = r sin(theta) sin(phi) – z = r cos(theta) u Texture Parameterization Example (Rogers) Part of a sphere x(θ,φ) = sin θ sin φ 0 ≤θ≤π/2 y(θ,φ) = cos φ • Mapping to a Sphere π/4 ≤φ≤π/2 z(θ,φ) = cos θ sin φ (u,v) = (0,0) ⇔ (θ,φ) = (0, π/2) (u,v) = (1,0) ⇔ (θ,φ) = (π/2, π/2) v (u,v) = (0,1) ⇔ (θ,φ) = (0, π/4) (u,v) = (1,1) ⇔ (θ,φ) = (π/2, π/4) • Setup up surface, define correspondence, and voila! u Example Continued All Is Not Good • Can even solve for (θ,φ) and (u,v) • Let’s take a closer look: – A= π/2, B=0, C=-π/4, D=π/2 π π π θ (u,v) = u φ(u,v) = − v 2 2 4 θ π −φ u(θ ,φ) = v(θ ,φ) = 2 π π 2 4 So looks like we have the texture space ⇔ object space part done! Started with squares and ended with curves / It only gets worse for larger parts of the sphere Texture Parameterization Two-pass Mappings v • Mapping to a Cube • Map texture to: – Plane – Cylinder u – Sphere u-axis v – Box common seam • Map object to same. u S and O Mapping S Mapping Example • Pre-distort the texture by mapping it onto a • Cylindrical Mapping simple surface like a plane, cylinder, sphere, or box • Map the result of that onto the surface • Texture → Intermediate is S mapping • Intermediate → Object is O mapping S T (u,v) (xi,yi)(xo,yo,zo) θ −θ z − z s = A t = A Texture space Intermediate space Object space θ B −θ A zB − z A O Mapping O Mappings Cont’d • A method to relate the surface to the cylinder • Bier and Sloan defined 4 main ways Reflected ray Object normal or or Object centroid Intermediate surface normal Texture Parameterization Texture Parameterization • Plane/ISN (projector) • Plane/ISN – Works well for planar objects • Cylinder/ISN (shrink-wrap) – Works well for solids of revolution • Box/ISN Works well for roughly • Sphere/Centroid spherical shapes • Box/Centroid Texture Parameterization Texture Parameterization • Plane/ISN • Plane/ISN – Resembles a slide – Draw vector from point (vertex or object space projector pixel point) in the direction of the texture plane. – Distortions on surfaces perpendicular to the – The vector will intersect plane. the plane at some point depending on the coordinate system Watt Texture Parameterization Texture Parameterization • Cylinder/ISN • Sphere/ISN – Distortions on – Small distortion horizontal planes everywhere. – Draw vector from – Draw vector from point to cylinder sphere center through – Vector connects point point on the surface to cylinder axis and intersect it with the sphere. Watt Watt Texture Parameterization Texture Parameterization • What is this ISN? • Flip direction of – Intermediate surface vector such that it normal. points in the same – Needed to handle half-space as the concave objects outward surface properly. normal. – Sudden flip in texture coordinates when the object crosses the axis. Triangle Mapping Triangle Mapping • Given: a triangle with texture coordinates at • Given: a triangle with texture coordinates at each vertex. each vertex. • Find the texture coordinates at each point • Find the texture coordinates at each point within the triangle. within the triangle. U=1,v=1 U=0,v=1 U=0,v=0 U=1,v=0 U=0,v=0 U=1,v=0 Triangle Mapping Triangle Interpolation • Triangles define linear mappings. • The equation: f(x,y) = Ax+By+C defines a linear • u(x,y,z) = Ax + By + Cz + D function in 2D. • v(x,y,z) = Ex + Fy + Gz + H • Knowing the values of f() at three locations gives us • Plug in the each point and corresponding enough information texture coordinate. to solve for A, B • Three equations and three unknowns and C. • Need to handle special cases: u==u(x,y) or • Provided the triangle v==v(x), etc. lies in the xy-plane. Triangle Interpolation Triangle Interpolation • We need to find two 3D functions: u(x,y,z) • A linear function in 3D is defined as and v(x,y,z). – f(x,y,z) = Ax + By + Cz + D • However, there is a relationship between x, • Note, four points uniquely determine this y and z, so they are not independent. equation, hence a tetrahedron has a unique • The plane equation of the triangle yields: linear function through it.