Texture Mapping

Texture Mapping Aaron Bloomfield CS 445: Introduction to Graphics Fall 2006 (Slide set originally by David Luebke)

Overview • Motivation and Examples • Fundamentals • Algorithms • Perspective-Correct Texturing • Transparency and Anti-Aliasing • MIP-Maps • Bump and Displacement Maps • Illumination Maps

Texture Mapping: Motivation • Scenes created with diffuse lighting look convincingly three-dimensional, but are flat, chalky, and “cartoonish” • Phong lighting lets us simulate materials like plastic and (to a lesser extent) metal, but scenes still seem very cartoonish and unreal • Big problem: polygons are too coarse-grained to usefully model fine surface detail • Solution: texture mapping

Texture Mapping: Motivation • Adding surface detail helps keep CG images from looking simple and sterile • Explicitly modeling this detail in geometry can be very expensive • Zebra stripes, wood grain, writing on a whiteboard • Texture mapping pastes images onto the surfaces in the scene, adding realistic fine detail without exploding the geometry

Texture Mapping: Examples

Texture Mapping: Examples Doom III (ID Software)

Texture Mapping • In short: it is impractical to explicitly model fine surface detail with geometry • Solution: use images to capture the “texture” of surfaces • Texture maps can modulate many factors that affect the rendering of a surface • Color or reflectance (diffuse, ambient, specular) • Transparency (smoke effects) • What else?

Texture Mapping: Fundamentals • A texture is typically a 2-D image • Can also be • A 2-D procedural texture • A 3-D procedural texture • See video • A 3-D volumetric texture (rarely used)

Texture Mapping: Fundamentals • A texture is typically a 2-D image • Image elements are called texels • Value stored at a texel affects surface appearance in some way • Example: diffuse reflectance, shininess, transparency… • The mapping of the texture to the surface determines the correspondence, i.e., how the texture lies on the surface • Mapping a texture to a triangle is easy (why?) • Mapping a texture to an arbitrary 3-D shape is more complicated (why?)

Texturing Fundamentals • A texture is typically a 2-D array of texels • Mapping the texture to an arbitrary 3-D shape is complex:

Texturing Fundamentals • A texture is typically a 2-D array of texels • Mapping the texture to an arbitrary 3-D shape is complex! http://www.cs.virginia.edu/~gfx/Courses/2003/Intro.spring.03/animations/unfold.mov

How to map a texture • There are a number of possibilities: • Flat • Cube • Tube (i.e. cylinder) • Sphere • The slides on the next slide are from http://mediawiki.blender.org/index.php/Manual/Map_Input#2D_to_3D_Mapping

Texture Mapping: Rendering • Rendering uses the mapping: • Find the visible surface at a pixel • Find the point on that surface corresponding to that pixel • Find the point in the texture corresponding to that point on the surface • Use the parameters associated with that point on the texture to shade the pixel • Using triangulated meshes reduces the problem to mapping a portion of the image to each triangle:

Texture Mapping: Rendering

Texture Mapping:User-Generated Mappings • For complex 3-D objects, mapping textures is still something of an art…so we often let the user do it

Texture Mapping: Rendering • We typically parameterize the texture as a function in (u, v) • For simplicity, normalize u & v to [0, 1] • Associate each triangle with a texture • Give each vertex of the triangle a texture coordinate (u, v) • For other points on the triangle, interpolate texture coordinate from the vertices • Much like interpolating color or depth • But there’s a catch...

Naïve Texture Mapping • A first cut at a texture-mapping rasterizer: • For each pixel: • Interpolate u & v down edges and across spans • Look up nearest texel in texture map • Color pixel according to texel color (possibly modulated by lighting calculations) • McMillan’s demo of this is at http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide05.html • What artifacts do you see in this demo?

Naïve Texturing Artifacts • Another serious artifact is warping at the edges of triangles making up the mesh • A more obvious example:http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide06.html • To address this, need to consider the geometry of interpolating parameters more carefully

Interpolating Parameters • The problem turns out to be fundamental to interpolating parameters in screen-space • Uniform steps in screen space  uniform steps in world coords

Interpolating Parameters • Perspective foreshortening is not getting applied to our interpolated parameters • Parameters should be compressed with distance • Linearly interpolating them in screen-space doesn’t do this • Is this a problem with Gouraud shading? • Is everything I taught wrong? • A: It can be, but we usually don’t notice (why?)

Perspective-Correct Interpolation • Skipping a bit of math to make a long story short… • Rather than interpolating u and v directly, interpolate u/z and v/z • These do interpolate correctly in screen space • Also need to interpolate z and multiply per-pixel • Problem: we don’t know z anymore • Solution: we do know w  1/z • So…interpolate uw and vw and w, and compute u = uw/w and v = vw/w for each pixel • This unfortunately involves a divide per pixel (Just 1?)

Perspective-Correct Texturing • Known as perspective-correct texture mapping • Early PC cards and game consoles didn’t support it • So how did they avoid the warping problem? • http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide15.html • As mentioned, other interpolation schemes really ought to use perspective correction • E.g., Gouraud shading • Generally get away without it because it is more important to be smooth than correct • Java code fragment from McMillan’s edge-equation triangle rasterizer:

Perspective-Correct Texturing: Code ... PlaneEqn(uPlane, (u0*w0), (u1*w1), (u2*w2)); PlaneEqn(vPlane, (v0*w0), (v1*w1), (v2*w2)); PlaneEqn(wPlane, w0, w1, w2); ... for (y = yMin; y <= yMax; y += raster.width) { e0 = t0; e1 = t1; e2 = t2; u = tu; v = tv; w = tw; z = tz; boolean beenInside = false; for (x = xMin; x <= xMax; x++) { if ((e0 >= 0) && (e1 >= 0) && (e2 >= 0))) { int iz = (int) z; if (iz <= raster.zbuff[y+x]) { float denom = 1.0f / w; int uval = (int) (u * denom + 0.5f); uval = tile(uval, texture.width); int vval = (int) (v * denom + 0.5f); vval = tile(vval, texture.height); int pix = texture.getPixel(uval, vval); if ((pix & 0xff000000) != 0) { raster.pixel[y+x] = pix; raster.zbuff[y+x] = iz; } } beenInside = true; } else if (beenInside) break; e0 += A0; e1 += A1; e2 += A2; z += Az; u += Au; v += Av; w += Aw; } t0 += B0; t1 += B1; t2 += B2; tz += Bz; tu += Bu; tv += Bv; tw += Bw; }

Texture Tiling • It is often handy to tile a repeating texture pattern onto a surface • The previous code does this via tile(): int uval = (int) (u * denom + 0.5f); uval = tile(uval, texture.width); int vval = (int) (v * denom + 0.5f); vval = tile(vval, texture.height); int pix = texture.getPixel(uval, vval); int tile(int val, int size) { while (val >= size) val -= size; while (val < 0) val += size; } See http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide18.html

Texture Transparency • McMillan’s code also includes a “quick fix” for handling transparent texture: if ((pix & 0xff000000) != 0) { raster.pixel[y+x] = pix; raster.zbuff[y+x] = iz; } • Note that this doesn’t handle partial transparency (How might such partial transparency arise?) • Demo at: http://graphics.lcs.mit.edu/classes/6.837/F98/Lecture21/Slide19.html

Texture Map Aliasing • Naive texture mapping looks blocky, pixelated • Problem: using a single texel to color each pixel: int uval = (int) (u * denom + 0.5f); int vval = (int) (v * denom + 0.5f); int pix = texture.getPixel(uval, vval); • Actually, each pixel maps to a region in texture • If the pixel is larger than a texel, we should average the contribution from multiple texels somehow • If the pixel is smaller than a texel, we should interpolate between texel values somehow • Even if pixel size  texel size, a pixel will in general fall between four texels • An example of a general problem called aliasing

Texture Map Antialiasing • Use bilinear interpolation to average nearby texel values into a single pixel value (Draw it) • Find 4 nearest texture samples • Round u & v up and down • Interpolate texel values in u • Interpolate resulting values in v • Also addresses the problem of many pixels projecting to a single texel (Why?)

Texture Map Antialiasing • What if a single pixel covers many texels? • Problem: sampling those texels at a single point (the center of the pixel): • Produces Moire patterns in coherent texture (checkers) • Leads to flicker or texture crawling as the texture moves

Moire Patterns

Texture Map Antialiasing • What if a single pixel covers many texels? • Approach: blur the image under the pixel, averaging the contributions of the covered texels • But calculating which texels every pixel covers is way too expensive, especially as the texture is compressed • Solution: pre-calculate lower-resolution versions of the texture that incorporate this averaging

Original Texture Lower Resolution Versions MIP-maps • For a texture of 2n x 2n pixels, compute n-1 textures, each at ½ the resolution of previous: • This multiresolution texture is called a MIP-map

Generating MIP-maps • Generating a MIP-map from a texture is easy • For each texel in level i, average the values of the four corresponding texels in level i-1 • If a texture requires n bytes of storage, how much storage will a MIP-map require? • Answer: 4n/3

R G R G B R G B R G B B Representing MIP-maps Trivia: MIP = Multum In Parvo(many things in a small place)

Using MIP-maps • Each level of the MIP-map represents a pre-blurred version of multiple texels • A texel at level n represents 2n original texels • When rendering: • Figure out the texture coverage of the pixel (i.e., the size of the pixel in texels of the original map) • Find the level of the MIP map in which texels average approximately that many original texels • Interpolate the value of the four nearest texels

Using MIP-maps • Even better: • Likely, the coverage of the pixel will fall somewhere between the coverage of texels in two adjacent levels of the MIP map • Find the pixel’s value in each of the two textures using two bilinear interpolations • Using a third interpolation, find a value in between these two values, based on the coverage of the pixel versus each of the MIP-map levels • This is (misleadingly?) called trilinear interpolation

MIP-map Example • No filtering: • MIP-map texturing:

Can We Do Better? • What assumption does MIP-mapping implicitly make? • A: The pixel covers a square region of the texture • More exactly, the compression or oversampling rate is the same in uandv • Is this a valid assumption? Why or why not?

MIP-maps and Signal Processing • An aside: aliasing and antialiasing are properly topics in sampling theory • Nyquist theorem, convolution and reconstruction, filters and filter widths • Textures are particularly difficult because a tiled texture can easily generate infinite frequencies • E.g., a checkered plane receding to an infinite horizon • Using a MIP-map amounts to prefiltering the texture image to reduce artifacts caused by sampling at too low a rate

Summed-Area Tables • A technique called summed-area tableslets us integrate texels covered by the pixel more exactly (but still quickly) • Details in the book • Example: MIP-map texturing Summed-area table texturing

Texture Mapping

Texture Mapping

Presentation Transcript

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping

Texture Mapping