Texture Mapping with Bilinear Filtering

Topic Overview

UV Coordinates on Triangle Surfaces

Texture mapping applies a 2D image to a 3D surface. Edwin Catmull introduced the idea in 1974; today every real-time renderer depends on it. Each vertex carries texture coordinates (u, v) in the range 0 to 1. During rasterization, UVs interpolate across the triangle (with perspective correction) and each fragment looks up a texel color.

Texture mapping decouples surface detail from geometry. A flat triangle can look like brick, wood, or skin without extra vertices.

Nearest vs Bilinear Sampling

Nearest-neighbor rounds (u, v) to the closest texel. Fast but blocky when magnified.

Bilinear filtering blends four neighboring texels:

cCopy
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For example, sampling (u=0.375, v=0.625) on a 4x4 texture blends texels at (1,2), (2,2), (1,3), and (2,3). Mipmaps (Lance Williams, 1983) add precomputed smaller versions to fix minification aliasing when distant textures shrink to a few pixels.

• Nearest: one texel lookup, sharp but pixelated
• Bilinear: four lookups + three lerps, smoother
• Trilinear: bilinear across two mipmap levels

For This Exercise

You will sample a 4x4 gradient texture using nearest-neighbor and bilinear filtering at test UV coordinates. This task requires you to print both results so the quality difference is visible. Understanding these two modes is essential for software renderers, image resizers, and any GLSL texture() call where you choose sampler settings.

Your Task

Implement a GPU texture sampler for a small grayscale texture:

• nearest and bilinear filtering;
• the repeat, clamp-to-edge and mirrored-repeat wrap modes.

Texture coordinates arrive in normalized units (0..1 across the texture). Converting them to texel space correctly, with texel centres at half-integers, is where most sampler bugs live. Everything is fixed-point integer arithmetic, so results are exact.

Input

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cCopy
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Rules

• Wrap (for index i in a dimension of size n):
  • repeat: ((i mod n) + n) mod n;
  • clamp: min(max(i, 0), n-1);
  • mirror: p = i mod 2n (made non-negative), then p < n ? p : 2n-1-p.
• Nearest: the texel containing the point, i = floor(U·W / 1000) and j = floor(V·H / 1000), then wrapped.
• Bilinear: in texel space the centres are at half-integers, so use a denominator of 2000:
  • xs = 2·U·W - 1000, i0 = floor(xs / 2000), fx = xs - 2000·i0 (0..1999), and the same for y;
  • the four texels are (i0, j0), (i0+1, j0), (i0, j0+1) and (i0+1, j0+1), each index wrapped separately;
  • value = ((2000-fx)(2000-fy)·t00 + fx(2000-fy)·t10 + (2000-fx)fy·t01 + fx·fy·t11 + 2000000) / 4000000, using integer division, so rounding is half up.
• Layout: texel (x, y) is the x-th value on row y.

Output

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• Nearest prints the unwrapped texel, then -> (x, y) if wrapping changed either index.
• Bilinear prints the wrapped column pair, the wrapped row pair, fx and fy, the four texel values (t00 t10 t01 t11) and the result.
• Errors:
  • texture: W H (1-16)
  • texture row R: expected W hex bytes (the texture is discarded)
  • sample: U V nearest|bilinear repeat|clamp|mirror (U V in 1/1000)
  • sample: wrap mode must be repeat, clamp or mirror
  • sample: load a texture first
  • unknown command X

Example

Input:

cCopy
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Output:

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Requirements

1. Wrap each of the four bilinear indices separately. At a repeat edge the filter blends texels from opposite sides of the texture, while clamp repeats the edge texel.
2. Use floor division for negative coordinates. C's / truncates toward zero and would pick the wrong texel.

Hidden tests cover samples at exact texel centres and edges, negative and out-of-range coordinates with every wrap mode, a non-square texture, bilinear filtering across the repeat seam, and malformed input.