Image Mipmapping

August 19

1. Mipmap Basics

Item	Description
Purpose	Pre-generate multiple lower-resolution versions of textures to reduce aliasing and moiré patterns during distant sampling, and improve cache hits and performance.
Pyramid Levels	If the original image size is $(W \times H)$ , the $(l)$ -th level is usually $(\max(1, \lfloor W/2^l \rfloor) \times \max(1, \lfloor H/2^l \rfloor))$ , down to 1x1.
Storage Overhead	The total area of all levels is about $(\frac{1}{3})$ of the original image (more precisely, $(\sum_{k=1}^{\infty} 4^{-k} = \frac{1}{3})$ ), so the total texture memory is about $(\frac{4}{3})$ of the original image.

2. Common Downsampling Algorithms

Algorithm/Type	Approach	Advantages	Disadvantages/Notes
Box Filter (Mean/Box Filter)	Average every 2x2 pixels to generate the next level. Formula: $C' = \frac{1}{4}\sum_{i=0}^{1}\sum_{j=0}^{1} C_{2x+i,2y+j}$	Simple, fast	Poor frequency roll-off, prone to aliasing; not smooth for high-frequency textures
Triangle/Bilinear Filter (Linear Kernel)	Downsample after low-pass filtering with linear weights, or use triangle kernel	Smoother than box, slightly better edge preservation	Slightly higher cost than box
Gaussian Filter	Apply Gaussian blur before downsampling	Closer to ideal low-pass, better anti-aliasing	Heavier computation, may blur details
Lanczos (Windowed sinc)	Resample using Lanczos kernel (commonly a=2 or 3)	Sharp and excellent anti-aliasing	Highest cost, may cause slight ringing
Mitchell-Netravali / Catmull-Rom (Cubic Family)	Resample with cubic convolution kernel	Good balance of sharpness and smoothness, moderate cost	Common in offline pipelines, texture baking
Gamma/Color Space Handling	Average in linear space to avoid darkening in sRGB. Process: sRGB → linear; filter/downsample; linear → sRGB. Alpha: use premultiplied alpha before filtering/downsampling.	Physically correct color	-
Normal Map Specific	Transform to tangent space unit vectors, accumulate then normalize, remap offset if needed. Can weight (e.g. Z or curvature) to reduce bias.	Physically correct normals	Cannot linearly average RGB normals

3. Common Runtime Filtering Algorithms

Algorithm/Type	Approach	Advantages	Disadvantages/Notes
Nearest Mipmap Nearest (MMN)	Selects the nearest mip level, uses nearest point sampling	Fastest	Blocky appearance
Nearest Mipmap Linear (NML)	Selects the nearest mip level, uses bilinear sampling	Smoother than MMN	May jump when switching levels
Linear Mipmap Nearest (LMN)	Samples two adjacent mip levels with nearest point, then linearly interpolates	-	Rarely used alone
Trilinear Filtering	Bilinear sampling on two levels, then linear interpolation by LOD fraction	Good balance, commonly default	-
Anisotropic Filtering	Multi-point sampling along major axis in oblique view, suppresses blur/flicker. With mipmap: each candidate mip level samples along anisotropic direction. Parameter: anisotropy level (e.g. 4x, 8x, 16x) affects quality and performance.	High quality for oblique views	Cost increases with level
LOD Control & Bias	Hardware estimates LOD by screen-space derivatives: $\lambda = \log_2 \left(\max \left(\sqrt{(\partial u/\partial x)^2 + (\partial v/\partial x)^2},\ \sqrt{(\partial u/\partial y)^2 + (\partial v/\partial y)^2}\right) \right)$ . Bias can be added to control sharpness/noise tradeoff.	Adjustable sharpness	-

4. Special Mipmap Variants

Type	Description
Ripmaps	Independently downsample u and v to get a rectangular pyramid, suitable for strong anisotropic scaling.
Summed-Area Tables (SAT)	Used for fast box filtering and variable-size region averaging, high storage cost.
Sparse/Virtual Textures	Tile-based storage and on-demand loading of mip tiles. MegaTexture, Tiled Resources.
Specular/Gloss mip chain	Pre-filter roughness-related maps by physical model (e.g. prefiltered mipmap for environment reflection, BRDF-Convolved Env Map).
Alpha Coverage Preserving Mipmaps	Adjust alpha during downsampling of transparent textures to preserve coverage, reduce distant flicker.

Common Issues

Averaging in sRGB space: causes color shift and darkening.
Directly averaging normal or specular roughness maps: leads to physical inconsistency.
Not preserving coverage for transparent textures: distant flicker.
Mipmap level boundary seams: handle wrap/clamp and border pixel copy during generation to avoid seams.