Sixteen probes for colour, four for data

How many probes a pixel takes

A pixel's footprint on the floor has two lengths in texels, across and along. Their ratio is the footprint's anisotropy. The sampler takes about as many probes as the ratio, up to its maximum, from the mip whose texels match the footprint's width, and averages them. Past the maximum it reads a coarser mip, so that its probes still cover the length, and the texture blurs across.

The extension EXT_texture_filter_anisotropic wrote the rule down in 1999 as one possible approach, and the Vulkan specification keeps its steps. They start from how fast the texture coordinates change from one pixel to the next along the screen's x and y, in texels per pixel:

Each probe is an ordinary trilinear read at λ, so it blends the texels of mips ⌊λ⌋ and ⌊λ⌋ + 1. While the ratio stays under the maximum, Pmax / N comes out a little under Pmin: the mip matches the footprint's width, and the probes tile its length. Past the maximum, the mip follows the length divided by the maximum, coarser than the width needs. Vulkan writes the level with the unrounded ratio, λ = log2(Pmax / η) with η = min(Pmax / Pmin, maxAniso), less than one level from the extension's.

Both specifications leave the scheme to the implementation. It may round N up, to powers of two for instance, and approximate Px and Py. So the demos here model the specifications' rule, not the M4 Pro's hardware. In wgpu the maximum is anisotropy_clamp on the sampler descriptor, which "must be at least 1" and needs every filter mode linear above 1. The Capability floor requires the ANISOTROPIC_FILTERING downlevel flag, without which wgpu silently clamps every sampler's anisotropy to 1 (ADR 0021).

On minification's floor

For a pixel straight ahead, this guide's arithmetic gives the footprint in closed form. A pixel at depth z along the view axis spans z / f metres across and z² / (f h) metres along, where h is the camera's height and f the focal length in pixels: 1,738 at 1440p for the 45° vertical field of view (src/camera/mod.rs), 435 at 640 × 360. So the ratio is z / h, the depth over the camera's height. The floor holds 256 texels a metre, since its 256-texel Textures repeat every metre (src/scenes/mod.rs).

16 m out at 1440p, a pixel spans 2.4 texels across and 31 along, a ratio of 13.3: 14 probes at 16x, from mips 1 and 2. At 640 × 360, the Golden's size, the same pixel spans 9.4 by 126 texels, the figures Distant Textures Explained gave, and its probes read mips 3 and 4. The model's geometry matches the captures: it puts the floor's far edge, 96 m out, at row 481.4 of 1440 and 120.4 of 360, and the captures' first floor rows are 481 and 120.

What 16x costs

At 1440p, 16x on every Channel costs minification's lit pass 214 µs over 1x, 38% of the pass's 557.2 µs. The lit pass does all the sampling: measured against 064d2ba, every setting moves the GPU frame total by what it moves the lit pass, within 2.1 µs. Lower settings cost less, but not in step with the probes they save: over 1x, 2x costs nothing, 4x 26 µs and 8x 104 µs.

The A/B ran release builds of 064d2ba, main at the time, and of 064d2ba with a measurement patch, not committed, that gives each Material Channel its own sampler and reads each one's anisotropy from an environment variable. praxis bench ran minification at 2560 × 1440 with 4x MSAA, 248 warm-up and 1,800 measured frames, in three rounds, each running every variant once in rotated order, with the load average about 13 from other builds and a browser running on the machine. At 16x the patch times as one sampler does, and its capture matches the old Golden pixel for pixel.

• Three Textures pay. The floor samples three Textures, in the albedo, normal and metallic-roughness Channels; occlusion and emissive read Layer 0's 1×1 Textures, whose anisotropy costs nothing: those two at 1x leave the pass at 557.5 µs.
• Channels overlap. Albedo alone at 16x costs 42 µs over 1x everywhere, yet dropping albedo alone to 1x from 16x everywhere saves 84 µs. Costs measured one Channel at a time do not add up.
• MSAA does not multiply it. The measurements at the end of milestone 2 (M2's exit) found 222 µs of a 478 µs main pass, before the pass had MSAA. The lit pass reads no sample index, so it shades once per pixel, and 4x MSAA leaves its texture reads as they were (this guide's reading of main.wgsl).
• Surfaces that face the camera do not pay. In those measurements, clamping the anisotropy to 1 took the helmet from 357 to 356 µs and left normal-tangent at 202 µs. They face the camera, so most of their footprints are close to square.
• Against 064d2ba. 8x everywhere saves 110 µs; colour 16x with data 8x saves 74, with data 4x 126 and with data 2x 158. Data 4x's 126 µs is 1.1% of ADR 0013's 12 ms GPU Budget.

Probe counts do not explain the cost. By the specifications' rule, the floor fills 66.5% of the 1440p frame, and its average pixel takes 2.00 probes at 2x, 3.59 at 4x, 5.00 at 8x and 6.25 at 16x (this guide's model). Yet over 1x, 2x costs nothing measurable, and 16x costs twice what 8x does. The specifications leave the scheme to the implementation, so on the M4 Pro the cost of a probe is something to measure, not to model.

Where the image changes

Every difference sits in one band below the horizon, where a pixel's footprint is longer, in texels, than the lower anisotropy times its width. Nearer the camera, both settings take the same probes from the same mips. For the candidates that lower only the data Channels, the band also ends further out: beyond about 23 m, 16x itself reads mips in which the bumps have averaged away (chapter 4).

By this guide's count of the 1440p captures made for PR #257, which the PR publishes only as halved difference images and zooms, colour 16x with data 4x differs from 16x in rows 552 to 870: 567,416 pixels, 15.4% of the frame. The model puts row 552 at 22.8 m straight ahead and row 870 at 5.0 m, where the ratio is 4.3. 8x everywhere differs in rows 492 to 749, through the checker too, and its differences of more than 7 steps all lie above row 680; the model puts a ratio of 8 straight ahead at row 679. Of the candidates put to the owner, 8x everywhere moves 10.9% of the 1440p frame (400,069 pixels), colour 16x with data 4x 15.4% (567,416) and with data 2x 25.0% (920,160), all of it in that band.

The Golden's size hides some of it

The Golden is 640 × 360. There each pixel's footprint is four times wider and longer in texels, so its probes read mips two levels coarser, and most of the band reads mips where the bumps have already averaged away. Colour 16x with data 8x changes 221 pixels of the Golden, 0.1%, but 273,950 at 1440p, 7.4%, whose pixels sample finer mips that still hold the bumps. Against the Golden of 064d2ba:

Why fewer probes removed the sparkle

The cooker widens each mip's roughness by the spread of the normals inside one of that mip's texels. The sampler picks the mip by the footprint's width, so the widening covers the width. Along the length, the probes average the normals, and the shader rebuilds a unit normal from that average, so the spread along the length never reaches roughness. Reading the data Channels at 4x makes their probes read coarser mips, whose texels hold whole bumps.

What each mip holds

Toksvig's widening runs per mip in the cooker (src/cook/texture.rs; ADR 0018). For each texel of a mip, it takes the unit normals the texel covers, their mean m and their spread 1 − |m|², and widens the roughness so that α′² = α² + 2σ², where α, the highlight's width in the GGX model, is the roughness squared, and σ² = (1 − |m|) / |m| after Toksvig, as Distant Textures Explained introduced. This guide ran the cooker's algorithm on minification's normal and roughness Textures, in f64 before BC5 encoding. It gives three kinds of mip:

Mip 2 is flat because of where the bumps sit. A bump repeats every 8 texels, so a mip 2 texel covers half a bump each way, and the slopes at the four texel centres across half a bump cancel: cos(π/8) + cos(3π/8) + cos(5π/8) + cos(7π/8) = 0. From mip 2 on, every texel holds a flat normal and the bumps' whole spread as roughness.

Where each sampler reaches mip 2

So what decides the shading is the mip the data probes read. By the model at 1440p, 16x reads only mip 2 and coarser from 22.9 m out, a boundary at row 551.6 of the screen; 8x from 16.1 m (row 589.9), 4x from 11.4 m (row 644.0), 2x from 8.0 m (row 720.5) and 1x from 5.6 m (row 828.7). Two measurements match it to within a row: every data-Channel candidate's differences from 16x begin at row 552, and data 2x and data 1x render identical pixels above row 720 (this guide's count of the captures made for PR #257).

What the shader does with the probes

Between about 5 and 23 m, 16x's probes read mip 0 or mip 1, blended with the next: tilted normals and roughness under 0.585. At the light's mirror point, 16.4 m out by this guide's arithmetic, it takes 14 probes at λ 1.23, mostly from mip 1, and reads roughness 0.456; data at 4x takes 4 probes at λ 3.04 and reads 0.585 on flat normals. The 14 probes average their tilted normals along the length, and tangent_space_normal rebuilds z = √(1 − x² − y²) from the averaged x and y (src/render/shaders/shading.wgsl), so the normal comes back to unit length. The spread the averaging removed is lost: it is in neither the normal nor the roughness. Toksvig's 2004 brief names the result: "Most algorithms simply renormalize, but this paper explores how the shortening can be used as a measure of normal variation to eliminate the common problem of strobing/sparkling noise due to aliasing of specular highlights."

That is what the captures show. By eye, 16x and 8x render the highlight as a broad white column with single-pixel sparkle along its flanks and lower end, specular aliasing; the data Channels at 4x or 2x render a smoother, narrower V without the sparkle. Under ADR 0008 nothing the image depends on reads screen-space history, the pixels of earlier frames, so a temporal mode could only ever be an option, and nothing can be counted on to average that sparkle away. No candidate shows the light's mirror image, which the reference does: Toksvig widens one lobe and keeps no sharp peak.

The trade

Lowering the data Channels costs some detail. Beyond a footprint 4 texels long per texel wide, the bumps shade flatter than at 16x. Data at 2x saves 32 µs more and smooths further, but flattens the distant bumps' shading enough to move the checker 19% further from the reference than 16x everywhere does. The colour Channels stay at 16x because the checker is albedo, sharp detail that a coarser mip blurs across: 8x everywhere moves the checker 15% further from the reference, and 4x 81% (chapter 5).

Judging the images against a reference

To score each candidate, the work on #184 rendered the frame with nine samples per pixel and measured how far each one-sample capture lies from it. The score is fair on the checker, a fixed pattern. In the highlight, the reference resolves glints that no one-sample render can place, and the score rewards blur there, so the highlight was judged by eye.

How the reference was made

The reference is minification at 16x, captured at 7680 × 4320, three times 1440p each way, and box-filtered to 2560 × 1440 in linear light, so each pixel averages nine. The command:

magick main-7680x4320.png -alpha off -colorspace RGB -filter box \
  -resize 2560x1440! -colorspace sRGB -depth 8 reference-2560x1440.png

It averages the displayed colours, after the tonemapping pass, decoded to linear light. It is Praxis's own 16x render with more samples, so it measures how each candidate filters against more samples of the same shading, not against the physics of a real floor.

The score is the RMSE in 8-bit steps over the band of rows 484 to 759, the floor from about 86 m to 7 m out by the model, and within it over the highlight's columns 1000 to 1559 and the checker either side, columns 0 to 899 and 1660 to 2559:

The checker, a fair test

The checker is fixed in texture space, so the reference and every candidate show it in the same place, and its error measures filtering. 16x everywhere is closest, at 2.98 steps, as is colour 16x with data 8x. 8x is 15% further off, and 4x 81%. Keeping the albedo at 16x keeps the checker within 3% of 16x with data at 4x, and within 19% with data at 2x, through the bumps' shading.

The highlight, where RMSE favours blur

In the highlight the reference holds glints: it samples each pixel nine times, so it catches the bumps' reflections where they are. A one-sample render's glints land wherever its single sample happens to fall. RMSE squares every difference, so a glint in the wrong place costs twice, once where it should be and once where it is, while a blur that never commits costs less. That is why the smoothest settings score best there: data at 1x 27.65 steps, data at 2x 28.09 and 2x everywhere 28.98, against 33.89 for 16x. The highlight was judged by eye instead.

The decision and the change

On 2026-10-22 the owner chose candidate C: the colour Channels, albedo and emissive, at 16x, and the data Channels, normal, metallic-roughness and occlusion, at 4x. It was judged on minification then, rather than with town's first Goldens (#206) as the milestone had paired them, so those Goldens were made under the settled sampler when PR #273 added town.

Two samplers

The Material bind group now carries one sampler per kind of Channel, as the glossary says: "Albedo and emissive are the colour Channels, encoded sRGB; normal, metallic-roughness and occlusion are the data Channels, encoded linear. Each kind has its own sampler." (CONTEXT.md, Channel). In src/assets/material.rs:

const COLOUR_ANISOTROPY: u16 = 16;
const DATA_ANISOTROPY: u16 = 4;

let colour_sampler = trilinear_sampler(device, "Colour Channel Sampler", COLOUR_ANISOTROPY);
let data_sampler = trilinear_sampler(device, "Data Channel Sampler", DATA_ANISOTROPY);

/// A trilinear, repeating sampler (#144) at `anisotropy`, which the
/// Capability floor's anisotropic filtering keeps wgpu from clamping to 1
/// (ADR 0021).
fn trilinear_sampler(device: &wgpu::Device, label: &str, anisotropy: u16) -> wgpu::Sampler {
    device.create_sampler(&wgpu::SamplerDescriptor {
        label: Some(label),
        address_mode_u: wgpu::AddressMode::Repeat,
        address_mode_v: wgpu::AddressMode::Repeat,
        mag_filter: wgpu::FilterMode::Linear,
        min_filter: wgpu::FilterMode::Linear,
        mipmap_filter: wgpu::MipmapFilterMode::Linear,
        anisotropy_clamp: anisotropy,
        ..Default::default()
    })
}

The lit pass's fragment shader picks the sampler by Channel (src/render/shaders/main.wgsl):

textureSample(albedo_textures[material.layers.x], colour_sampler, in.uv),
textureSample(metallic_roughness_textures[material.layers.z], data_sampler, in.uv).rg,
textureSample(occlusion_textures[material.layers.w], data_sampler, in.uv).r,
textureSample(emissive_textures[material.extra.x], colour_sampler, in.uv).rgb
…
let normal_sample = textureSample(normal_textures[material.layers.y], data_sampler, in.uv).rg;

The bind group's definition declares the two as colour_sampler and data_sampler (src/render/bind_groups.rs), in place of the one pbr_sampler. ADR 0017 now records the 4x as "a choice of cost and image the samplers make, not a cap the streaming design imposes", and ADR 0021 the two settings.

The Metal Goldens moved with it. minification's new Golden is candidate C's capture, pixel for pixel, and differs from the old one in 18,119 pixels (this guide's count). normal-tangent moves 5 pixels by one step at its quads' grazing edges. Of the other scenes with Goldens, none samples a data Channel Texture: uv-grid, orrery, texture-wall, pbr-sweep and far-coplanar capture identically at 1440p, and the shadows scene, which #259 added just before the merge, has no Textures and kept its Goldens. The helmet, which has no Golden, samples all three data Channels through SciFiHelmet's normal, metallic-roughness and occlusion Textures: at 1280 × 720 its capture changed in 702 pixels, 50 of them by more than one step and none by more than 6, all on the helmet's grazing silhouette (PR #257's description). The merge regenerated minification's and normal-tangent's software-Vulkan Goldens too, and minification's Metal Golden on main is still candidate C's capture.

What stays open

The decision settles the two samplers every Material uses, judged on minification. It does not make the cooked roughness see a footprint's length.

• town's first Goldens (#206) were made under these samplers when PR #273 added town (c8f7071).
• The length's spread. Past a ratio of 4, each data probe's mip texel spans a quarter of the footprint's length, so most of the spread along it reaches roughness. Up to a ratio of 4, the spread along the length beyond the width still does not. The lit pass computes no screen-space derivatives of its own: its shaders call no dpdx, dpdy or fwidth. The visibility survey (#45) noted that MSAA anti-aliases coverage, not shading, and that a forward pass has the quad derivatives that geometric specular anti-aliasing needs.
• Reading, not plans. LEAN mapping stores the bumps' normal distribution "in a linearly-filterable form compatible with standard MIP and anisotropic filtering hardware", so averaging probes would average the spread as well. Kaplanyan, Hill, Patney and Lefohn filter the normal distribution over a region estimated from effects "such as pixel footprint".

Reading list

Grouped by source, each with what it is good for here. Repo links that #184 changed are pinned to its merge, f4c7442; the others to 064d2ba, main when #184 was measured; they are unchanged at the merge. Each outside citation was checked against its publisher, registry or the book's own contents.

Glossary