# Sparse GP noise A single-octave experiment in smooth noise with sparse handles. Bridson Poisson-disk locations get IID Gaussian values; a Gaussian process completes the field. A posterior draw retains variation between constraints. ## Run Requires Python 3, NumPy, SciPy, and Pillow (`requirements.txt`). ```bash OPENBLAS_NUM_THREADS=1 python3 gp_noise.py OPENBLAS_NUM_THREADS=1 python3 gp_noise.py --compare --resolution 128 --out output_nosync/comparison OPENBLAS_NUM_THREADS=1 python3 -m unittest -v ``` If dependencies are missing, create a virtual environment named `.venv_nosync` and use its `pip3 install -r requirements.txt`. Limiting BLAS threads avoids thread overhead on these small matrix operations. [Comparison sheet](http://bigger/art/gaussian-process-noise/output_nosync/comparison/comparison.png) [Example draw](http://bigger/art/gaussian-process-noise/output_nosync/example/sample.png) Each run saves mean, sample, and residual images, unchanged floating-point fields in NPZ, and the CLI parameters in JSON. Comparison runs save one NPZ per case. All image colors use the same fixed symmetric display range; saturation is clipping for display only. Array row zero is y=0; image bottom is y=0. ## Parameters | Flag | Default | Meaning | | ------------------ | --------------------- | --------------------------------------------------------------- | | `--radius` | 0.18 | Minimum Bridson spacing on the unit square | | `--count` | automatic | Random subset of a saturated set; errors if insufficient points | | `--seed` | 7 | Locations and IID handle values | | `--sample-seed` | seed + 1 | Independent posterior randomness | | `--kernel` | matern32 | rbf, matern32, or matern52 | | `--length` | 0.10 | Kernel correlation length | | `--value-variance` | 1 | Variance of random handle values | | `--variance` | 1 | Prior marginal field variance | | `--noise` | 0 | Observation noise variance; positive values soften constraints | | `--alpha` | 1 | Residual multiplier; zero gives mean | | `--features` | 2048 | Number of random spectral frequency pairs | | `--resolution` | 256 | Pixels along each edge | | `--display-limit` | 3 | Colors map from minus to plus this value | | `--controls` | none | JSON file of additional `[x, y, value]` triples | | `--out` | output_nosync/example | Output directory | | `--compare` | off | Fixed sweep: 2 spacings, 3 lengths, 3 kernels, mean/draw pairs | `--count 0 --controls controls.json` allows a field driven entirely by explicit handles. For example, `[[0.25, 0.5, 1], [0.75, 0.5, -1]]`. Coordinates use the unit square. Conflicting duplicate controls should be avoided with zero noise. This is a data hook, not an interactive editor. ## Architecture and approximation `poisson_disk` implements Bridson's active-list algorithm with a sparse spatial hash and uniform-area annulus proposals. Optional count selection happens after saturation, avoiding a partial cluster grown from the initial seed. Small radii can still produce very large observation sets; use modest spacing for this prototype. `GP` accepts arbitrary points and scalar values. It uses analytic stationary kernels and a Cholesky factorization of the observation covariance only: ```text A = K(X, X) + (noise_variance + jitter) I mean(x) = K(x, X) A^-1 values residual(x) = prior_draw(x) - K(x, X) A^-1 (prior_draw(X) + noise_draw) field(x) = mean(x) + alpha * residual(x) ``` The mean is exact up to a fixed 1e-10 numerical nugget. The posterior draw is **approximate**: the prior uses random Fourier features, with Gaussian frequencies for RBF and multivariate Student-t frequencies with 3 or 5 degrees of freedom for Matern. Sine/cosine coefficients are Gaussian. Increasing feature count improves the prior covariance approximation. This is pathwise conditioning with an approximate prior and analytic correction, following [Wilson et al.](https://proceedings.mlr.press/v119/wilson20a.html). Alpha=1 approximates the specified posterior. Other alpha values deliberately alter its covariance by alpha squared while preserving its mean. With zero observation noise, constraints hold up to numerical nugget effects. With positive noise, draws and means can depart from observed values. `PosteriorDraw` stores frequencies and coefficients, so changing resolution, chunk size, or evaluation order evaluates the same function. No periodic tiling is imposed. Seed reproducibility assumes the same numerical library environment. For N handles, M pixels, and F frequency pairs, fitting costs O(N^3), and a draw costs O(M(N+F)) after fitting. Evaluation is chunked: memory is O(N^2 + M + chunk\*(N+F)). There is no M-by-M covariance or dense grid factorization. Tests cover point separation, interpolation, deterministic chunked evaluation, alpha=0, empty observations, and empirical posterior covariance at off-grid points for all three kernels, including noisy observations. ## First visual findings - Sparse handles with length much smaller than spacing produce isolated bumps in the mean. Posterior samples fill the blank regions with coherent variation. - The sparse RBF row at length 0.12 is a promising smooth direction: broad, irregular patches without an obvious lattice. Matern 5/2 adds finer structure; Matern 3/2 is visibly rougher at the same nominal length. - Long correlation lengths plus dense IID handles can fight each other. RBF at radius 0.12 and length 0.24 reaches absolute values around 18 between handles. The fixed color range intentionally exposes this overshoot. More smoothing does not necessarily mean calmer fields. - Dense constraints suppress the difference between mean and draw. Sparse constraints leave more scope for changing the sample seed without moving the handles. - This does not yet establish a visual advantage over Perlin. There is no matched-spectrum Perlin comparison here; avoiding a lattice alone does not guarantee a distinct aesthetic. The most promising distinction is the ability to preserve a few deliberate landmarks across different random completions. A useful next experiment is holding three deliberate landmarks fixed and making a sheet of different posterior seeds. That isolates artistic steering from the particular layout of random handles. The Python harness stays single-octave; the browser now includes the separate two-layer demo described below. ## Browser playground [Open the playground](http://bigger/art/gaussian-process-noise/browser/). The static `browser/` directory needs only an HTTP server. It runs entirely in JavaScript, with a Web Worker keeping the interface responsive. It shows the mean and posterior draw together, supports all three kernels, and exposes spacing, correlation length, residual amplitude, variances, seeds, and quality. Changing residual amplitude or display settings only repaints cached fields. Other changes produce a 64-pixel preview followed by the selected resolution. Obsolete workers are terminated when parameters change. The browser uses the same mathematical model but its own seeded random number generator, so Python and browser seeds do not produce identical fields. The browser defaults to 1,024 frequency pairs. Sine/cosine recurrence along scanlines reduces trigonometric evaluation cost without changing the spectral model. Initial timings on bigger (single run, not a rigorous benchmark): | Resolution | Python, 2,048 pairs, 24 handles | JS engine in Node, 1,024 pairs, 15 handles | | ---------- | ------------------------------- | ------------------------------------------ | | 64 | 0.34 s | 0.034 s | | 128 | 1.35 s | 0.066 s | | 256 | 5.30 s | 0.257 s | | 512 | 20.75 s | 0.956 s | These are field evaluation times, excluding image drawing. Python fit took about 3 ms, while the JS fit including Bridson sampling took about 10 ms. Browser status reports measured fit and field times on your device. Different settings, hardware, and browser engines change performance. The two columns use different feature counts and handle sets, so they are not an equal-work speed comparison. ### Editing handles Handles are separate HTML overlays over the clean canvas images. Select a handle in either view, drag it to move, and adjust its value with the sidebar slider or numeric field. Arrow keys move it by 0.005; Shift + arrow moves it by 0.05. Double-click empty space to add a zero-valued handle. Use Remove handle or Delete to remove the selection. Nearly coincident handles are rejected to avoid conflicting pinned observations. Removing all handles leaves an unconstrained prior draw. Edited handles survive changes to kernel, correlation length, posterior seed, and render quality. New handles, spacing, handle seed, and handle value variance regenerate the observations and replace edits. Edits are held for the current page session only. Hide handles hides the overlay without modifying image pixels. Hover a parameter control or its question-mark button for an explanation. The question-mark buttons also work by keyboard focus or tap. Explanations describe visual effects and flag controls that replace edited handles. ### Density-gradient demo [Open the density-gradient demo](http://bigger/art/gaussian-process-noise/browser/?demo=density). Handle values remain IID Gaussian. Only the spatial density changes: a dense Bridson set is thinned with retention probability `0.06 + 0.94*x^4`, making the left moderately dense and the right much denser. The minimum spacing still holds, but the thinned set is not saturated. Defaults use spacing 0.04 and Matern 5/2 at length 0.10. The link opens a separate tab so the uniform playground keeps its edits. Manual additions are unrestricted by the density profile. New manual handles use a separate deterministic random sequence derived from the handle seed and placement count, scaled by handle value variance. Deleting and replacing a handle gives a new value. Regenerating the layout resets that sequence. Selecting a handle also reveals a Delete button directly over each image, in addition to the sidebar button and keyboard shortcut. ## Hosting and source - [Playground](https://noise.joshu.dev/) - [Density-gradient demo](https://noise.joshu.dev/?demo=density) - [Private GitHub repository](https://github.com/jschachter/gaussian-process-noise) The Cloudflare Pages project uses direct upload. To publish the current browser code with Python 3, npm, and Wrangler installed and Cloudflare authentication configured (and Chromium installed for the live-browser check): ```bash npm run deploy ``` `npm run build` puts only the browser files and README into `site_nosync/`. JavaScript filenames include a content hash shared by the interface, worker, and engine so cached scripts from different builds cannot get mixed. The deployment does not contain generated media, field arrays, or credentials. GitHub pushes and Cloudflare deployments are separate steps; pushing to GitHub alone does not automatically deploy the site. ## Octave editor [Open the two-layer demo](https://noise.joshu.dev/?demo=octaves). The broad-structure layer starts with RBF at length 0.18 and amplitude 1. The fine-detail layer starts with Matern 5/2 at length 0.05 and amplitude 0.3. Scales are freely adjustable; there is no forced frequency ratio. Select a layer to edit its handles, kernel, spacing, length, seeds, variance, randomness, and spectral quality. The selected layer retains its own edits and selection when switching. Resolution and display range are shared. Only the selected layer's handles appear over the combined mean and combined draw. Handle values constrain their own layer; other layers can change the final sum at those locations. Each enabled layer contributes `amplitude * (mean + randomness * residual)`. Disabling a layer mutes it. If any enabled layers are soloed, only those layers contribute. Amplitude, mute, solo, and randomness changes repaint cached fields. Other edits currently recompute both independent fields in the worker, using sparse solves for each layer. The two existing single-layer demos remain available. Layer edits last for the current page session. `npm run verify:live` checks the deployed octave, density, and single-layer demos in a fresh Chromium browser, waits for completed renders, and checks layer switching. Deployment runs this check automatically.