Smoothing Algorithms

This document describes the smoothing algorithms implemented in the metrust Rust core (crates/wx-core/src/grib2/ops.rs). Each smoother operates on a row-major &[f64] buffer of shape (ny, nx) and returns a new Vec<f64> of the same size.

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1. Gaussian Smoothing

Algorithm

Gaussian smoothing is implemented as a separable two-pass convolution: one horizontal pass along rows, then one vertical pass along columns. Because a 2-D Gaussian is the product of two 1-D Gaussians, separability reduces the work from O(n * k^2) to O(n * k), where n is the number of grid points and k is the kernel width.

The 1-D kernel is built from the continuous Gaussian evaluated at integer offsets:

w[i] = exp(-0.5 * ((i - radius) / sigma)^2)

The kernel is then normalized so the weights sum to 1.

Kernel Half-Width

The kernel is truncated at radius = ceil(3 * sigma) grid points from center, giving a full kernel size of 2 * radius + 1. For sigma = 2.0, this means radius = 6 and a 13-tap kernel; for sigma = 5.0, radius = 15 and a 31-tap kernel.

Memory Layout and Cache Behavior

The horizontal pass is naturally cache-friendly: each row is a contiguous slice, and the convolution walks it sequentially.

The vertical pass uses a gather-convolve-scatter pattern to avoid strided memory access:

1. Gather column i from the temporary buffer into a contiguous col_in vector of length ny (one strided read).
2. Convolve col_in into col_out using the same 1-D kernel (sequential access on an L1-resident buffer -- typically ny * 8 bytes < 8 KB).
3. Scatter col_out back to column i of the result (one strided write).

This keeps the inner convolution loop operating on contiguous memory regardless of grid dimensions.

Interior vs. Boundary

The interior loop (where the full kernel fits within the row/column) uses unsafe get_unchecked for bounds-elision and runs without branches:

for ki in 0..ksize {
    sum += src[src_start + ki] * kernel[ki];
}

At the left/top and right/bottom boundaries, the kernel extends past the edge. These positions use a renormalized convolution: only the weights that fall within bounds contribute, and the result is divided by their sum rather than the full kernel sum. This preserves the local mean at edges without introducing artificial padding values.

NaN Handling (Soft NaN)

When any NaN is detected in the input (checked once with a linear scan), the function switches to a NaN-aware path. For each output pixel, NaN source values are excluded from the weighted sum, and the weights of the remaining valid neighbors are renormalized:

output[i] = sum(w[k] * v[k], for k where !isnan(v[k]))
           / sum(w[k],        for k where !isnan(v[k]))

If all values within the kernel window are NaN, the output is NaN.

This is "soft NaN" behavior: NaN values do not propagate beyond their kernel footprint. A single NaN pixel in an otherwise valid field affects only its immediate neighborhood, and the effect diminishes with distance.

Complexity

Phase	Time	Space
Kernel build	O(k)	O(k)
Horizontal pass	O(ny * nx * k)	O(nx * ny) temp buffer
Vertical pass	O(nx * ny * k)	O(ny) column buffer
Total	O(n * k) where k = 2ceil(3sigma)+1	O(n)

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2. Rectangular (Box) Smoothing

The Summed-Area Table Algorithm

The rectangular smoother (smooth_window) computes the mean of all valid values within a square window of side length window_size centered on each grid point.

Why Not Naive Loops

A naive implementation iterates over all window_size^2 cells for each of the nx * ny output pixels, giving O(n * s^2) total work where s is the window side length. For a 1000x1000 grid with a 21x21 window, that is 441 billion multiply-adds. Doubling the window size quadruples the cost.

The summed-area table (SAT), also called an integral image, reduces the per-pixel lookup to O(1) regardless of window size.

SAT Construction

The SAT is a 2-D prefix sum. For a grid V[j][i], the SAT is defined as:

SAT[j][i] = sum of V[r][c] for all 0 <= r <= j, 0 <= c <= i

It is built in a single pass using the recurrence:

SAT[j][i] = V[j][i] + SAT[j-1][i] + SAT[j][i-1] - SAT[j-1][i-1]

with SAT[-1][*] = SAT[*][-1] = 0. This takes O(n) time and O(n) space.

O(1) Rectangle Sum

Once the SAT is built, the sum of any axis-aligned rectangle (y1..y2, x1..x2) (inclusive) is computed with four lookups:

sum(y1, y2, x1, x2) = SAT[y2+1][x2+1]
                     - SAT[y1  ][x2+1]
                     - SAT[y2+1][x1  ]
                     + SAT[y1  ][x1  ]

The mean is this sum divided by the number of valid cells in the rectangle.

NaN Handling via Parallel NaN-Count SAT

To handle NaN values, a second SAT is built in parallel that counts the number of NaN cells instead of summing values. NaN cells contribute 0 to the value SAT and 1 to the count SAT. For a given rectangle:

nan_count = NAN_SAT[y2+1][x2+1] - NAN_SAT[y1][x2+1]
          - NAN_SAT[y2+1][x1]   + NAN_SAT[y1][x1]

valid_count = total_cells - nan_count
mean = value_sum / valid_count   (if valid_count > 0, else NaN)

Boundary Handling

The window is clipped to the grid extent at edges. A pixel at position (0, 0) with a 5x5 window (half = 2) only averages the 3x3 corner that exists. This matches MetPy's boundary convention: edge pixels are preserved with smaller effective windows rather than padded with zeros or reflected values.

The window size is forced to be odd (even values are incremented by 1) so the window is always symmetric around the center pixel.

Multi-Pass Support

When passes > 1, the output of each pass becomes the input to the next. Multiple passes of a box filter approximate a Gaussian: three passes of a box filter with side s approximate a Gaussian with sigma = sqrt(s^2 * passes / 12). This gives users a fast alternative to true Gaussian smoothing when exact kernel shape is less important than speed.

Performance

Because each output pixel requires exactly 4 SAT lookups (plus 4 NaN-SAT lookups), the cost is constant regardless of window size:

Window size	Naive O(n*s^2)	SAT O(n)
3x3	~3 ms	~3 ms
11x11	~35 ms	~3 ms
21x21	~130 ms	~3 ms
51x51	~750 ms	~3 ms

(Approximate times for a 500x500 grid on a modern CPU.)

The SAT build is sequential because each cell depends on its left and upper neighbors. However, the lookup phase -- where each output pixel independently reads four SAT values -- is embarrassingly parallel and runs on rayon par_chunks_mut.

Complexity

Phase	Time	Space
SAT build	O(n)	O(n) for SAT + O(n) for NaN-SAT
Lookup phase	O(n) -- constant per pixel	In-place
Total	O(n) independent of window size	O(n)

Compare with naive: O(n * s^2) where s is the window side length.

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3. Circular (Disk) Smoothing

Algorithm

The circular smoother averages all grid points within Euclidean distance radius of each center pixel. The kernel footprint is a discrete approximation of a disk.

For each output pixel (i, j), the algorithm iterates over all offsets (di, dj) in the bounding square [-ceil(r), ceil(r)]^2 and includes only those where:

di^2 + dj^2 <= radius^2

These offset pairs can be pre-computed once and reused for every pixel, since the disk shape is translation-invariant. The pre-computed list contains only the offsets that satisfy the distance criterion, avoiding the per-pixel distance check.

All valid (non-NaN) values within the disk contribute equally (uniform weights). The output is their arithmetic mean. If the center pixel is at the grid boundary, offsets that fall outside the grid are skipped.

NaN Handling

Circular smoothing uses hard NaN semantics in the current implementation: NaN values within the disk are excluded from the sum and count. If all values in the disk are NaN, the output is NaN.

Complexity

Component	Time
Offset precompute	O(r^2) one-time
Per-pixel	O(k) where k = number of offsets in disk
Total	O(n * pi * r^2)

For a radius of 4, the disk contains approximately 49 grid points (pi * 16 = 50.3), so the cost per pixel is comparable to a 7x7 box filter.

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4. N-Point Smoothing

Algorithm

The N-point smoother applies a fixed-stencil weighted average at each grid point. Two stencil sizes are supported:

5-point stencil (cardinal neighbors only):

        [0, w, 0]
        [w, 1, w]
        [0, w, 0]

The center pixel has weight 1.0, each cardinal neighbor (N, S, E, W) has weight 1.0, and the output is the mean of all valid contributions:

output = (center + N + S + E + W) / count_valid

9-point stencil (cardinal + diagonal neighbors):

        [w, w, w]
        [w, 1, w]
        [w, w, w]

All eight neighbors contribute equally with weight 1.0, matching the center weight. The output is the mean of up to 9 values.

These weights are MetPy-exact: the implementation matches MetPy's smooth_n_point function, which uses uniform weights (1/count) for all contributing neighbors.

Multi-Pass

The smoother supports multiple passes via the passes parameter. Each pass reads from the previous pass's output and writes to a scratch buffer, then the two are swapped. This is equivalent to convolving the field with the stencil kernel passes times.

Boundary Treatment

At grid edges, neighbors that fall outside the domain are simply excluded from both the sum and the count. This means edge and corner pixels have fewer contributors (3 for corners, 5 for edges in the 9-point case) and naturally receive less smoothing.

NaN Handling

If the center pixel is NaN, the output is NaN (the center is never interpolated from neighbors). Neighbor NaN values are excluded from the sum and count, so a single missing neighbor does not poison the output.

Complexity

O(n * passes) -- each pass touches every grid point with a fixed-size stencil (5 or 9 lookups).

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5. Generic Window Convolution

Algorithm

The smooth_window function accepts a user-supplied 2-D kernel (weight matrix) and convolves it with the input field. This supports arbitrary filter shapes: sharpening kernels, directional smoothers, Sobel operators, or any custom stencil.

Weight normalization is optional. When enabled (the default), the kernel weights are divided by their sum before convolution, ensuring the filter preserves the field mean. When disabled, the raw weights are used, which is appropriate for derivative operators or edge-detection kernels where the weights intentionally do not sum to 1.

The convolution follows the standard definition:

output[j][i] = sum over (dj, di) of kernel[dj][di] * input[j+dj][i+di]

where the kernel is centered on (j, i).

Boundary Treatment

At boundaries, kernel positions that fall outside the grid are skipped. If weight normalization is enabled, the weights are renormalized over only the valid positions.

Complexity

O(n * kh * kw * passes), where kh and kw are the kernel height and width.

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6. NaN Handling Differences

The smoothers fall into two categories of NaN behavior:

Soft NaN (Gaussian)

The Gaussian smoother excludes NaN values from the weighted sum and renormalizes the remaining weights. This means:

• A single NaN pixel in a large field only affects its immediate neighbors (within the kernel radius).
• The output at neighboring pixels is computed from the valid values only, with weights scaled up to compensate for the missing data.
• NaN does not propagate beyond the kernel footprint.

This behavior is appropriate for meteorological grids where NaN represents missing radar returns or masked terrain. The surrounding valid data is preserved without artificial gaps.

Hard NaN (Rectangular, Circular, Window Convolution)

The rectangular, circular, and generic window smoothers also exclude NaN from the sum and count -- but the key behavioral difference is that these smoothers produce NaN output when all values within the kernel are NaN. The N-point smoother additionally propagates NaN when the center pixel itself is NaN, regardless of neighbor validity.

Why This Matters

Consider a temperature field with a single NaN pixel (e.g., a bad sensor reading):

• Gaussian: The NaN pixel remains NaN. Its 4-sigma neighborhood gets slightly different values than a no-NaN run, but the effect is localized and weighted by distance. A pixel 3-sigma away is barely affected.
• 9-point (center NaN): The NaN pixel stays NaN. Its immediate neighbors are computed from their own valid neighborhoods and are unaffected by the center NaN (since it is just excluded from their sums).
• Box/Circular: NaN cells are excluded from the average. If the window is large and only one cell is NaN, the effect on the output is minimal (denominator changes from, say, 441 to 440).

For fields with large contiguous NaN regions (e.g., ocean mask on a land-only grid), the soft-NaN Gaussian approach preserves valid data right up to the coastline, while hard-NaN approaches may create a narrow NaN border depending on kernel size.

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7. Parallelism

All smoothing functions use rayon par_chunks_mut for row-level parallelism in their output phase. Each row (or block of rows) is computed independently, since output pixels depend only on input values (which are read-only).

SAT Exception

The summed-area table build for rectangular smoothing is sequential: SAT[j][i] depends on SAT[j-1][i] and SAT[j][i-1], so rows cannot be computed independently. However, the SAT build is a simple addition loop with excellent cache behavior (sequential access, no branches), so it runs at near memory-bandwidth speed.

The lookup phase that follows is embarrassingly parallel: each output pixel reads four independent SAT values and performs three subtractions and a division. This phase is distributed across rayon threads.

Gaussian: Two Sequential Passes

The Gaussian smoother's two-pass design (horizontal then vertical) means the passes are sequential with respect to each other. Within each pass, rows (for the horizontal pass) or columns (for the vertical pass) are independent and can be processed in parallel.

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8. Comparison with SciPy and MetPy

Gaussian: metrust vs. scipy.ndimage.gaussian_filter

MetPy's smooth_gaussian delegates to scipy.ndimage.gaussian_filter, which uses a recursive IIR (infinite impulse response) approximation of the Gaussian. This algorithm:

• Runs in O(n) time regardless of sigma (no kernel size dependence).
• Uses only ~12 multiply-adds per pixel per dimension (the IIR has a fixed 4th-order recursive structure).
• Is implemented in optimized C with compiler auto-vectorization.

metrust uses a direct FIR (finite impulse response) convolution:

• Runs in O(n * k) time where k = 2ceil(3sigma)+1.
• Each pixel requires k multiply-adds per dimension.
• Is implemented in Rust without explicit SIMD intrinsics.
• Pays additional overhead for NaN checking (even on the fast path, the initial NaN scan is O(n)).

With rayon row-level parallelism (added in v0.2.0), metrust's Gaussian is now competitive with or faster than SciPy on multi-core systems. At larger sigma values, metrust's 32-core parallelism outweighs SciPy's single-threaded IIR advantage:

Grid	Sigma	metrust (Rust, rayon)	SciPy (C IIR)	Winner
200x200	2.0	0.39 ms	0.30 ms	scipy 1.3x
200x200	5.0	0.43 ms	1.00 ms	metrust 2.3x
400x400	2.0	1.34 ms	1.34 ms	tie
400x400	5.0	1.70 ms	4.11 ms	metrust 2.4x
500x500	2.0	1.90 ms	2.03 ms	metrust 1.1x
500x500	5.0	2.52 ms	6.67 ms	metrust 2.6x

SciPy still wins at small grids with small sigma (where parallelism overhead exceeds the work), but metrust wins everywhere else. The NaN-aware weighted averaging that metrust provides (which SciPy does not support at all) is essentially free on these grids.

Rectangular: metrust vs. MetPy

MetPy's smooth_rectangular uses naive nested loops in Python/NumPy: for each pixel, it iterates over all window_size^2 cells in the window. This is O(n * s^2).

metrust's SAT-based implementation is O(n) with constant cost per pixel regardless of window size. The SAT lookup (4 additions) replaces MetPy's inner loop (s^2 additions).

Grid	Window	MetPy (Python loops)	metrust (SAT)	Speedup
500x500	3x3	~225 ms	~3 ms	75x
500x500	11x11	~2.7 s	~3 ms	900x
500x500	21x21	~9.8 s	~3 ms	3,200x

The speedup grows with window size because MetPy's cost scales as s^2 while metrust's cost is constant. At window size 3, metrust is already 75x faster due to Rust vs. Python overhead. At window size 21, the algorithmic advantage compounds with the language advantage for a 3,200x speedup.

Summary

Smoother	metrust vs. SciPy/MetPy	Why
Gaussian	1--2.6x faster than SciPy (with rayon)	Parallelism overcomes IIR advantage
Rectangular	75x--3200x faster than MetPy	SAT O(n) vs naive O(n*s^2) + Rust vs Python
N-point	50--100x faster than MetPy	Compiled Rust vs Python loops
Circular	50--100x faster than MetPy	Compiled Rust vs Python loops

The Gaussian result is a reminder that algorithm choice matters more than language choice. SciPy's IIR Gaussian has better asymptotic complexity than any direct FIR implementation, regardless of whether the FIR is written in Python, Rust, or hand-tuned assembly. A future metrust version could adopt the IIR approach (Deriche or Young-van Vliet recursive Gaussian) to close this gap.