How Can I Efficiently Perform A 3D Convolution Operation On A Large, Irregularly-gridded Dataset Stored In An IDL Array, While Simultaneously Handling The Varying Grid Spacings And Ensuring Accurate Interpolation Across The Dataset's Irregular Boundaries, Using Only IDL's Built-in Functions And Without Relying On External Libraries Or Regridding The Data To A Regular Grid?

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To perform a 3D convolution on a large, irregularly-gridded dataset in IDL without external libraries, follow these steps:

Approach

  1. Sort Coordinates and Store Indices:

    • Sort the x, y, and z coordinates of your dataset and keep track of their original indices. This allows efficient range queries using binary search.

  2. Define the Convolution Kernel:

    • Create a 3D kernel (e.g., Gaussian) and its corresponding coordinate grids. This kernel will be used to weight the neighboring points.

  3. Loop Through Each Data Point:

    • For each point, determine its coordinates and find all neighboring points within the kernel's radius using binary search on the sorted coordinates.

  4. Interpolate Kernel Weights:

    • For each neighbor, compute their relative position to the current point and interpolate the kernel's weight at that position.

  5. Compute the Convolved Value:

    • Sum the products of the neighbors' values and the interpolated kernel weights to get the convolved value at the current point.

  6. Handle Boundaries and Interpolation:

    • Use IDL's interpolation functions to handle points beyond the dataset boundaries, ensuring accurate results without regridding.

Solution Code

function convolve_3d_irregular, data, x, y, z, kernel, kernel_radius_x, kernel_radius_y, kernel_radius_z) {

; Get the dimensions of the data
n = n_elements(data)

; Sort the coordinates and store the original indices
sorted_x = sort(x, /index)
idx_x = sorted_x.index
sorted_y = sort(y, /index)
idx_y = sorted_y.index
sorted_z = sort(z, /index)
idx_z = sorted_z.index

; Initialize the result array
result = data * 0

; Define the kernel grid coordinates
kernel_size = size(kernel, /dimension)
kernel_x = findgen(kernel_size[0]) - (kernel_size[0]-1)/2
kernel_y = findgen(kernel_size[1]) - (kernel_size[1]-1)/2
kernel_z = findgen(kernel_size[2]) - (kernel_size[2]-1)/2

; Loop through each point in the dataset
for i = 0, n-1 do begin
    xi = x[i]
    yi = y[i]
    zi = z[i]

    ; Find indices in x within kernel_radius_x
    idx_min_x = bisect_left(sorted_x, xi - kernel_radius_x)
    idx_max_x = bisect_right(sorted_x, xi + kernel_radius_x)
    indices_x = idx_x[idx_min_x:idx_max_x-1]

    ; Find indices in y within kernel_radius_y
    idx_min_y = bisect_left(sorted_y, yi - kernel_radius_y)
    idx_max_y = bisect_right(sorted_y, yi + kernel_radius_y)
    indices_y = idx_y[idx_min_y:idx_max_y-1]

    ; Find indices in z within kernel_radius_z
    idx_min_z = bisect_left(sorted_z, zi - kernel_radius_z)
    idx_max_z = bisect_right(sorted_z, zi + kernel_radius_z)
    indices_z = idx_z[idx_min_z:idx_max_z-1]

    ; Generate all combinations of indices
    indices = array_indices(indices_x, indices_y, indices_z)

    ; Initialize sum
    sum = 0

    ; Loop through each neighbor
    for j = 0, n_elements(indices) - 1 do begin
        idx = indices[j]
        xj = x[idx]
        yj = y[idx]
        zj = z[idx]

        ; Compute relative position
        dx = xj - xi
        dy = yj - yi
        dz = zj - zi

        ; Interpolate kernel weight
        weight = interpolate(kernel, dx, dy, dz, $
                            x_coords=kernel_x, y_coords=kernel_y, z_coords=kernel_z, $
                            method='linear')

        ; Accumulate the product
        sum += data[idx] * weight
    endfor

    ; Assign the result
    result[i] = sum
endfor

return result



}

Explanation

  • Sorting and Indexing: By sorting the coordinates and storing their original indices, we can efficiently find the range of points within the kernel's radius using binary search.
  • Kernel Application: For each point, the kernel is applied by considering all neighbors within the kernel's radius. The kernel's weight at each neighbor's relative position is interpolated using IDL's INTERPOLATE function.
  • Efficiency Considerations: While this approach involves nested loops, using vectorized operations and efficient lookups minimizes computational overhead. The solution handles irregular boundaries by interpolating values beyond the dataset edges.

This method ensures accurate convolution while respecting the dataset's irregular grid structure, using only IDL's built-in functions.