Fast convolutions (II)

I will analyze the algorithmic complexity of the convolution algorithms described in my previous posts.

To make things simpler, let’s assume that the dimensions of the image are >= the dimensions of the convolution kernel, and that both are square, with dimensions SxS and sxs, respectively.

1- Naive algorithm — O(n^4)

“w·h operations for each of the W·H image pixels”.

i.e., S·S·s·s operations. This is quadratic with a heavy constant for tiny kernels, but quickly becomes quartic for medium-to-large kernels.

The auxiliary buffer can be identical in size and bit-depth to the original image. So the memory usage factor is 2x.

2- Separable convolution — O(n^3)

“One 1D convolution for each row + One 1D convolution for each column”.

i.e., 2·S·S·s operations. This is quadratic with a bad constant for small kernels, but becomes cubic for large kernels.

Remarkably, the total running time depends on the dimensions of the image -and- the dimensions of the kernel.

Again, the auxiliary buffer can be identical in size and bit-depth to the original image. So the memory usage factor is 2x.

3- Convolution theorem — O(n^2·log(n))

“Two FFTs + One point-wise product + One IFFT”.

Let’s call S’ to the closest power-of-2 such that S’>=S. Then a proper implementation of the FFT/IFFT does (approx.) 2·S’·S’·log(S`) operations, while the point-wise product does S’·S’ operations. This makes the algorithm O(S’·S’·log(S’)) with some heavy (quadratic) overhead due to the memory copying, padding, and the point-wise product.

Remarkably, the total running time is independent of the size of the kernel.

This algorithm is quite memory hungry, though, because two S’xS’ complex-number buffers are required. This means two floating-point numbers per entry (i.e., the real/imaginary coefficients). The algorithm starts by copying the image/kernel to the real part of the corresponding complex-number buffer, leaving the imaginary coefficient and the surface excess filled with zeroes. Then the FFTs/product/IFFT happen in-place.

So the auxiliary memory required is 4·S’·S’ floating-point numbers.

In the worst case where S is a power-of-2 plus 1, S’ gets nearly twice as large as S. If the original image is 8-bit and we are using single-precision floating-point math for the FFT/IFFT, this means a memory usage factor of 64x. In the case of an HDR (single-precision floating-point) grayscale image, we are speaking of a worst case scenario of 16x. In average, however, the memory usage factor is around 8x. If S is a power-of-2, then the memory usage factor goes down to 4x.

Heavy glare using the FFT-based method in an HDR image by Paul Debevec

This image with heavy glare has been output with some Arion-related experimental tonemapping code I am working on these days.

Conclusions:

Assuming that we are only interested in sheer performance:

1- The FFT-based method is (by far) the fastest for large images/kernels. Interestingly, the algorithm is not affected by the size of the kernel, which can be as large as the (padded) image itself without a penalty.
2- The FFT-based method becomes even faster if the same kernel is applied multiple times. The kernel FFT can be calculated just once, and then be re-used.
3- Due to the heavy setup overhead in the FFT-based method, the separable method can be faster for small (separable) kernels where s is in the range of log(S’).

Last, but certainly not least, there is a much faster and more light-weight algorithm for the special case of Box/Gaussian Blur. I will talk about this in a separate blog entry.

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