Fast implementation of covariance of two 8-bit arrays

Question

I need to compare a big amount of similar images of small size (up to 200x200). So I try to implement SSIM (Structural similarity see https://en.wikipedia.org/wiki/Structural_similarity ) algorithm. SSIM requires calculation of covariance of two 8-bit gray images. A trivial implementation look like:

float SigmaXY(const uint8_t * x, const uint8_t * y, size_t size, float averageX, float averageY)
{
    float sum = 0;
    for(size_t i = 0; i < size; ++i)
        sum += (x[i] - averageX) * (y[i] - averageY);
    return sum / size;
}

But it has poor performance. So I hope to improve it with using SIMD or CUDA (I heard that it can be done). Unfortunately I have no experience to do this. How it will look? And where I have to go?

Answer 1

I have another nice solution!

At first I want to mention some mathematical formulas:

averageX = Sum(x[i])/size;
averageY = Sum(y[i])/size;

And therefore:

Sum((x[i] - averageX)*(y[i] - averageY))/size = 

Sum(x[i]*y[i])/size - Sum(x[i]*averageY)/size - 
Sum(averageX*y[i])/size + Sum(averageX*averageY)/size = 

Sum(x[i]*y[i])/size - averageY*Sum(x[i])/size - 
averageX*Sum(y[i])/size + averageX*averageY*Sum(1)/size =   

Sum(x[i]*y[i])/size - averageY*averageX - 
averageX*averageY + averageX*averageY = 

Sum(x[i]*y[i])/size - averageY*averageX;     

It allows to modify our algorithm:

float SigmaXY(const uint8_t * x, const uint8_t * y, size_t size, float averageX, float averageY)
{
    uint32_t sum = 0; // If images will have size greater then 256x256 than you have to use uint64_t.
    for(size_t i = 0; i < size; ++i)
        sum += x[i]*y[i];
    return sum / size - averageY*averageX;
} 

And only after that we can use SIMD (I used SSE2):

#include <emmintrin.h>

inline __m128i SigmaXY(__m128i x, __m128i y)
{
    __m128i lo = _mm_madd_epi16(_mm_unpacklo_epi8(x, _mm_setzero_si128()), _mm_unpacklo_epi8(y, _mm_setzero_si128()));
    __m128i hi = _mm_madd_epi16(_mm_unpackhi_epi8(y, _mm_setzero_si128()), _mm_unpackhi_epi8(y, _mm_setzero_si128()));
    return _mm_add_epi32(lo, hi);
}

float SigmaXY(const uint8_t * x, const uint8_t * y, size_t size, float averageX, float averageY)
{
    uint32_t sum = 0;
    size_t i = 0, alignedSize = size/16*16;
    if(size >= 16)
    {
        __m128i sums = _mm_setzero_si128();
        for(; i < alignedSize; i += 16)
        {
            __m128i _x = _mm_loadu_si128((__m128i*)(x + i));
            __m128i _y = _mm_loadu_si128((__m128i*)(y + i));
            sums = _mm_add_epi32(sums, SigmaXY(_x, _y));
        }
        uint32_t _sums[4];
        _mm_storeu_si128(_sums, sums);
        sum = _sums[0] + _sums[1] + _sums[2] + _sums[3];
    } 
    for(; i < size; ++i)
        sum += x[i]*y[i];
    return sum / size - averageY*averageX;
}