Fast implementation of covariance of two 8-bit arrays

Question

I need to compare a large number of similar images of a small size (up to 200x200). Therefore, I am trying to implement the SSIM (structural similarities see https://en.wikipedia.org/wiki/Structural_similarity ) algorithm. SSIM requires the calculation of the covariance of two 8-bit gray images. The trivial implementation looks like this:

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 SIMD or CUDA (I heard that it can be done). Unfortunately, I have no experience for this. How will it look like? And where should I go?

Answer 1

!

:

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

:

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;     

:

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;
} 

SIMD ( 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;
}

Answer 2

SIMD ( SSE4.1):

#include <smmintrin.h>

template <int shift> inline __m128 SigmaXY(const __m128i & x, const __m128i & y, __m128 & averageX, __m128 & averageY)
{
    __m128 _x = _mm_cvtepi32_ps(_mm_cvtepu8_epi32(_mm_srli_si128(x, shift)));
    __m128 _y = _mm_cvtepi32_ps(_mm_cvtepu8_epi32(_mm_srli_si128(y, shift)));
    return _mm_mul_ps(_mm_sub_ps(_x, averageX), _mm_sub_ps(_y, averageY))
}

float SigmaXY(const uint8_t * x, const uint8_t * y, size_t size, float averageX, float averageY)
{
    float sum = 0;
    size_t i = 0, alignedSize = size/16*16;
    if(size >= 16)
    {
        __m128 sums = _mm_setzero_ps();
        __m128 avgX = _mm_set1_ps(averageX);
        __m128 avgY = _mm_set1_ps(averageY);
        for(; i < alignedSize; i += 16)
        {
            __m128i _x = _mm_loadu_si128((__m128i*)(x + i));
            __m128i _y = _mm_loadu_si128((__m128i*)(y + i));
            sums = _mm_add_ps(sums, SigmaXY<0>(_x, _y, avgX, avgY);
            sums = _mm_add_ps(sums, SigmaXY<4>(_x, _y, avgX, avgY);
            sums = _mm_add_ps(sums, SigmaXY<8>(_x, _y, avgX, avgY);
            sums = _mm_add_ps(sums, SigmaXY<12>(_x, _y, avgX, avgY);
        }
        float _sums[4];
        _mm_storeu_ps(_sums, sums);
        sum = _sums[0] + _sums[1] + _sums[2] + _sums[3];
    }
    for(; i < size; ++i)
        sum += (x[i] - averageX) * (y[i] - averageY);
    return sum / size;
}

, .