// Assumes that f is no smaller than x.
func copyRealTo(f *rimg64.Image, x *fftw.Array2) {
for u := 0; u < f.Width; u++ {
for v := 0; v < f.Height; v++ {
f.Set(u, v, real(x.At(u, v)))
}
}
}
// CorrBankStrideFFT computes the strided correlation of
// an image with a bank of filters.
// h_p[u, v] = (f corr g_p)[stride*u, stride*v]
func CorrBankStrideFFT(f *rimg64.Image, g *Bank, stride int) (*rimg64.Multi, error) {
out := ValidSizeStride(f.Size(), g.Size(), stride)
if out.X <= 0 || out.Y <= 0 {
return nil, nil
}
// Compute strided convolution as the sum over
// a stride x stride grid of small convolutions.
grid := image.Pt(stride, stride)
// But do not divide into a larger grid than the size of the filter.
// If the filter is smaller than the stride,
// then some pixels in the image will not affect the output.
grid.X = min(grid.X, g.Width)
grid.Y = min(grid.Y, g.Height)
// Determine the size of the sub-sampled filter.
gsub := image.Pt(ceilDiv(g.Width, grid.X), ceilDiv(g.Height, grid.Y))
// The sub-sampled size of the image should be such that
// the output size is attained.
fsub := image.Pt(out.X+gsub.X-1, out.Y+gsub.Y-1)
// Determine optimal size for FFT.
work, _ := FFT2Size(fsub)
// Cache FFT of image for convolving with multiple filters.
// Re-use plan for multiple convolutions too.
fhat := fftw.NewArray2(work.X, work.Y)
ffwd := fftw.NewPlan2(fhat, fhat, fftw.Forward, fftw.Estimate)
defer ffwd.Destroy()
// FFT for current filter.
ghat := fftw.NewArray2(work.X, work.Y)
gfwd := fftw.NewPlan2(ghat, ghat, fftw.Forward, fftw.Estimate)
defer gfwd.Destroy()
// Allocate one array per output channel.
hhat := make([]*fftw.Array2, len(g.Filters))
for k := range hhat {
hhat[k] = fftw.NewArray2(work.X, work.Y)
}
// Normalization factor.
alpha := complex(1/float64(work.X*work.Y), 0)
// Add the convolutions over channels and strides.
for i := 0; i < grid.X; i++ {
for j := 0; j < grid.Y; j++ {
// Take transform of downsampled image given offset (i, j).
copyStrideTo(fhat, f, stride, image.Pt(i, j))
ffwd.Execute()
// Take transform of each downsampled channel given offset (i, j).
for q := range hhat {
copyStrideTo(ghat, g.Filters[q], stride, image.Pt(i, j))
gfwd.Execute()
addMul(hhat[q], ghat, fhat)
}
}
}
// Take the inverse transform of each channel.
h := rimg64.NewMulti(out.X, out.Y, len(g.Filters))
for q := range hhat {
scale(alpha, hhat[q])
fftw.IFFT2To(hhat[q], hhat[q])
copyRealToChannel(h, q, hhat[q])
}
return h, nil
}
// CorrBankFFT computes the correlation of an image with a bank of filters.
// h_p[u, v] = (f corr g_p)[u, v]
func CorrBankFFT(f *rimg64.Image, g *Bank) (*rimg64.Multi, error) {
out := ValidSize(f.Size(), g.Size())
if out.X <= 0 || out.Y <= 0 {
return nil, nil
}
// Determine optimal size for FFT.
work, _ := FFT2Size(f.Size())
// Re-use FFT of image.
fhat := fftw.NewArray2(work.X, work.Y)
copyImageTo(fhat, f)
fftw.FFT2To(fhat, fhat)
// Transform of each filter.
curr := fftw.NewArray2(work.X, work.Y)
fwd := fftw.NewPlan2(curr, curr, fftw.Forward, fftw.Estimate)
defer fwd.Destroy()
bwd := fftw.NewPlan2(curr, curr, fftw.Backward, fftw.Estimate)
defer bwd.Destroy()
h := rimg64.NewMulti(out.X, out.Y, len(g.Filters))
alpha := complex(1/float64(work.X*work.Y), 0)
// For each output channel.
for p, gp := range g.Filters {
// Take FFT.
copyImageTo(curr, gp)
fwd.Execute()
// h_p[x] = (G_p corr F)[x]
// H_p[x] = conj(G_p[x]) F[x]
scaleMul(curr, alpha, curr, fhat)
bwd.Execute()
copyRealToChannel(h, p, curr)
}
return h, nil
}
// Flip mirrors an image in x and y.
func Flip(f *rimg64.Image) *rimg64.Image {
g := rimg64.New(f.Width, f.Height)
for i := 0; i < f.Width; i++ {
for j := 0; j < f.Height; j++ {
g.Set(f.Width-1-i, f.Height-1-j, f.At(i, j))
}
}
return g
}
func square(f *rimg64.Image) *rimg64.Image {
g := rimg64.New(f.Width, f.Height)
for i := 0; i < f.Width; i++ {
for j := 0; j < f.Height; j++ {
g.Set(i, j, sqr(f.At(i, j)))
}
}
return g
}
// CorrBankBLAS computes the correlation of an image with a bank of filters.
// h_p[u, v] = (f corr g_p)[u, v]
func CorrBankBLAS(f *rimg64.Image, g *Bank) (*rimg64.Multi, error) {
out := ValidSize(f.Size(), g.Size())
if out.X <= 0 || out.Y <= 0 {
return nil, nil
}
// Express as dense matrix multiplication.
// h_p[u, v] = (f corr g_q)[u, v]
// Y(h) = A(f) X(g)
// If the number of output channels is k, then
// A is (M-m+1)(N-n+1) x mn and
// X is mn x k, so that
// Y is (M-m+1)(N-n+1) x k.
h := rimg64.NewMulti(out.X, out.Y, len(g.Filters))
m, n, k := g.Width, g.Height, len(g.Filters)
a := blas.NewMat(out.X*out.Y, m*n)
{
var r int
for u := 0; u < h.Width; u++ {
for v := 0; v < h.Height; v++ {
var s int
for i := 0; i < g.Width; i++ {
for j := 0; j < g.Height; j++ {
a.Set(r, s, f.At(i+u, j+v))
s++
}
}
r++
}
}
}
x := blas.NewMat(m*n, k)
{
var r int
for i := 0; i < g.Width; i++ {
for j := 0; j < g.Height; j++ {
for p := 0; p < h.Channels; p++ {
x.Set(r, p, g.Filters[p].At(i, j))
}
r++
}
}
}
y := blas.MatMul(1, a, x)
{
var r int
for u := 0; u < h.Width; u++ {
for v := 0; v < h.Height; v++ {
for p := 0; p < h.Channels; p++ {
h.Set(u, v, p, y.At(r, p))
}
r++
}
}
}
return h, nil
}
// CorrBankStrideBLAS computes the strided correlation of
// an image with a bank of filters.
// h_p[u, v] = (f corr g_p)[stride*u, stride*v]
func CorrBankStrideBLAS(f *rimg64.Image, g *Bank, stride int) (*rimg64.Multi, error) {
out := ValidSizeStride(f.Size(), g.Size(), stride)
if out.X <= 0 || out.Y <= 0 {
return nil, nil
}
h := rimg64.NewMulti(out.X, out.Y, len(g.Filters))
// Size of filters.
m, n := g.Width, g.Height
// Express as dense matrix multiplication.
// h_p[u, v] = sum_q (f_q corr g_pq)[u, v]
// h = A(f) X(g)
// where A is whk by mnk
// with w = ceil[(M-m+1)/stride],
// h = ceil[(N-n+1)/stride].
a := blas.NewMat(h.Width*h.Height, m*n)
{
var r int
for u := 0; u < h.Width; u++ {
for v := 0; v < h.Height; v++ {
var s int
for i := 0; i < g.Width; i++ {
for j := 0; j < g.Height; j++ {
a.Set(r, s, f.At(stride*u+i, stride*v+j))
s++
}
}
r++
}
}
}
x := blas.NewMat(m*n, h.Channels)
{
var r int
for i := 0; i < g.Width; i++ {
for j := 0; j < g.Height; j++ {
for p := 0; p < h.Channels; p++ {
x.Set(r, p, g.Filters[p].At(i, j))
}
r++
}
}
}
y := blas.MatMul(1, a, x)
{
var r int
for u := 0; u < h.Width; u++ {
for v := 0; v < h.Height; v++ {
for p := 0; p < h.Channels; p++ {
h.Set(u, v, p, y.At(r, p))
}
r++
}
}
}
return h, nil
}
// Decimate takes every r-th sample starting at (0, 0).
func Decimate(f *rimg64.Image, r int) *rimg64.Image {
out := ceilDivPt(f.Size(), r)
g := rimg64.New(out.X, out.Y)
for i := 0; i < g.Width; i++ {
for j := 0; j < g.Height; j++ {
g.Set(i, j, f.At(r*i, r*j))
}
}
return g
}
func copyImageTo(x *fftw.Array2, f *rimg64.Image) {
w, h := x.Dims()
for u := 0; u < w; u++ {
for v := 0; v < h; v++ {
if u < f.Width && v < f.Height {
x.Set(u, v, complex(f.At(u, v), 0))
} else {
x.Set(u, v, 0)
}
}
}
}
func invNorm(f *rimg64.Image) float64 {
var norm float64
for i := 0; i < f.Width; i++ {
for j := 0; j < f.Height; j++ {
norm += sqr(f.At(i, j))
}
}
norm = math.Sqrt(norm) // This will never be negative.
if norm == 0 {
return 0
}
return 1 / norm
}
// CorrStrideFFT computes the strided correlation of an image with a filter.
// h[u, v] = (f corr g)[stride*u, stride*v]
func CorrStrideFFT(f, g *rimg64.Image, stride int) (*rimg64.Image, error) {
out := ValidSizeStride(f.Size(), g.Size(), stride)
if out.X <= 0 || out.Y <= 0 {
return nil, nil
}
// Compute strided convolution as the sum over
// a stride x stride grid of small convolutions.
grid := image.Pt(stride, stride)
// But do not divide into a larger grid than the size of the filter.
// If the filter is smaller than the stride,
// then some pixels in the image will not affect the output.
grid.X = min(grid.X, g.Width)
grid.Y = min(grid.Y, g.Height)
// Determine the size of the sub-sampled filter.
gsub := image.Pt(ceilDiv(g.Width, grid.X), ceilDiv(g.Height, grid.Y))
// The sub-sampled size of the image should be such that
// the output size is attained.
fsub := image.Pt(out.X+gsub.X-1, out.Y+gsub.Y-1)
// Determine optimal size for FFT.
work, _ := FFT2Size(fsub)
// Cache FFT of each channel of image for convolving with multiple filters.
// Re-use plan for multiple convolutions too.
fhat := fftw.NewArray2(work.X, work.Y)
ffwd := fftw.NewPlan2(fhat, fhat, fftw.Forward, fftw.Estimate)
defer ffwd.Destroy()
// FFT for current filter.
curr := fftw.NewArray2(work.X, work.Y)
gfwd := fftw.NewPlan2(curr, curr, fftw.Forward, fftw.Estimate)
defer gfwd.Destroy()
// Normalization factor.
alpha := complex(1/float64(work.X*work.Y), 0)
// Add the convolutions over strides.
hhat := fftw.NewArray2(work.X, work.Y)
for i := 0; i < grid.X; i++ {
for j := 0; j < grid.Y; j++ {
// Copy each downsampled channel and take its transform.
copyStrideTo(fhat, f, stride, image.Pt(i, j))
ffwd.Execute()
copyStrideTo(curr, g, stride, image.Pt(i, j))
gfwd.Execute()
addMul(hhat, curr, fhat)
}
}
// Take the inverse transform.
h := rimg64.New(out.X, out.Y)
scale(alpha, hhat)
fftw.IFFT2To(hhat, hhat)
copyRealTo(h, hhat)
return h, nil
}
// dst[i, j] = src[i*stride + offset.X, j*stride + offset.Y],
// or zero if this is outside the boundary.
func copyStrideTo(dst *fftw.Array2, src *rimg64.Image, stride int, offset image.Point) {
m, n := dst.Dims()
bnds := image.Rect(0, 0, src.Width, src.Height)
for i := 0; i < m; i++ {
for j := 0; j < n; j++ {
p := image.Pt(i, j).Mul(stride).Add(offset)
var val complex128
if p.In(bnds) {
val = complex(src.At(p.X, p.Y), 0)
}
dst.Set(i, j, val)
}
}
}
// Tests whether (u, v) is a local maximum.
// Pixels at the edge can be maxima.
func notLocalMax(r *rimg64.Image, u, v int) bool {
uv := r.At(u, v)
if u > 0 && r.At(u-1, v) > uv {
return true
}
if u < r.Width-1 && r.At(u+1, v) > uv {
return true
}
if v > 0 && r.At(u, v-1) > uv {
return true
}
if v < r.Height-1 && r.At(u, v+1) > uv {
return true
}
return false
}
// CorrStrideBLAS computes the strided correlation of an image with a filter.
// h[u, v] = (f corr g)[stride*u, stride*v]
func CorrStrideBLAS(f, g *rimg64.Image, stride int) (*rimg64.Image, error) {
out := ValidSizeStride(f.Size(), g.Size(), stride)
if out.X <= 0 || out.Y <= 0 {
return nil, nil
}
h := rimg64.New(out.X, out.Y)
// Size of filters.
m, n := g.Width, g.Height
// Express as dense matrix multiplication.
// h[u, v] = (f corr g)[stride*u, stride*v]
// y(h) = A(f) x(g)
// where A is wh by mn
// with w = ceil[(M-m+1)/stride],
// h = ceil[(N-n+1)/stride].
a := blas.NewMat(h.Width*h.Height, m*n)
{
var r int
for u := 0; u < h.Width; u++ {
for v := 0; v < h.Height; v++ {
var s int
for i := 0; i < g.Width; i++ {
for j := 0; j < g.Height; j++ {
a.Set(r, s, f.At(stride*u+i, stride*v+j))
s++
}
}
r++
}
}
}
x := blas.NewMat(m*n, 1)
{
var r int
for i := 0; i < g.Width; i++ {
for j := 0; j < g.Height; j++ {
x.Set(r, 0, g.At(i, j))
r++
}
}
}
y := blas.MatMul(1, a, x)
{
var r int
for u := 0; u < h.Width; u++ {
for v := 0; v < h.Height; v++ {
h.Set(u, v, y.At(r, 0))
r++
}
}
}
return h, nil
}
// CorrBankStrideNaive computes the strided correlation of
// an image with a bank of filters.
// h_p[u, v] = (f corr g_p)[stride*u, stride*v]
func CorrBankStrideNaive(f *rimg64.Image, g *Bank, stride int) (*rimg64.Multi, error) {
out := ValidSizeStride(f.Size(), g.Size(), stride)
if out.X <= 0 || out.Y <= 0 {
return nil, nil
}
h := rimg64.NewMulti(out.X, out.Y, len(g.Filters))
for u := 0; u < h.Width; u++ {
for v := 0; v < h.Height; v++ {
for p := 0; p < h.Channels; p++ {
var sum float64
for i := 0; i < g.Width; i++ {
for j := 0; j < g.Height; j++ {
sum += f.At(stride*u+i, stride*v+j) * g.Filters[p].At(i, j)
}
}
h.Set(u, v, p, sum)
}
}
}
return h, nil
}
// CorrBankNaive computes the correlation of an image with a bank of filters.
// h_p[u, v] = (f corr g_p)[u, v]
func CorrBankNaive(f *rimg64.Image, g *Bank) (*rimg64.Multi, error) {
out := ValidSize(f.Size(), g.Size())
if out.X <= 0 || out.Y <= 0 {
return nil, nil
}
h := rimg64.NewMulti(out.X, out.Y, len(g.Filters))
for u := 0; u < h.Width; u++ {
for v := 0; v < h.Height; v++ {
for p := 0; p < h.Channels; p++ {
var total float64
for i := 0; i < g.Width; i++ {
for j := 0; j < g.Height; j++ {
total += f.At(i+u, j+v) * g.Filters[p].At(i, j)
}
}
h.Set(u, v, p, total)
}
}
}
return h, nil
}
// CorrStrideNaive computes the strided correlation of an image with a filter.
// h[u, v] = (f corr g)[stride*u, stride*v]
func CorrStrideNaive(f, g *rimg64.Image, stride int) (*rimg64.Image, error) {
out := ValidSizeStride(f.Size(), g.Size(), stride)
h := rimg64.New(out.X, out.Y)
for i := 0; i < h.Width; i++ {
for j := 0; j < h.Height; j++ {
var total float64
for u := 0; u < g.Width; u++ {
for v := 0; v < g.Height; v++ {
p := image.Pt(i, j).Mul(stride).Add(image.Pt(u, v))
total += f.At(p.X, p.Y) * g.At(u, v)
}
}
h.Set(i, j, total)
}
}
return h, nil
}
// CorrNaive computes the correlation of an image with a filter.
// h[u, v] = (f corr g)[u, v]
func CorrNaive(f, g *rimg64.Image) (*rimg64.Image, error) {
out := ValidSize(f.Size(), g.Size())
if out.X <= 0 || out.Y <= 0 {
return nil, nil
}
h := rimg64.New(out.X, out.Y)
for i := 0; i < out.X; i++ {
for j := 0; j < out.Y; j++ {
var total float64
for u := 0; u < g.Width; u++ {
for v := 0; v < g.Height; v++ {
total += f.At(i+u, j+v) * g.At(u, v)
}
}
h.Set(i, j, total)
}
}
return h, nil
}
// CorrAuto computes the correlation of an image with a filter.
// h[u, v] = (f corr g)[u, v]
// Automatically selects between naive and Fourier-domain convolution.
func CorrAuto(f, g *rimg64.Image) (*rimg64.Image, error) {
// Size of output.
size := ValidSize(f.Size(), g.Size())
// Return empty image if that's the result.
if size.Eq(image.ZP) {
return nil, nil
}
// Need to compute one inner product per output element.
naiveMuls := size.X * size.Y * g.Width * g.Height
// Optimal FFT size and number of multiplications.
_, fftMuls := FFT2Size(f.Size())
// Need to perform two forward and one inverse transform.
fftMuls *= 3
// Switch implementation based on image size.
if fftMuls < naiveMuls {
return CorrFFT(f, g)
}
return CorrNaive(f, g)
}
// CorrFFT computes the correlation of an image with a filter.
// h[u, v] = (f corr g)[u, v]
func CorrFFT(f, g *rimg64.Image) (*rimg64.Image, error) {
out := ValidSize(f.Size(), g.Size())
if out.X <= 0 || out.Y <= 0 {
return nil, nil
}
// Determine optimal size for FFT.
work, _ := FFT2Size(f.Size())
fhat := fftw.NewArray2(work.X, work.Y)
ghat := fftw.NewArray2(work.X, work.Y)
// Take forward transforms.
copyImageTo(fhat, f)
fftw.FFT2To(fhat, fhat)
copyImageTo(ghat, g)
fftw.FFT2To(ghat, ghat)
// Scale such that convolution theorem holds.
n := float64(work.X * work.Y)
scaleMul(fhat, complex(1/n, 0), ghat, fhat)
// Take inverse transform.
h := rimg64.New(out.X, out.Y)
fftw.IFFT2To(fhat, fhat)
copyRealTo(h, fhat)
return h, nil
}
func adjSum(f *rimg64.Image, x1, y1, x2, y2 int) float64 {
return f.At(x1, y1) + f.At(x1, y2) + f.At(x2, y1) + f.At(x2, y2)
}
// Avoids f.Set(x, y, f.Get(x, y, ...)).
func addTo(f *rimg64.Image, x, y int, v float64) {
f.Set(x, y, f.At(x, y)+v)
}
func errIfNotEqImage(f, g *rimg64.Image, eps float64) error {
if !f.Size().Eq(g.Size()) {
return fmt.Errorf("different size: %v, %v", f.Size(), g.Size())
}
for i := 0; i < f.Width; i++ {
for j := 0; j < f.Height; j++ {
a, b := f.At(i, j), g.At(i, j)
if math.Abs(a-b) > eps*math.Max(math.Abs(a), math.Abs(b)) {
return fmt.Errorf("different at x %d, y %d: %g, %g", i, j, a, b)
}
}
}
return nil
}