func (phi *AdjChanNorm) Apply(x *rimg64.Multi) (*rimg64.Multi, error) {
	y := rimg64.NewMulti(x.Width, x.Height, x.Channels)
	r := (phi.Num - 1) / 2
	for i := 0; i < x.Width; i++ {
		for j := 0; j < x.Height; j++ {
			k := 0
			// Range over which to compute sum.
			a, b := k-r, k+r+1
			// Take sum excluding leading element.
			var t float64
			for p := 0; p < min(b, x.Channels); p++ {
				t += sqr(x.At(i, j, p))
			}
			for ; k < x.Channels; k++ {
				a, b = k-r, k+r+1
				// Set element.
				norm := math.Pow(phi.K+phi.Alpha*t, phi.Beta)
				y.Set(i, j, k, x.At(i, j, k)/norm)
				// Subtract trailing element.
				if a >= 0 {
					t -= sqr(x.At(i, j, a))
				}
				// Add leading element.
				if b < x.Channels {
					t += sqr(x.At(i, j, b))
				}
			}
		}
	}
	return y, nil
}

func (phi SumPool) Apply(x *rimg64.Multi) (*rimg64.Multi, error) {
	if phi.Field.X <= 0 || phi.Field.Y <= 0 {
		err := fmt.Errorf("invalid field size: %v", phi.Field)
		return nil, err
	}
	if phi.Stride <= 0 {
		err := fmt.Errorf("invalid stride: %d", phi.Stride)
		return nil, err
	}
	size := image.Pt(
		ceilDiv(x.Width-phi.Field.X+1, phi.Stride),
		ceilDiv(x.Height-phi.Field.Y+1, phi.Stride),
	)
	y := rimg64.NewMulti(size.X, size.Y, x.Channels)
	for i := 0; i < y.Width; i++ {
		for j := 0; j < y.Height; j++ {
			for k := 0; k < x.Channels; k++ {
				// Position in original image.
				p := image.Pt(i, j).Mul(phi.Stride)
				var t float64
				for u := p.X; u < p.X+phi.Field.X; u++ {
					for v := p.Y; v < p.Y+phi.Field.Y; v++ {
						t += x.At(u, v, k)
					}
				}
				y.Set(i, j, k, t)
			}
		}
	}
	return y, 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
}

// EvalFunc evaluates a function on every window in an image.
// If the input image is M x N and the window size is m x n,
// then the output is (M-m+1) x (N-n+1).
// If the window size is larger than the image size in either dimension,
// a nil image is returned with no error.
func EvalFunc(im *rimg64.Multi, size image.Point, f ScoreFunc) (*rimg64.Image, error) {
	if im.Width < size.X || im.Height < size.Y {
		return nil, nil
	}
	r := rimg64.New(im.Width-size.X+1, im.Height-size.Y+1)
	x := rimg64.NewMulti(size.X, size.Y, im.Channels)
	for i := 0; i < r.Width; i++ {
		for j := 0; j < r.Height; j++ {
			// Copy window into x.
			for u := 0; u < size.X; u++ {
				for v := 0; v < size.Y; v++ {
					for p := 0; p < im.Channels; p++ {
						x.Set(u, v, p, im.At(i+u, j+v, p))
					}
				}
			}
			y, err := f(x)
			if err != nil {
				return nil, err
			}
			r.Set(i, j, y)
		}
	}
	return r, nil
}

// CorrMultiBankStrideBLAS computes the strided correlation of
// a multi-channel image with a bank of multi-channel filters.
// 	h_p[u, v] = sum_q (f_q corr g_pq)[stride*u, stride*v]
func CorrMultiBankStrideBLAS(f *rimg64.Multi, g *MultiBank, 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, k := g.Width, g.Height, g.Channels
	// 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*k)
	{
		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++ {
						for q := 0; q < g.Channels; q++ {
							a.Set(r, s, f.At(stride*u+i, stride*v+j, q))
							s++
						}
					}
				}
				r++
			}
		}
	}
	x := blas.NewMat(m*n*k, h.Channels)
	{
		var r int
		for i := 0; i < g.Width; i++ {
			for j := 0; j < g.Height; j++ {
				for q := 0; q < g.Channels; q++ {
					for p := 0; p < h.Channels; p++ {
						x.Set(r, p, g.Filters[p].At(i, j, q))
					}
					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
}

// 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
}

func (phi *MaxPool) Apply(x *rimg64.Multi) (*rimg64.Multi, error) {
	if phi.Field.X <= 0 || phi.Field.Y <= 0 {
		err := fmt.Errorf("invalid field size: %v", phi.Field)
		return nil, err
	}
	if phi.Stride <= 0 {
		err := fmt.Errorf("invalid stride: %d", phi.Stride)
		return nil, err
	}
	size := image.Pt(
		ceilDiv(x.Width-phi.Field.X+1, phi.Stride),
		ceilDiv(x.Height-phi.Field.Y+1, phi.Stride),
	)
	y := rimg64.NewMulti(size.X, size.Y, x.Channels)
	for i := 0; i < y.Width; i++ {
		for j := 0; j < y.Height; j++ {
			for k := 0; k < x.Channels; k++ {
				// Position in original image.
				p := image.Pt(i, j).Mul(phi.Stride)
				max := math.Inf(-1)
				for u := 0; u < phi.Field.X; u++ {
					for v := 0; v < phi.Field.Y; v++ {
						q := p.Add(image.Pt(u, v))
						max = math.Max(max, x.At(q.X, q.Y, k))
					}
				}
				y.Set(i, j, k, max)
			}
		}
	}
	return y, nil
}

// 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
}

func (phi *PosPart) Apply(x *rimg64.Multi) (*rimg64.Multi, error) {
	y := rimg64.NewMulti(x.Width, x.Height, x.Channels)
	for i := 0; i < x.Width; i++ {
		for j := 0; j < x.Height; j++ {
			for k := 0; k < x.Channels; k++ {
				y.Set(i, j, k, math.Max(0, x.At(i, j, k)))
			}
		}
	}
	return y, nil
}

// FlipMulti mirrors a multi-channel image in x and y.
func FlipMulti(f *rimg64.Multi) *rimg64.Multi {
	g := rimg64.NewMulti(f.Width, f.Height, f.Channels)
	for i := 0; i < f.Width; i++ {
		for j := 0; j < f.Height; j++ {
			for k := 0; k < f.Channels; k++ {
				g.Set(f.Width-1-i, f.Height-1-j, k, f.At(i, j, k))
			}
		}
	}
	return g
}

func randMulti(width, height, channels int) *rimg64.Multi {
	f := rimg64.NewMulti(width, height, channels)
	for i := 0; i < width; i++ {
		for j := 0; j < height; j++ {
			for k := 0; k < channels; k++ {
				f.Set(i, j, k, rand.NormFloat64())
			}
		}
	}
	return f
}

func (phi *ChannelInterval) Apply(f *rimg64.Multi) (*rimg64.Multi, error) {
	g := rimg64.NewMulti(f.Width, f.Height, phi.B-phi.A)
	for u := 0; u < f.Width; u++ {
		for v := 0; v < f.Height; v++ {
			for p := phi.A; p < phi.B; p++ {
				g.Set(u, v, p-phi.A, f.At(u, v, p))
			}
		}
	}
	return g, nil
}

func (phi *Scale) Apply(x *rimg64.Multi) (*rimg64.Multi, error) {
	y := rimg64.NewMulti(x.Width, x.Height, x.Channels)
	for u := 0; u < x.Width; u++ {
		for v := 0; v < x.Height; v++ {
			for p := 0; p < x.Channels; p++ {
				y.Set(u, v, p, float64(*phi)*x.At(u, v, p))
			}
		}
	}
	return y, nil
}

// DecimateMulti takes every r-th sample starting at (0, 0).
func DecimateMulti(f *rimg64.Multi, r int) *rimg64.Multi {
	out := ceilDivPt(f.Size(), r)
	g := rimg64.NewMulti(out.X, out.Y, f.Channels)
	for i := 0; i < g.Width; i++ {
		for j := 0; j < g.Height; j++ {
			for k := 0; k < g.Channels; k++ {
				g.Set(i, j, k, f.At(r*i, r*j, k))
			}
		}
	}
	return g
}

func parseImageCSV(rows [][]string) (*rimg64.Multi, error) {
	// Convert to numbers.
	var (
		u, v, w []int
		f       []float64
	)
	for _, r := range rows {
		if len(r) == 0 {
			continue
		}
		if len(r) != 4 {
			panic("wrong number of elements in row")
		}
		ui, err := strconv.ParseInt(r[0], 10, 32)
		if err != nil {
			return nil, err
		}
		vi, err := strconv.ParseInt(r[1], 10, 32)
		if err != nil {
			return nil, err
		}
		wi, err := strconv.ParseInt(r[2], 10, 32)
		if err != nil {
			return nil, err
		}
		fi, err := strconv.ParseFloat(r[3], 64)
		if err != nil {
			return nil, err
		}
		u = append(u, int(ui))
		v = append(v, int(vi))
		w = append(w, int(wi))
		f = append(f, fi)
	}

	// Take max over u, v, w.
	var width, height, channels int
	for i := range u {
		width = max(u[i]+1, width)
		height = max(v[i]+1, height)
		channels = max(w[i]+1, channels)
	}

	// Set pixels in image.
	im := rimg64.NewMulti(width, height, channels)
	for i := range u {
		im.Set(u[i], v[i], w[i], f[i])
	}
	return im, nil
}

// CorrMultiBankFFT computes the correlation of
// a multi-channel image with a bank of multi-channel filters.
// 	h_p[u, v] = sum_q (f_q corr g_pq)[u, v]
func CorrMultiBankFFT(f *rimg64.Multi, g *MultiBank) (*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())
	// Cache FFT of each channel of image.
	fhat := make([]*fftw.Array2, f.Channels)
	for i := range fhat {
		fhat[i] = fftw.NewArray2(work.X, work.Y)
		copyChannelTo(fhat[i], f, i)
		fftw.FFT2To(fhat[i], fhat[i])
	}

	curr := fftw.NewArray2(work.X, work.Y)
	fwd := fftw.NewPlan2(curr, curr, fftw.Forward, fftw.Estimate)
	defer fwd.Destroy()
	sum := fftw.NewArray2(work.X, work.Y)
	bwd := fftw.NewPlan2(sum, sum, 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 {
		zero(sum)
		// For each input channel.
		for q := 0; q < f.Channels; q++ {
			// Take FFT of this input channel.
			copyChannelTo(curr, gp, q)
			fwd.Execute()
			// h_p[x] = (G_qp corr F_p)[x]
			// H_p[x] = conj(G_qp[x]) F_p[x]
			addScaleMul(sum, alpha, curr, fhat[q])
		}
		bwd.Execute()
		copyRealToChannel(h, p, sum)
	}
	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
}

// Flattens 31 (or 27) channels down to 9 for visualization.
func compress(src *rimg64.Multi, weights WeightSet) *rimg64.Multi {
	dst := rimg64.NewMulti(src.Width, src.Height, 9)
	for i := 0; i < 27; i++ {
		for x := 0; x < src.Width; x++ {
			for y := 0; y < src.Height; y++ {
				v := src.At(x, y, i)
				switch weights {
				default:
				case Pos:
					v = math.Max(0, v)
				case Neg:
					v = math.Min(0, v)
				case Abs:
					v = math.Abs(v)
				}
				dst.Set(x, y, i%9, dst.At(x, y, i%9)+v)
			}
		}
	}
	return dst
}

// CorrMultiBankStrideNaive computes the strided correlation of
// a multi-channel image with a bank of multi-channel filters.
// 	h_p[u, v] = sum_q (f_q corr g_pq)[stride*u, stride*v]
func CorrMultiBankStrideNaive(f *rimg64.Multi, g *MultiBank, 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++ {
				for i := 0; i < g.Width; i++ {
					for j := 0; j < g.Height; j++ {
						for q := 0; q < g.Channels; q++ {
							val := f.At(stride*u+i, stride*v+j, q) * g.Filters[p].At(i, j, q)
							h.Set(u, v, p, h.At(u, v, p)+val)
						}
					}
				}
			}
		}
	}
	return h, nil
}