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
}

// Assumes that f is no smaller than x.
func copyRealToChannel(f *rimg64.Multi, p int, x *fftw.Array2) {
	for u := 0; u < f.Width; u++ {
		for v := 0; v < f.Height; v++ {
			f.Set(u, v, p, real(x.At(u, v)))
		}
	}
}

// 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
}

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
}

// 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
}

// CorrMultiStrideFFT computes the correlation of
// a multi-channel image with a multi-channel filter.
// 	h[u, v] = sum_q (f_q corr g_q)[u, v]
func CorrMultiStrideFFT(f, g *rimg64.Multi, stride int) (*rimg64.Image, error) {
	if err := errIfChannelsNotEq(f, g); err != nil {
		panic(err)
	}
	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()
	ghat := fftw.NewArray2(work.X, work.Y)
	gfwd := fftw.NewPlan2(ghat, ghat, fftw.Forward, fftw.Estimate)
	defer gfwd.Destroy()
	// Normalization factor.
	alpha := complex(1/float64(work.X*work.Y), 0)
	// Add the convolutions over channels and strides.
	hhat := fftw.NewArray2(work.X, work.Y)
	for k := 0; k < f.Channels; k++ {
		for i := 0; i < grid.X; i++ {
			for j := 0; j < grid.Y; j++ {
				// Copy each downsampled channel and take its transform.
				copyChannelStrideTo(fhat, f, k, stride, image.Pt(i, j))
				ffwd.Execute()
				copyChannelStrideTo(ghat, g, k, stride, image.Pt(i, j))
				gfwd.Execute()
				addMul(hhat, ghat, 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
}

// 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 (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
}

// Takes the sum over channels.
func squareMulti(f *rimg64.Multi) *rimg64.Image {
	g := rimg64.New(f.Width, f.Height)
	for i := 0; i < f.Width; i++ {
		for j := 0; j < f.Height; j++ {
			for k := 0; k < f.Channels; k++ {
				g.Set(i, j, g.At(i, j)+sqr(f.At(i, j, k)))
			}
		}
	}
	return g
}

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

func dot(x, y *rimg64.Multi) float64 {
	var d float64
	for i := 0; i < x.Width; i++ {
		for j := 0; j < x.Height; j++ {
			for k := 0; k < x.Channels; k++ {
				d += x.At(i, j, k) * y.At(i, j, k)
			}
		}
	}
	return d
}

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
}

// 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
}

// Assumes that f is no smaller than x.
// Pads with zeros.
func copyChannelTo(x *fftw.Array2, f *rimg64.Multi, p int) {
	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, p), 0))
			} else {
				x.Set(u, v, 0)
			}
		}
	}
}

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
}

// dst[i, j] = src[i*stride + offset.X, j*stride + offset.Y],
// or zero if this is outside the boundary.
func copyChannelStrideTo(dst *fftw.Array2, src *rimg64.Multi, channel, 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, channel), 0)
			}
			dst.Set(i, j, val)
		}
	}
}

func drawCell(feat *rimg64.Multi, i, j int, gc *draw2d.ImageGraphicContext, cell int) {
	u := (float64(i) + 0.5) * float64(cell)
	v := (float64(j) + 0.5) * float64(cell)
	r := float64(cell) / 2

	for k := 0; k < Orientations; k++ {
		x := feat.At(i, j, k)
		x = math.Max(x, 0)
		x = math.Min(x, 1)
		gc.SetStrokeColor(color.Gray{uint8(x*254 + 1)})
		theta := (0.5 + float64(k)/float64(Orientations)) * math.Pi
		drawOrientedLine(gc, u, v, theta, r)
	}
}

func invNormMulti(f *rimg64.Multi) float64 {
	var norm float64
	for i := 0; i < f.Width; i++ {
		for j := 0; j < f.Height; j++ {
			for k := 0; k < f.Channels; k++ {
				norm += sqr(f.At(i, j, k))
			}
		}
	}
	norm = math.Sqrt(norm) // This cannot be negative.
	if norm == 0 {
		return 0
	}
	return 1 / norm
}

func formatImageCSV(im *rimg64.Multi) [][]string {
	var rows [][]string
	for u := 0; u < im.Width; u++ {
		for v := 0; v < im.Height; v++ {
			for w := 0; w < im.Channels; w++ {
				r := make([]string, 4)
				r[0] = strconv.FormatInt(int64(u), 10)
				r[1] = strconv.FormatInt(int64(v), 10)
				r[2] = strconv.FormatInt(int64(w), 10)
				r[3] = strconv.FormatFloat(im.At(u, v, w), 'g', -1, 64)
				rows = append(rows, r)
			}
		}
	}
	return rows
}

// Returns gradient with greatest magnitude across all channels.
// 1 <= x <= width-2, 1 <= y <= height-2
func maxGrad(f *rimg64.Multi, x, y int) (point, float64) {
	var (
		grad point
		max  float64
	)
	for d := 0; d < f.Channels; d++ {
		p := point{
			f.At(x+1, y, d) - f.At(x-1, y, d),
			f.At(x, y+1, d) - f.At(x, y-1, d),
		}
		v := p.X*p.X + p.Y*p.Y
		if v > max {
			grad, max = p, v
		}
	}
	return grad, max
}