Fix for EWA (elliptical weighted average) sampling in the compositor.

EWA sampling is designed for downsampling images, i.e. scaling down the size of
input image pixels, which happens regularly in compositing. While the standard
sampling methods (linear, cubic) work reasonably well for linear
transformations, they don't yield good results in non-linear cases like
perspective projection or arbitrary displacement. EWA sampling is comparable to
mipmapping, but avoids problems with discontinuities.

To work correctly the EWA algorithm needs partial derivatives of the mapping
functions which convert output pixel coordinates back into the input image
space (2x2 Jacobian matrix). With these derivatives the EWA algorithm
projects ellipses into the input space and accumulates colors over their
area. This calculation was not done correctly in the compositor, only the
derivatives du/dx and dv/dy were calculation, basically this means it only
worked for non-rotated input images.

The patch introduces full derivative calculations du/dx, du/dy, dv/dx, dv/dy for
the 3 nodes which use EWA sampling currently: PlaneTrackWarp, MapUV and
Displace. In addition the calculation of ellipsis area and axis-aligned
bounding boxes has been fixed.

For the MapUV and Displace nodes the derivatives have to be estimated by
evaluating the UV/displacement inputs with 1-pixel offsets, which can still have
problems on discontinuities and sub-pixel variations. These potential problems
can only be alleviated by more radical design changes in the compositor
functions, which are out of scope for now. Basically the values passed to the
UV/Displacement inputs would need to be associated with their 1st order
derivatives, which requires a general approach to derivatives in all nodes.

1 files changed, 83 insertions, 111 deletions

diff --git a/source/blender/compositor/intern/COM_MemoryBuffer.cpp b/source/blender/compositor/intern/COM_MemoryBuffer.cpp
index f10e6696c6a..8a010561adf 100644
--- a/source/blender/compositor/intern/COM_MemoryBuffer.cpp
+++ b/source/blender/compositor/intern/COM_MemoryBuffer.cpp
@@ -214,136 +214,108 @@ static const float EWA_WTS[EWA_MAXIDX + 1] = {
 	0.00754159f, 0.00625989f, 0.00498819f, 0.00372644f, 0.00247454f, 0.00123242f, 0.f
 };
 
-static void radangle2imp(float a2, float b2, float th, float *A, float *B, float *C, float *F)
+static void ellipse_bounds(float A, float B, float C, float F, float &xmax, float &ymax)
 {
-	float ct2 = cosf(th);
-	const float st2 = 1.f - ct2 * ct2;    // <- sin(th)^2
-	ct2 *= ct2;
-	*A = a2 * st2 + b2 * ct2;
-	*B = (b2 - a2) * sinf(2.f * th);
-	*C = a2 * ct2 + b2 * st2;
-	*F = a2 * b2;
-}
-
-// all tests here are done to make sure possible overflows are hopefully minimized
-static void imp2radangle(float A, float B, float C, float F, float *a, float *b, float *th, float *ecc)
-{
-	if (F <= 1e-5f) {   // use arbitrary major radius, zero minor, infinite eccentricity
-		*a = sqrtf(A > C ? A : C);
-		*b = 0.f;
-		*ecc = 1e10f;
-		*th = 0.5f * (atan2f(B, A - C) + (float)M_PI);
+	float denom = 4.0f*A*C - B*B;
+	if (denom > 0.0f && A != 0.0f && C != 0.0f) {
+		xmax = sqrt(F)/(2.0f*A) * (sqrt(F*(4.0f*A - B*B/C)) + B*B*sqrt(F/(C*denom)));
+		ymax = sqrt(F)/(2.0f*C) * (sqrt(F*(4.0f*C - B*B/A)) + B*B*sqrt(F/(A*denom)));
 	}
 	else {
-		const float AmC = A - C, ApC = A + C, F2 = F * 2.f;
-		const float r = sqrtf(AmC * AmC + B * B);
-		float d = ApC - r;
-		*a = (d <= 0.f) ? sqrtf(A > C ? A : C) : sqrtf(F2 / d);
-		d = ApC + r;
-		if (d <= 0.f) {
-			*b = 0.f;
-			*ecc = 1e10f;
-		}
-		else {
-			*b = sqrtf(F2 / d);
-			*ecc = *a / *b;
-		}
-		/* incr theta by 0.5 * pi (angle of major axis) */
-		*th = 0.5f * (atan2f(B, AmC) + (float)M_PI);
+		xmax = 0.0f;
+		ymax = 0.0f;
 	}
 }
 
-static float clipuv(float x, float limit)
+static void ellipse_params(float Ux, float Uy, float Vx, float Vy, float &A, float &B, float &C, float &F, float &umax, float &vmax)
 {
-	x = (x < 0) ? 0 : ((x >= limit) ? (limit - 1) : x);
-	return x;
+	A = Vx*Vx + Vy*Vy;
+	B = -2.0f * (Ux*Vx + Uy*Vy);
+	C = Ux*Ux + Uy*Uy;
+	F = A*C - B*B * 0.25f;
+
+	float factor = (F != 0.0f ? (float)(EWA_MAXIDX+1) / F : 0.0f);
+	A *= factor;
+	B *= factor;
+	C *= factor;
+	F = (float)(EWA_MAXIDX+1);
+
+	ellipse_bounds(A, B, C, sqrtf(F), umax, vmax);
 }
 
 /**
- * \note \a sampler at the moment is either 'COM_PS_NEAREST' or not, other values won't matter.
+ * Filtering method based on
+ * "Creating raster omnimax images from multiple perspective views using the elliptical weighted average filter"
+ * by Ned Greene and Paul S. Heckbert (1986)
  */
-void MemoryBuffer::readEWA(float result[4], float fx, float fy, float dx, float dy, PixelSampler sampler)
+void MemoryBuffer::readEWA(float result[4], const float uv[2], const float derivatives[2][2], PixelSampler sampler)
 {
-	const int width = this->getWidth(), height = this->getHeight();
-	
-	// scaling dxt/dyt by full resolution can cause overflow because of huge A/B/C and esp. F values,
-	// scaling by aspect ratio alone does the opposite, so try something in between instead...
-	const float ff2 = width, ff = sqrtf(ff2), q = height / ff;
-	const float Ux = dx * ff, Vx = dx * q, Uy = dy * ff, Vy = dy * q;
-	float A = Vx * Vx + Vy * Vy;
-	float B = -2.f * (Ux * Vx + Uy * Vy);
-	float C = Ux * Ux + Uy * Uy;
-	float F = A * C - B * B * 0.25f;
-	float a, b, th, ecc, a2, b2, ue, ve, U0, V0, DDQ, U, ac1, ac2, BU, d;
-	int u, v, u1, u2, v1, v2;
-	// The so-called 'high' quality ewa method simply adds a constant of 1 to both A & C,
-	// so the ellipse always covers at least some texels. But since the filter is now always larger,
-	// it also means that everywhere else it's also more blurry then ideally should be the case.
-	// So instead here the ellipse radii are modified instead whenever either is too low.
-	// Use a different radius based on interpolation switch, just enough to anti-alias when interpolation is off,
-	// and slightly larger to make result a bit smoother than bilinear interpolation when interpolation is on
-	// (minimum values: const float rmin = intpol ? 1.f : 0.5f;)
-
-	/* note: 0.765625f is too sharp, 1.0 will not blur with an exact pixel sample
-	 * useful to avoid blurring when there is no distortion */
-#if 0
-	const float rmin = ((sampler != COM_PS_NEAREST) ? 1.5625f : 0.765625f) / ff2;
-#else
-	const float rmin = ((sampler != COM_PS_NEAREST) ? 1.5625f : 1.0f     ) / ff2;
-#endif
-	imp2radangle(A, B, C, F, &a, &b, &th, &ecc);
-	if ((b2 = b * b) < rmin) {
-		if ((a2 = a * a) < rmin) {
-			B = 0.f;
-			A = C = rmin;
-			F = A * C;
-		}
-		else {
-			b2 = rmin;
-			radangle2imp(a2, b2, th, &A, &B, &C, &F);
-		}
-	}
+	zero_v4(result);
+	int width = this->getWidth(), height = this->getHeight();
+	if (width == 0 || height == 0)
+		return;
 
-	ue = ff * sqrtf(C);
-	ve = ff * sqrtf(A);
-	d = (float)(EWA_MAXIDX + 1) / (F * ff2);
-	A *= d;
-	B *= d;
-	C *= d;
-
-	U0 = fx;
-	V0 = fy;
-	u1 = (int)(floorf(U0 - ue));
-	u2 = (int)(ceilf(U0 + ue));
-	v1 = (int)(floorf(V0 - ve));
-	v2 = (int)(ceilf(V0 + ve));
-	U0 -= 0.5f;
-	V0 -= 0.5f;
-	DDQ = 2.f * A;
-	U = u1 - U0;
-	ac1 = A * (2.f * U + 1.f);
-	ac2 = A * U * U;
-	BU = B * U;
-
-	d = result[0] = result[1] = result[2] = result[3] = 0.f;
-	for (v = v1; v <= v2; ++v) {
-		const float V = v - V0;
-		float DQ = ac1 + B * V;
-		float Q = (C * V + BU) * V + ac2;
-		for (u = u1; u <= u2; ++u) {
-			if (Q < (float)(EWA_MAXIDX + 1)) {
+	float u = uv[0], v = uv[1];
+	float Ux = derivatives[0][0], Vx = derivatives[1][0], Uy = derivatives[0][1], Vy = derivatives[1][1];
+	float A, B, C, F, ue, ve;
+	ellipse_params(Ux, Uy, Vx, Vy, A, B, C, F, ue, ve);
+
+	/* Note: highly eccentric ellipses can lead to large texture space areas to filter!
+	 * This is limited somewhat by the EWA_WTS size in the loop, but a nicer approach
+	 * could be the one found in
+	 * "High Quality Elliptical Texture Filtering on GPU"
+	 * by Pavlos Mavridis and Georgios Papaioannou
+	 * in which the eccentricity of the ellipse is clamped.
+	 */
+
+	int U0 = (int)u;
+	int V0 = (int)v;
+	/* pixel offset for interpolation */
+	float ufac = u - floor(u), vfac = v - floor(v);
+	/* filter size */
+	int u1 = (int)(u - ue);
+	int u2 = (int)(u + ue);
+	int v1 = (int)(v - ve);
+	int v2 = (int)(v + ve);
+
+	/* sane clamping to avoid unnecessarily huge loops */
+	/* note: if eccentricity gets clamped (see above),
+	 * the ue/ve limits can also be lowered accordingly
+	 */
+	if (U0-u1 > EWA_MAXIDX) u1 = U0 - EWA_MAXIDX;
+	if (u2-U0 > EWA_MAXIDX) u2 = U0 + EWA_MAXIDX;
+	if (V0-v1 > EWA_MAXIDX) v1 = V0 - EWA_MAXIDX;
+	if (v2-V0 > EWA_MAXIDX) v2 = V0 + EWA_MAXIDX;
+
+	float DDQ = 2.f * A;
+	float U = u1 - U0;
+	float ac1 = A * (2.f*U + 1.f);
+	float ac2 = A * U*U;
+	float BU = B * U;
+
+	float sum = 0.0f;
+	for (int v = v1; v <= v2; ++v) {
+		float V = v - V0;
+
+		float DQ = ac1 + B*V;
+		float Q = (C*V + BU)*V + ac2;
+		for (int u = u1; u <= u2; ++u) {
+			if (Q < F) {
 				float tc[4];
-				const float wt = EWA_WTS[(Q < 0.f) ? 0 : (unsigned int)Q];
-				read(tc, clipuv(u, width), clipuv(v, height));
+				const float wt = EWA_WTS[CLAMPIS((int)Q, 0, EWA_MAXIDX)];
+				switch (sampler) {
+					case COM_PS_NEAREST: read(tc, u, v); break;
+					case COM_PS_BILINEAR: readBilinear(tc, (float)u+ufac, (float)v+vfac); break;
+					case COM_PS_BICUBIC: readBilinear(tc, (float)u+ufac, (float)v+vfac); break; /* XXX no readBicubic method yet */
+					default: zero_v4(tc); break;
+				}
 				madd_v4_v4fl(result, tc, wt);
-				d += wt;
+				sum += wt;
 			}
 			Q += DQ;
 			DQ += DDQ;
 		}
 	}
 	
-	// d should hopefully never be zero anymore
-	d = 1.f / d;
-	mul_v4_fl(result, d);
+	mul_v4_fl(result, (sum != 0.0f ? 1.0f / sum : 0.0f));
 }