--- ray/src/rt/ambcomp.c	2005/05/28 22:27:54	2.15
+++ ray/src/rt/ambcomp.c	2014/05/07 16:02:26	2.50
@@ -1,44 +1,899 @@
 #ifndef lint
-static const char	RCSid[] = "$Id: ambcomp.c,v 2.15 2005/05/28 22:27:54 greg Exp $";
+static const char	RCSid[] = "$Id: ambcomp.c,v 2.50 2014/05/07 16:02:26 greg Exp $";
 #endif
 /*
  * Routines to compute "ambient" values using Monte Carlo
  *
+ *  Hessian calculations based on "Practical Hessian-Based Error Control
+ *	for Irradiance Caching" by Schwarzhaupt, Wann Jensen, & Jarosz
+ *	from ACM SIGGRAPH Asia 2012 conference proceedings.
+ *
+ *  Added book-keeping optimization to avoid calculations that would
+ *	cancel due to traversal both directions on edges that are adjacent
+ *	to same-valued triangles.  This cuts about half of Hessian math.
+ *
  *  Declarations of external symbols in ambient.h
  */
 
 #include "copyright.h"
 
 #include  "ray.h"
-
 #include  "ambient.h"
-
 #include  "random.h"
 
+#ifdef NEWAMB
 
+extern void		SDsquare2disk(double ds[2], double seedx, double seedy);
+
+				/* vertex direction bit positions */
+#define	VDB_xy	0
+#define VDB_y	01
+#define VDB_x	02
+#define VDB_Xy	03
+#define VDB_xY	04
+#define VDB_X	05
+#define VDB_Y	06
+#define VDB_XY	07
+				/* get opposite vertex direction bit */
+#define VDB_OPP(f)	(~(f) & 07)
+				/* adjacent triangle vertex flags */
+static const int  adjacent_trifl[8] = {
+			0,			/* forbidden diagonal */
+			1<<VDB_x|1<<VDB_y|1<<VDB_Xy,
+			1<<VDB_y|1<<VDB_x|1<<VDB_xY,
+			1<<VDB_y|1<<VDB_Xy|1<<VDB_X,
+			1<<VDB_x|1<<VDB_xY|1<<VDB_Y,
+			1<<VDB_Xy|1<<VDB_X|1<<VDB_Y,
+			1<<VDB_xY|1<<VDB_Y|1<<VDB_X,
+			0,			/* forbidden diagonal */
+		};
+
+typedef struct {
+	COLOR	v;		/* hemisphere sample value */
+	float	d;		/* reciprocal distance (1/rt) */
+	FVECT	p;		/* intersection point */
+} AMBSAMP;		/* sample value */
+
+typedef struct {
+	RAY	*rp;		/* originating ray sample */
+	FVECT	ux, uy;		/* tangent axis unit vectors */
+	int	ns;		/* number of samples per axis */
+	COLOR	acoef;		/* division contribution coefficient */
+	AMBSAMP	sa[1];		/* sample array (extends struct) */
+}  AMBHEMI;		/* ambient sample hemisphere */
+
+#define ambndx(h,i,j)	((i)*(h)->ns + (j))
+#define ambsam(h,i,j)	(h)->sa[ambndx(h,i,j)]
+
+typedef struct {
+	FVECT	r_i, r_i1, e_i, rcp, rI2_eJ2;
+	double	I1, I2;
+	int	valid;
+} FFTRI;		/* vectors and coefficients for Hessian calculation */
+
+
+/* Get index for adjacent vertex */
+static int
+adjacent_verti(AMBHEMI *hp, int i, int j, int dbit)
+{
+	int	i0 = i*hp->ns + j;
+
+	switch (dbit) {
+	case VDB_y:	return(i0 - hp->ns);
+	case VDB_x:	return(i0 - 1);
+	case VDB_Xy:	return(i0 - hp->ns + 1);
+	case VDB_xY:	return(i0 + hp->ns - 1);
+	case VDB_X:	return(i0 + 1);
+	case VDB_Y:	return(i0 + hp->ns);
+				/* the following should never occur */
+	case VDB_xy:	return(i0 - hp->ns - 1);
+	case VDB_XY:	return(i0 + hp->ns + 1);
+	}
+	return(-1);
+}
+
+
+/* Get vertex direction bit for the opposite edge to complete triangle */
+static int
+vdb_edge(int db1, int db2)
+{
+	switch (db1) {
+	case VDB_x:	return(db2==VDB_y ? VDB_Xy : VDB_Y);
+	case VDB_y:	return(db2==VDB_x ? VDB_xY : VDB_X);
+	case VDB_X:	return(db2==VDB_Xy ? VDB_y : VDB_xY);
+	case VDB_Y:	return(db2==VDB_xY ? VDB_x : VDB_Xy);
+	case VDB_xY:	return(db2==VDB_x ? VDB_y : VDB_X);
+	case VDB_Xy:	return(db2==VDB_y ? VDB_x : VDB_Y);
+	}
+	error(INTERNAL, "forbidden diagonal in vdb_edge()");
+	return(-1);
+}
+
+
+static AMBHEMI *
+inithemi(			/* initialize sampling hemisphere */
+	COLOR	ac,
+	RAY	*r,
+	double	wt
+)
+{
+	AMBHEMI	*hp;
+	double	d;
+	int	n, i;
+					/* set number of divisions */
+	if (ambacc <= FTINY &&
+			wt > (d = 0.8*intens(ac)*r->rweight/(ambdiv*minweight)))
+		wt = d;			/* avoid ray termination */
+	n = sqrt(ambdiv * wt) + 0.5;
+	i = 1 + 5*(ambacc > FTINY);	/* minimum number of samples */
+	if (n < i)
+		n = i;
+					/* allocate sampling array */
+	hp = (AMBHEMI *)malloc(sizeof(AMBHEMI) + sizeof(AMBSAMP)*(n*n - 1));
+	if (hp == NULL)
+		return(NULL);
+	hp->rp = r;
+	hp->ns = n;
+					/* assign coefficient */
+	copycolor(hp->acoef, ac);
+	d = 1.0/(n*n);
+	scalecolor(hp->acoef, d);
+					/* make tangent plane axes */
+	hp->uy[0] = 0.5 - frandom();
+	hp->uy[1] = 0.5 - frandom();
+	hp->uy[2] = 0.5 - frandom();
+	for (i = 3; i--; )
+		if ((-0.6 < r->ron[i]) & (r->ron[i] < 0.6))
+			break;
+	if (i < 0)
+		error(CONSISTENCY, "bad ray direction in inithemi");
+	hp->uy[i] = 1.0;
+	VCROSS(hp->ux, hp->uy, r->ron);
+	normalize(hp->ux);
+	VCROSS(hp->uy, r->ron, hp->ux);
+					/* we're ready to sample */
+	return(hp);
+}
+
+
+/* Sample ambient division and apply weighting coefficient */
+static int
+getambsamp(RAY *arp, AMBHEMI *hp, int i, int j, int n)
+{
+	int	hlist[3], ii;
+	double	spt[2], zd;
+					/* ambient coefficient for weight */
+	if (ambacc > FTINY)
+		setcolor(arp->rcoef, AVGREFL, AVGREFL, AVGREFL);
+	else
+		copycolor(arp->rcoef, hp->acoef);
+	if (rayorigin(arp, AMBIENT, hp->rp, arp->rcoef) < 0)
+		return(0);
+	if (ambacc > FTINY) {
+		multcolor(arp->rcoef, hp->acoef);
+		scalecolor(arp->rcoef, 1./AVGREFL);
+	}
+	hlist[0] = hp->rp->rno;
+	hlist[1] = j;
+	hlist[2] = i;
+	multisamp(spt, 2, urand(ilhash(hlist,3)+n));
+	if (!n) {			/* avoid border samples for n==0 */
+		if ((spt[0] < 0.1) | (spt[0] >= 0.9))
+			spt[0] = 0.1 + 0.8*frandom();
+		if ((spt[1] < 0.1) | (spt[1] >= 0.9))
+			spt[1] = 0.1 + 0.8*frandom();
+	}
+	SDsquare2disk(spt, (j+spt[1])/hp->ns, (i+spt[0])/hp->ns);
+	zd = sqrt(1. - spt[0]*spt[0] - spt[1]*spt[1]);
+	for (ii = 3; ii--; )
+		arp->rdir[ii] =	spt[0]*hp->ux[ii] +
+				spt[1]*hp->uy[ii] +
+				zd*hp->rp->ron[ii];
+	checknorm(arp->rdir);
+	dimlist[ndims++] = ambndx(hp,i,j) + 90171;
+	rayvalue(arp);			/* evaluate ray */
+	ndims--;			/* apply coefficient */
+	multcolor(arp->rcol, arp->rcoef);
+	return(1);
+}
+
+
+static AMBSAMP *
+ambsample(				/* initial ambient division sample */
+	AMBHEMI	*hp,
+	int	i,
+	int	j
+)
+{
+	AMBSAMP	*ap = &ambsam(hp,i,j);
+	RAY	ar;
+					/* generate hemispherical sample */
+	if (!getambsamp(&ar, hp, i, j, 0) || ar.rt <= FTINY) {
+		memset(ap, 0, sizeof(AMBSAMP));
+		return(NULL);
+	}
+	ap->d = 1.0/ar.rt;		/* limit vertex distance */
+	if (ar.rt > 10.0*thescene.cusize)
+		ar.rt = 10.0*thescene.cusize;
+	VSUM(ap->p, ar.rorg, ar.rdir, ar.rt);
+	copycolor(ap->v, ar.rcol);
+	return(ap);
+}
+
+
+/* Estimate errors based on ambient division differences */
+static float *
+getambdiffs(AMBHEMI *hp)
+{
+	float	*earr = (float *)calloc(hp->ns*hp->ns, sizeof(float));
+	float	*ep;
+	AMBSAMP	*ap;
+	double	b, d2;
+	int	i, j;
+
+	if (earr == NULL)		/* out of memory? */
+		return(NULL);
+					/* compute squared neighbor diffs */
+	for (ap = hp->sa, ep = earr, i = 0; i < hp->ns; i++)
+	    for (j = 0; j < hp->ns; j++, ap++, ep++) {
+		b = bright(ap[0].v);
+		if (i) {		/* from above */
+			d2 = b - bright(ap[-hp->ns].v);
+			d2 *= d2;
+			ep[0] += d2;
+			ep[-hp->ns] += d2;
+		}
+		if (j) {		/* from behind */
+			d2 = b - bright(ap[-1].v);
+			d2 *= d2;
+			ep[0] += d2;
+			ep[-1] += d2;
+		}
+	    }
+					/* correct for number of neighbors */
+	earr[0] *= 2.f;
+	earr[hp->ns-1] *= 2.f;
+	earr[(hp->ns-1)*hp->ns] *= 2.f;
+	earr[(hp->ns-1)*hp->ns + hp->ns-1] *= 2.f;
+	for (i = 1; i < hp->ns-1; i++) {
+		earr[i*hp->ns] *= 4./3.;
+		earr[i*hp->ns + hp->ns-1] *= 4./3.;
+	}
+	for (j = 1; j < hp->ns-1; j++) {
+		earr[j] *= 4./3.;
+		earr[(hp->ns-1)*hp->ns + j] *= 4./3.;
+	}
+	return(earr);
+}
+
+
+/* Perform super-sampling on hemisphere (introduces bias) */
+static void
+ambsupersamp(double acol[3], AMBHEMI *hp, int cnt)
+{
+	float	*earr = getambdiffs(hp);
+	double	e2sum = 0.0;
+	AMBSAMP	*ap;
+	RAY	ar;
+	double	asum[3];
+	float	*ep;
+	int	i, j, n;
+
+	if (earr == NULL)		/* just skip calc. if no memory */
+		return;
+					/* add up estimated variances */
+	for (ep = earr + hp->ns*hp->ns; ep-- > earr; )
+		e2sum += *ep;
+	ep = earr;			/* perform super-sampling */
+	for (ap = hp->sa, i = 0; i < hp->ns; i++)
+	    for (j = 0; j < hp->ns; j++, ap++) {
+		int	nss = *ep/e2sum*cnt + frandom();
+		asum[0] = asum[1] = asum[2] = 0.0;
+		for (n = 1; n <= nss; n++) {
+			if (!getambsamp(&ar, hp, i, j, n)) {
+				nss = n-1;
+				break;
+			}
+			addcolor(asum, ar.rcol);
+		}
+		if (nss) {		/* update returned ambient value */
+			const double	ssf = 1./(nss + 1);
+			for (n = 3; n--; )
+				acol[n] += ssf*asum[n] +
+						(ssf - 1.)*colval(ap->v,n);
+		}
+		e2sum -= *ep++;		/* update remainders */
+		cnt -= nss;
+	}
+	free(earr);
+}
+
+
+/* Compute vertex flags, indicating farthest in each direction */
+static uby8 *
+vertex_flags(AMBHEMI *hp)
+{
+	uby8	*vflags = (uby8 *)calloc(hp->ns*hp->ns, sizeof(uby8));
+	uby8	*vf;
+	AMBSAMP	*ap;
+	int	i, j;
+
+	if (vflags == NULL)
+		error(SYSTEM, "out of memory in vertex_flags()");
+	vf = vflags;
+	ap = hp->sa;		/* compute farthest along first row */
+	for (j = 0; j < hp->ns-1; j++, vf++, ap++)
+		if (ap[0].d <= ap[1].d)
+			vf[0] |= 1<<VDB_X;
+		else
+			vf[1] |= 1<<VDB_x;
+	++vf; ++ap;
+				/* flag subsequent rows */
+	for (i = 1; i < hp->ns; i++) {
+	    for (j = 0; j < hp->ns-1; j++, vf++, ap++) {
+		if (ap[0].d <= ap[-hp->ns].d)	/* row before */
+			vf[0] |= 1<<VDB_y;
+		else
+			vf[-hp->ns] |= 1<<VDB_Y;
+		if (ap[0].d <= ap[1-hp->ns].d)	/* diagonal we care about */
+			vf[0] |= 1<<VDB_Xy;
+		else
+			vf[1-hp->ns] |= 1<<VDB_xY;
+		if (ap[0].d <= ap[1].d)		/* column after */
+			vf[0] |= 1<<VDB_X;
+		else
+			vf[1] |= 1<<VDB_x;
+	    }
+	    if (ap[0].d <= ap[-hp->ns].d)	/* final column edge */
+		vf[0] |= 1<<VDB_y;
+	    else
+		vf[-hp->ns] |= 1<<VDB_Y;
+	    ++vf; ++ap;
+	}
+	return(vflags);
+}
+
+
+/* Return brightness of farthest ambient sample */
+static double
+back_ambval(AMBHEMI *hp, int i, int j, int dbit1, int dbit2, const uby8 *vflags)
+{
+	const int	v0 = ambndx(hp,i,j);
+	const int	tflags = (1<<dbit1 | 1<<dbit2);
+	int		v1, v2;
+
+	if ((vflags[v0] & tflags) == tflags)	/* is v0 the farthest? */
+		return(colval(hp->sa[v0].v,CIEY));
+	v1 = adjacent_verti(hp, i, j, dbit1);
+	if (vflags[v0] & 1<<dbit2)		/* v1 farthest if v0>v2 */
+		return(colval(hp->sa[v1].v,CIEY));
+	v2 = adjacent_verti(hp, i, j, dbit2);
+	if (vflags[v0] & 1<<dbit1)		/* v2 farthest if v0>v1 */
+		return(colval(hp->sa[v2].v,CIEY));
+						/* else check if v1>v2 */
+	if (vflags[v1] & 1<<vdb_edge(dbit1,dbit2))
+		return(colval(hp->sa[v1].v,CIEY));
+	return(colval(hp->sa[v2].v,CIEY));
+}
+
+
+/* Compute vectors and coefficients for Hessian/gradient calcs */
+static void
+comp_fftri(FFTRI *ftp, AMBHEMI *hp, int i, int j, int dbit, const uby8 *vflags)
+{
+	const int	i0 = ambndx(hp,i,j);
+	double		rdot_cp, dot_e, dot_er, rdot_r, rdot_r1, J2;
+	int		i1, ii;
+
+	ftp->valid = 0;			/* check if we can skip this edge */
+	ii = adjacent_trifl[dbit];
+	if ((vflags[i0] & ii) == ii)	/* cancels if vertex used as value */
+		return;
+	i1 = adjacent_verti(hp, i, j, dbit);
+	ii = adjacent_trifl[VDB_OPP(dbit)];
+	if ((vflags[i1] & ii) == ii)	/* on either end (for both triangles) */
+		return;
+					/* else go ahead with calculation */
+	VSUB(ftp->r_i, hp->sa[i0].p, hp->rp->rop);
+	VSUB(ftp->r_i1, hp->sa[i1].p, hp->rp->rop);
+	VSUB(ftp->e_i, hp->sa[i1].p, hp->sa[i0].p);
+	VCROSS(ftp->rcp, ftp->r_i, ftp->r_i1);
+	rdot_cp = 1.0/DOT(ftp->rcp,ftp->rcp);
+	dot_e = DOT(ftp->e_i,ftp->e_i);
+	dot_er = DOT(ftp->e_i, ftp->r_i);
+	rdot_r = 1.0/DOT(ftp->r_i,ftp->r_i);
+	rdot_r1 = 1.0/DOT(ftp->r_i1,ftp->r_i1);
+	ftp->I1 = acos( DOT(ftp->r_i, ftp->r_i1) * sqrt(rdot_r*rdot_r1) ) *
+			sqrt( rdot_cp );
+	ftp->I2 = ( DOT(ftp->e_i, ftp->r_i1)*rdot_r1 - dot_er*rdot_r +
+			dot_e*ftp->I1 )*0.5*rdot_cp;
+	J2 =  ( 0.5*(rdot_r - rdot_r1) - dot_er*ftp->I2 ) / dot_e;
+	for (ii = 3; ii--; )
+		ftp->rI2_eJ2[ii] = ftp->I2*ftp->r_i[ii] + J2*ftp->e_i[ii];
+	ftp->valid++;
+}
+
+
+/* Compose 3x3 matrix from two vectors */
+static void
+compose_matrix(FVECT mat[3], FVECT va, FVECT vb)
+{
+	mat[0][0] = 2.0*va[0]*vb[0];
+	mat[1][1] = 2.0*va[1]*vb[1];
+	mat[2][2] = 2.0*va[2]*vb[2];
+	mat[0][1] = mat[1][0] = va[0]*vb[1] + va[1]*vb[0];
+	mat[0][2] = mat[2][0] = va[0]*vb[2] + va[2]*vb[0];
+	mat[1][2] = mat[2][1] = va[1]*vb[2] + va[2]*vb[1];
+}
+
+
+/* Compute partial 3x3 Hessian matrix for edge */
+static void
+comp_hessian(FVECT hess[3], FFTRI *ftp, FVECT nrm)
+{
+	FVECT	ncp;
+	FVECT	m1[3], m2[3], m3[3], m4[3];
+	double	d1, d2, d3, d4;
+	double	I3, J3, K3;
+	int	i, j;
+
+	if (!ftp->valid) {		/* preemptive test */
+		memset(hess, 0, sizeof(FVECT)*3);
+		return;
+	}
+					/* compute intermediate coefficients */
+	d1 = 1.0/DOT(ftp->r_i,ftp->r_i);
+	d2 = 1.0/DOT(ftp->r_i1,ftp->r_i1);
+	d3 = 1.0/DOT(ftp->e_i,ftp->e_i);
+	d4 = DOT(ftp->e_i, ftp->r_i);
+	I3 = ( DOT(ftp->e_i, ftp->r_i1)*d2*d2 - d4*d1*d1 + 3.0/d3*ftp->I2 )
+			/ ( 4.0*DOT(ftp->rcp,ftp->rcp) );
+	J3 = 0.25*d3*(d1*d1 - d2*d2) - d4*d3*I3;
+	K3 = d3*(ftp->I2 - I3/d1 - 2.0*d4*J3);
+					/* intermediate matrices */
+	VCROSS(ncp, nrm, ftp->e_i);
+	compose_matrix(m1, ncp, ftp->rI2_eJ2);
+	compose_matrix(m2, ftp->r_i, ftp->r_i);
+	compose_matrix(m3, ftp->e_i, ftp->e_i);
+	compose_matrix(m4, ftp->r_i, ftp->e_i);
+	d1 = DOT(nrm, ftp->rcp);
+	d2 = -d1*ftp->I2;
+	d1 *= 2.0;
+	for (i = 3; i--; )		/* final matrix sum */
+	    for (j = 3; j--; ) {
+		hess[i][j] = m1[i][j] + d1*( I3*m2[i][j] + K3*m3[i][j] +
+						2.0*J3*m4[i][j] );
+		hess[i][j] += d2*(i==j);
+		hess[i][j] *= -1.0/PI;
+	    }
+}
+
+
+/* Reverse hessian calculation result for edge in other direction */
+static void
+rev_hessian(FVECT hess[3])
+{
+	int	i;
+
+	for (i = 3; i--; ) {
+		hess[i][0] = -hess[i][0];
+		hess[i][1] = -hess[i][1];
+		hess[i][2] = -hess[i][2];
+	}
+}
+
+
+/* Add to radiometric Hessian from the given triangle */
+static void
+add2hessian(FVECT hess[3], FVECT ehess1[3],
+		FVECT ehess2[3], FVECT ehess3[3], double v)
+{
+	int	i, j;
+
+	for (i = 3; i--; )
+	    for (j = 3; j--; )
+		hess[i][j] += v*( ehess1[i][j] + ehess2[i][j] + ehess3[i][j] );
+}
+
+
+/* Compute partial displacement form factor gradient for edge */
+static void
+comp_gradient(FVECT grad, FFTRI *ftp, FVECT nrm)
+{
+	FVECT	ncp;
+	double	f1;
+	int	i;
+
+	if (!ftp->valid) {		/* preemptive test */
+		memset(grad, 0, sizeof(FVECT));
+		return;
+	}
+	f1 = 2.0*DOT(nrm, ftp->rcp);
+	VCROSS(ncp, nrm, ftp->e_i);
+	for (i = 3; i--; )
+		grad[i] = (0.5/PI)*( ftp->I1*ncp[i] + f1*ftp->rI2_eJ2[i] );
+}
+
+
+/* Reverse gradient calculation result for edge in other direction */
+static void
+rev_gradient(FVECT grad)
+{
+	grad[0] = -grad[0];
+	grad[1] = -grad[1];
+	grad[2] = -grad[2];
+}
+
+
+/* Add to displacement gradient from the given triangle */
+static void
+add2gradient(FVECT grad, FVECT egrad1, FVECT egrad2, FVECT egrad3, double v)
+{
+	int	i;
+
+	for (i = 3; i--; )
+		grad[i] += v*( egrad1[i] + egrad2[i] + egrad3[i] );
+}
+
+
+/* Compute anisotropic radii and eigenvector directions */
+static int
+eigenvectors(FVECT uv[2], float ra[2], FVECT hessian[3])
+{
+	double	hess2[2][2];
+	FVECT	a, b;
+	double	evalue[2], slope1, xmag1;
+	int	i;
+					/* project Hessian to sample plane */
+	for (i = 3; i--; ) {
+		a[i] = DOT(hessian[i], uv[0]);
+		b[i] = DOT(hessian[i], uv[1]);
+	}
+	hess2[0][0] = DOT(uv[0], a);
+	hess2[0][1] = DOT(uv[0], b);
+	hess2[1][0] = DOT(uv[1], a);
+	hess2[1][1] = DOT(uv[1], b);
+					/* compute eigenvalue(s) */
+	i = quadratic(evalue, 1.0, -hess2[0][0]-hess2[1][1],
+			hess2[0][0]*hess2[1][1]-hess2[0][1]*hess2[1][0]);
+	if (i == 1)			/* double-root (circle) */
+		evalue[1] = evalue[0];
+	if (!i || ((evalue[0] = fabs(evalue[0])) <= FTINY*FTINY) |
+			((evalue[1] = fabs(evalue[1])) <= FTINY*FTINY) )
+		error(INTERNAL, "bad eigenvalue calculation");
+
+	if (evalue[0] > evalue[1]) {
+		ra[0] = sqrt(sqrt(4.0/evalue[0]));
+		ra[1] = sqrt(sqrt(4.0/evalue[1]));
+		slope1 = evalue[1];
+	} else {
+		ra[0] = sqrt(sqrt(4.0/evalue[1]));
+		ra[1] = sqrt(sqrt(4.0/evalue[0]));
+		slope1 = evalue[0];
+	}
+					/* compute unit eigenvectors */
+	if (fabs(hess2[0][1]) <= FTINY)
+		return;			/* uv OK as is */
+	slope1 = (slope1 - hess2[0][0]) / hess2[0][1];
+	xmag1 = sqrt(1.0/(1.0 + slope1*slope1));
+	for (i = 3; i--; ) {
+		b[i] = xmag1*uv[0][i] + slope1*xmag1*uv[1][i];
+		a[i] = slope1*xmag1*uv[0][i] - xmag1*uv[1][i];
+	}
+	VCOPY(uv[0], a);
+	VCOPY(uv[1], b);
+}
+
+
+static void
+ambHessian(				/* anisotropic radii & pos. gradient */
+	AMBHEMI	*hp,
+	FVECT	uv[2],			/* returned */
+	float	ra[2],			/* returned (optional) */
+	float	pg[2]			/* returned (optional) */
+)
+{
+	static char	memerrmsg[] = "out of memory in ambHessian()";
+	FVECT		(*hessrow)[3] = NULL;
+	FVECT		*gradrow = NULL;
+	uby8		*vflags;
+	FVECT		hessian[3];
+	FVECT		gradient;
+	FFTRI		fftr;
+	int		i, j;
+					/* be sure to assign unit vectors */
+	VCOPY(uv[0], hp->ux);
+	VCOPY(uv[1], hp->uy);
+			/* clock-wise vertex traversal from sample POV */
+	if (ra != NULL) {		/* initialize Hessian row buffer */
+		hessrow = (FVECT (*)[3])malloc(sizeof(FVECT)*3*(hp->ns-1));
+		if (hessrow == NULL)
+			error(SYSTEM, memerrmsg);
+		memset(hessian, 0, sizeof(hessian));
+	} else if (pg == NULL)		/* bogus call? */
+		return;
+	if (pg != NULL) {		/* initialize form factor row buffer */
+		gradrow = (FVECT *)malloc(sizeof(FVECT)*(hp->ns-1));
+		if (gradrow == NULL)
+			error(SYSTEM, memerrmsg);
+		memset(gradient, 0, sizeof(gradient));
+	}
+					/* get vertex position flags */
+	vflags = vertex_flags(hp);
+					/* compute first row of edges */
+	for (j = 0; j < hp->ns-1; j++) {
+		comp_fftri(&fftr, hp, 0, j, VDB_X, vflags);
+		if (hessrow != NULL)
+			comp_hessian(hessrow[j], &fftr, hp->rp->ron);
+		if (gradrow != NULL)
+			comp_gradient(gradrow[j], &fftr, hp->rp->ron);
+	}
+					/* sum each row of triangles */
+	for (i = 0; i < hp->ns-1; i++) {
+	    FVECT	hesscol[3];	/* compute first vertical edge */
+	    FVECT	gradcol;
+	    comp_fftri(&fftr, hp, i, 0, VDB_Y, vflags);
+	    if (hessrow != NULL)
+		comp_hessian(hesscol, &fftr, hp->rp->ron);
+	    if (gradrow != NULL)
+		comp_gradient(gradcol, &fftr, hp->rp->ron);
+	    for (j = 0; j < hp->ns-1; j++) {
+		FVECT	hessdia[3];	/* compute triangle contributions */
+		FVECT	graddia;
+		double	backg;
+		backg = back_ambval(hp, i, j, VDB_X, VDB_Y, vflags);
+					/* diagonal (inner) edge */
+		comp_fftri(&fftr, hp, i, j+1, VDB_xY, vflags);
+		if (hessrow != NULL) {
+		    comp_hessian(hessdia, &fftr, hp->rp->ron);
+		    rev_hessian(hesscol);
+		    add2hessian(hessian, hessrow[j], hessdia, hesscol, backg);
+		}
+		if (gradrow != NULL) {
+		    comp_gradient(graddia, &fftr, hp->rp->ron);
+		    rev_gradient(gradcol);
+		    add2gradient(gradient, gradrow[j], graddia, gradcol, backg);
+		}
+					/* initialize edge in next row */
+		comp_fftri(&fftr, hp, i+1, j+1, VDB_x, vflags);
+		if (hessrow != NULL)
+		    comp_hessian(hessrow[j], &fftr, hp->rp->ron);
+		if (gradrow != NULL)
+		    comp_gradient(gradrow[j], &fftr, hp->rp->ron);
+					/* new column edge & paired triangle */
+		backg = back_ambval(hp, i+1, j+1, VDB_x, VDB_y, vflags);
+		comp_fftri(&fftr, hp, i, j+1, VDB_Y, vflags);
+		if (hessrow != NULL) {
+		    comp_hessian(hesscol, &fftr, hp->rp->ron);
+		    rev_hessian(hessdia);
+		    add2hessian(hessian, hessrow[j], hessdia, hesscol, backg);
+		    if (i < hp->ns-2)
+			rev_hessian(hessrow[j]);
+		}
+		if (gradrow != NULL) {
+		    comp_gradient(gradcol, &fftr, hp->rp->ron);
+		    rev_gradient(graddia);
+		    add2gradient(gradient, gradrow[j], graddia, gradcol, backg);
+		    if (i < hp->ns-2)
+			rev_gradient(gradrow[j]);
+		}
+	    }
+	}
+					/* release row buffers */
+	if (hessrow != NULL) free(hessrow);
+	if (gradrow != NULL) free(gradrow);
+	free(vflags);
+	
+	if (ra != NULL)			/* extract eigenvectors & radii */
+		eigenvectors(uv, ra, hessian);
+	if (pg != NULL) {		/* tangential position gradient */
+		pg[0] = DOT(gradient, uv[0]);
+		pg[1] = DOT(gradient, uv[1]);
+	}
+}
+
+
+/* Compute direction gradient from a hemispherical sampling */
+static void
+ambdirgrad(AMBHEMI *hp, FVECT uv[2], float dg[2])
+{
+	AMBSAMP	*ap;
+	double	dgsum[2];
+	int	n;
+	FVECT	vd;
+	double	gfact;
+
+	dgsum[0] = dgsum[1] = 0.0;	/* sum values times -tan(theta) */
+	for (ap = hp->sa, n = hp->ns*hp->ns; n--; ap++) {
+					/* use vector for azimuth + 90deg */
+		VSUB(vd, ap->p, hp->rp->rop);
+					/* brightness over cosine factor */
+		gfact = colval(ap->v,CIEY) / DOT(hp->rp->ron, vd);
+					/* sine = proj_radius/vd_length */
+		dgsum[0] -= DOT(uv[1], vd) * gfact;
+		dgsum[1] += DOT(uv[0], vd) * gfact;
+	}
+	dg[0] = dgsum[0] / (hp->ns*hp->ns);
+	dg[1] = dgsum[1] / (hp->ns*hp->ns);
+}
+
+
+/* Compute potential light leak direction flags for cache value */
+static uint32
+ambcorral(AMBHEMI *hp, FVECT uv[2], const double r0, const double r1)
+{
+	const double	max_d = 1.0/(minarad*ambacc + 0.001);
+	const double	ang_res = 0.5*PI/(hp->ns-1);
+	const double	ang_step = ang_res/((int)(16/PI*ang_res) + (1+FTINY));
+	uint32		flgs = 0;
+	int		i, j;
+					/* circle around perimeter */
+	for (i = 0; i < hp->ns; i++)
+	    for (j = 0; j < hp->ns; j += !i|(i==hp->ns-1) ? 1 : hp->ns-1) {
+		AMBSAMP	*ap = &ambsam(hp,i,j);
+		FVECT	vec;
+		double	u, v;
+		double	ang, a1;
+		int	abp;
+		if ((ap->d <= FTINY) | (ap->d >= max_d))
+			continue;	/* too far or too near */
+		VSUB(vec, ap->p, hp->rp->rop);
+		u = DOT(vec, uv[0]) * ap->d;
+		v = DOT(vec, uv[1]) * ap->d;
+		if ((r0*r0*u*u + r1*r1*v*v) * ap->d*ap->d <= 1.0)
+			continue;	/* occluder outside ellipse */
+		ang = atan2a(v, u);	/* else set direction flags */
+		for (a1 = ang-.5*ang_res; a1 <= ang+.5*ang_res; a1 += ang_step)
+			flgs |= 1L<<(int)(16/PI*(a1 + 2.*PI*(a1 < 0)));
+	    }
+	return(flgs);
+}
+
+
+int
+doambient(				/* compute ambient component */
+	COLOR	rcol,			/* input/output color */
+	RAY	*r,
+	double	wt,
+	FVECT	uv[2],			/* returned (optional) */
+	float	ra[2],			/* returned (optional) */
+	float	pg[2],			/* returned (optional) */
+	float	dg[2],			/* returned (optional) */
+	uint32	*crlp			/* returned (optional) */
+)
+{
+	AMBHEMI	*hp = inithemi(rcol, r, wt);
+	int	cnt;
+	FVECT	my_uv[2];
+	double	d, K, acol[3];
+	AMBSAMP	*ap;
+	int	i, j;
+					/* check/initialize */
+	if (hp == NULL)
+		return(0);
+	if (uv != NULL)
+		memset(uv, 0, sizeof(FVECT)*2);
+	if (ra != NULL)
+		ra[0] = ra[1] = 0.0;
+	if (pg != NULL)
+		pg[0] = pg[1] = 0.0;
+	if (dg != NULL)
+		dg[0] = dg[1] = 0.0;
+	if (crlp != NULL)
+		*crlp = 0;
+					/* sample the hemisphere */
+	acol[0] = acol[1] = acol[2] = 0.0;
+	cnt = 0;
+	for (i = hp->ns; i--; )
+		for (j = hp->ns; j--; )
+			if ((ap = ambsample(hp, i, j)) != NULL) {
+				addcolor(acol, ap->v);
+				++cnt;
+			}
+	if (!cnt) {
+		setcolor(rcol, 0.0, 0.0, 0.0);
+		free(hp);
+		return(0);		/* no valid samples */
+	}
+	if (cnt < hp->ns*hp->ns) {	/* incomplete sampling? */
+		copycolor(rcol, acol);
+		free(hp);
+		return(-1);		/* return value w/o Hessian */
+	}
+	cnt = ambssamp*wt + 0.5;	/* perform super-sampling? */
+	if (cnt > 8)
+		ambsupersamp(acol, hp, cnt);
+	copycolor(rcol, acol);		/* final indirect irradiance/PI */
+	if ((ra == NULL) & (pg == NULL) & (dg == NULL)) {
+		free(hp);
+		return(-1);		/* no radius or gradient calc. */
+	}
+	if ((d = bright(acol)) > FTINY) {	/* normalize Y values */
+		d = 0.99*(hp->ns*hp->ns)/d;
+		K = 0.01;
+	} else {			/* or fall back on geometric Hessian */
+		K = 1.0;
+		pg = NULL;
+		dg = NULL;
+	}
+	ap = hp->sa;			/* relative Y channel from here on... */
+	for (i = hp->ns*hp->ns; i--; ap++)
+		colval(ap->v,CIEY) = bright(ap->v)*d + K;
+
+	if (uv == NULL)			/* make sure we have axis pointers */
+		uv = my_uv;
+					/* compute radii & pos. gradient */
+	ambHessian(hp, uv, ra, pg);
+
+	if (dg != NULL)			/* compute direction gradient */
+		ambdirgrad(hp, uv, dg);
+
+	if (ra != NULL) {		/* scale/clamp radii */
+		if (pg != NULL) {
+			if (ra[0]*(d = fabs(pg[0])) > 1.0)
+				ra[0] = 1.0/d;
+			if (ra[1]*(d = fabs(pg[1])) > 1.0)
+				ra[1] = 1.0/d;
+			if (ra[0] > ra[1])
+				ra[0] = ra[1];
+		}
+		if (ra[0] < minarad) {
+			ra[0] = minarad;
+			if (ra[1] < minarad)
+				ra[1] = minarad;
+		}
+		ra[0] *= d = 1.0/sqrt(sqrt(wt));
+		if ((ra[1] *= d) > 2.0*ra[0])
+			ra[1] = 2.0*ra[0];
+		if (ra[1] > maxarad) {
+			ra[1] = maxarad;
+			if (ra[0] > maxarad)
+				ra[0] = maxarad;
+		}
+		if (crlp != NULL)	/* flag encroached directions */
+			*crlp = ambcorral(hp, uv, ra[0]*ambacc, ra[1]*ambacc);
+		if (pg != NULL) {	/* cap gradient if necessary */
+			d = pg[0]*pg[0]*ra[0]*ra[0] + pg[1]*pg[1]*ra[1]*ra[1];
+			if (d > 1.0) {
+				d = 1.0/sqrt(d);
+				pg[0] *= d;
+				pg[1] *= d;
+			}
+		}
+	}
+	free(hp);			/* clean up and return */
+	return(1);
+}
+
+
+#else /* ! NEWAMB */
+
+
 void
 inithemi(			/* initialize sampling hemisphere */
-	register AMBHEMI  *hp,
+	AMBHEMI  *hp,
+	COLOR ac,
 	RAY  *r,
-	COLOR ac, 
 	double  wt
 )
 {
-	register int  i;
+	double	d;
+	int  i;
 					/* set number of divisions */
-	hp->nt = sqrt(ambdiv * wt * (1./PI/AVGREFL)) + 0.5;
+	if (ambacc <= FTINY &&
+			wt > (d = 0.8*intens(ac)*r->rweight/(ambdiv*minweight)))
+		wt = d;			/* avoid ray termination */
+	hp->nt = sqrt(ambdiv * wt / PI) + 0.5;
 	i = ambacc > FTINY ? 3 : 1;	/* minimum number of samples */
 	if (hp->nt < i)
 		hp->nt = i;
 	hp->np = PI * hp->nt + 0.5;
 					/* set number of super-samples */
 	hp->ns = ambssamp * wt + 0.5;
-					/* assign coefficients */
+					/* assign coefficient */
 	copycolor(hp->acoef, ac);
-	if (wt >= r->rweight)
-		hp->drc = 1.;
-	else
-		hp->drc = wt / r->rweight;
+	d = 1.0/(hp->nt*hp->np);
+	scalecolor(hp->acoef, d);
 					/* make axes */
 	VCOPY(hp->uz, r->ron);
 	hp->uy[0] = hp->uy[1] = hp->uy[2] = 0.0;
@@ -56,7 +911,7 @@ inithemi(			/* initialize sampling hemisphere */
 
 int
 divsample(				/* sample a division */
-	register AMBSAMP  *dp,
+	AMBSAMP  *dp,
 	AMBHEMI  *h,
 	RAY  *r
 )
@@ -67,14 +922,18 @@ divsample(				/* sample a division */
 	double  xd, yd, zd;
 	double  b2;
 	double  phi;
-	register int  i;
+	int  i;
 					/* ambient coefficient for weight */
-	setcolor(ar.rcoef, h->drc, h->drc, h->drc);
+	if (ambacc > FTINY)
+		setcolor(ar.rcoef, AVGREFL, AVGREFL, AVGREFL);
+	else
+		copycolor(ar.rcoef, h->acoef);
 	if (rayorigin(&ar, AMBIENT, r, ar.rcoef) < 0)
 		return(-1);
-	copycolor(ar.rcoef, h->acoef);	/* correct coefficient for trace */
-	b2 = 1.0/(h->nt*h->np + h->ns);	/* XXX not uniform if ns > 0 */
-	scalecolor(ar.rcoef, b2);
+	if (ambacc > FTINY) {
+		multcolor(ar.rcoef, h->acoef);
+		scalecolor(ar.rcoef, 1./AVGREFL);
+	}
 	hlist[0] = r->rno;
 	hlist[1] = dp->t;
 	hlist[2] = dp->p;
@@ -88,9 +947,11 @@ divsample(				/* sample a division */
 		ar.rdir[i] =	xd*h->ux[i] +
 				yd*h->uy[i] +
 				zd*h->uz[i];
+	checknorm(ar.rdir);
 	dimlist[ndims++] = dp->t*h->np + dp->p + 90171;
 	rayvalue(&ar);
 	ndims--;
+	multcolor(ar.rcol, ar.rcoef);	/* apply coefficient */
 	addcolor(dp->v, ar.rcol);
 					/* use rt to improve gradient calc */
 	if (ar.rt > FTINY && ar.rt < FHUGE)
@@ -131,7 +992,7 @@ ambnorm(				/* standard order */
 {
 	const AMBSAMP	*d1 = (const AMBSAMP *)p1;
 	const AMBSAMP	*d2 = (const AMBSAMP *)p2;
-	register int	c;
+	int	c;
 
 	if ( (c = d1->t - d2->t) )
 		return(c);
@@ -141,67 +1002,86 @@ ambnorm(				/* standard order */
 
 double
 doambient(				/* compute ambient component */
-	COLOR  acol,
+	COLOR  rcol,
 	RAY  *r,
-	COLOR  ac,
 	double  wt,
 	FVECT  pg,
 	FVECT  dg
 )
 {
-	double  b, d;
+	double  b, d=0;
 	AMBHEMI  hemi;
 	AMBSAMP  *div;
 	AMBSAMP  dnew;
-	register AMBSAMP  *dp;
+	double  acol[3];
+	AMBSAMP  *dp;
 	double  arad;
-	int  ndivs;
-	register int  i, j;
-					/* initialize color */
-	setcolor(acol, 0.0, 0.0, 0.0);
+	int  divcnt;
+	int  i, j;
 					/* initialize hemisphere */
-	inithemi(&hemi, r, ac, wt);
-	ndivs = hemi.nt * hemi.np;
-	if (ndivs == 0)
+	inithemi(&hemi, rcol, r, wt);
+	divcnt = hemi.nt * hemi.np;
+					/* initialize */
+	if (pg != NULL)
+		pg[0] = pg[1] = pg[2] = 0.0;
+	if (dg != NULL)
+		dg[0] = dg[1] = dg[2] = 0.0;
+	setcolor(rcol, 0.0, 0.0, 0.0);
+	if (divcnt == 0)
 		return(0.0);
 					/* allocate super-samples */
 	if (hemi.ns > 0 || pg != NULL || dg != NULL) {
-		div = (AMBSAMP *)malloc(ndivs*sizeof(AMBSAMP));
+		div = (AMBSAMP *)malloc(divcnt*sizeof(AMBSAMP));
 		if (div == NULL)
 			error(SYSTEM, "out of memory in doambient");
 	} else
 		div = NULL;
 					/* sample the divisions */
 	arad = 0.0;
+	acol[0] = acol[1] = acol[2] = 0.0;
 	if ((dp = div) == NULL)
 		dp = &dnew;
+	divcnt = 0;
 	for (i = 0; i < hemi.nt; i++)
 		for (j = 0; j < hemi.np; j++) {
 			dp->t = i; dp->p = j;
 			setcolor(dp->v, 0.0, 0.0, 0.0);
 			dp->r = 0.0;
 			dp->n = 0;
-			if (divsample(dp, &hemi, r) < 0)
-				goto oopsy;
+			if (divsample(dp, &hemi, r) < 0) {
+				if (div != NULL)
+					dp++;
+				continue;
+			}
 			arad += dp->r;
+			divcnt++;
 			if (div != NULL)
 				dp++;
 			else
 				addcolor(acol, dp->v);
 		}
-	if (hemi.ns > 0 && arad > FTINY && ndivs/arad < minarad)
+	if (!divcnt) {
+		if (div != NULL)
+			free((void *)div);
+		return(0.0);		/* no samples taken */
+	}
+	if (divcnt < hemi.nt*hemi.np) {
+		pg = dg = NULL;		/* incomplete sampling */
+		hemi.ns = 0;
+	} else if (arad > FTINY && divcnt/arad < minarad) {
 		hemi.ns = 0;		/* close enough */
-	else if (hemi.ns > 0) {		/* else perform super-sampling */
+	} else if (hemi.ns > 0) {	/* else perform super-sampling? */
 		comperrs(div, &hemi);			/* compute errors */
-		qsort(div, ndivs, sizeof(AMBSAMP), ambcmp);	/* sort divs */
+		qsort(div, divcnt, sizeof(AMBSAMP), ambcmp);	/* sort divs */
 						/* super-sample */
 		for (i = hemi.ns; i > 0; i--) {
 			dnew = *div;
-			if (divsample(&dnew, &hemi, r) < 0)
-				goto oopsy;
-							/* reinsert */
-			dp = div;
-			j = ndivs < i ? ndivs : i;
+			if (divsample(&dnew, &hemi, r) < 0) {
+				dp++;
+				continue;
+			}
+			dp = div;		/* reinsert */
+			j = divcnt < i ? divcnt : i;
 			while (--j > 0 && dnew.k < dp[1].k) {
 				*dp = *(dp+1);
 				dp++;
@@ -209,12 +1089,12 @@ doambient(				/* compute ambient component */
 			*dp = dnew;
 		}
 		if (pg != NULL || dg != NULL)	/* restore order */
-			qsort(div, ndivs, sizeof(AMBSAMP), ambnorm);
+			qsort(div, divcnt, sizeof(AMBSAMP), ambnorm);
 	}
 					/* compute returned values */
 	if (div != NULL) {
-		arad = 0.0;
-		for (i = ndivs, dp = div; i-- > 0; dp++) {
+		arad = 0.0;		/* note: divcnt may be < nt*np */
+		for (i = hemi.nt*hemi.np, dp = div; i-- > 0; dp++) {
 			arad += dp->r;
 			if (dp->n > 1) {
 				b = 1.0/dp->n;
@@ -226,7 +1106,7 @@ doambient(				/* compute ambient component */
 		}
 		b = bright(acol);
 		if (b > FTINY) {
-			b = ndivs/b;
+			b = 1.0/b;	/* compute & normalize gradient(s) */
 			if (pg != NULL) {
 				posgradient(pg, div, &hemi);
 				for (i = 0; i < 3; i++)
@@ -237,22 +1117,14 @@ doambient(				/* compute ambient component */
 				for (i = 0; i < 3; i++)
 					dg[i] *= b;
 			}
-		} else {
-			if (pg != NULL)
-				for (i = 0; i < 3; i++)
-					pg[i] = 0.0;
-			if (dg != NULL)
-				for (i = 0; i < 3; i++)
-					dg[i] = 0.0;
 		}
 		free((void *)div);
 	}
-	b = 1.0/ndivs;
-	scalecolor(acol, b);
+	copycolor(rcol, acol);
 	if (arad <= FTINY)
 		arad = maxarad;
 	else
-		arad = (ndivs+hemi.ns)/arad;
+		arad = (divcnt+hemi.ns)/arad;
 	if (pg != NULL) {		/* reduce radius if gradient large */
 		d = DOT(pg,pg);
 		if (d*arad*arad > 1.0)
@@ -269,22 +1141,18 @@ doambient(				/* compute ambient component */
 	if ((arad /= sqrt(wt)) > maxarad)
 		arad = maxarad;
 	return(arad);
-oopsy:
-	if (div != NULL)
-		free((void *)div);
-	return(0.0);
 }
 
 
 void
 comperrs(			/* compute initial error estimates */
 	AMBSAMP  *da,	/* assumes standard ordering */
-	register AMBHEMI  *hp
+	AMBHEMI  *hp
 )
 {
 	double  b, b2;
 	int  i, j;
-	register AMBSAMP  *dp;
+	AMBSAMP  *dp;
 				/* sum differences from neighbors */
 	dp = da;
 	for (i = 0; i < hp->nt; i++)
@@ -332,14 +1200,14 @@ void
 posgradient(					/* compute position gradient */
 	FVECT  gv,
 	AMBSAMP  *da,			/* assumes standard ordering */
-	register AMBHEMI  *hp
+	AMBHEMI  *hp
 )
 {
-	register int  i, j;
+	int  i, j;
 	double  nextsine, lastsine, b, d;
 	double  mag0, mag1;
 	double  phi, cosp, sinp, xd, yd;
-	register AMBSAMP  *dp;
+	AMBSAMP  *dp;
 
 	xd = yd = 0.0;
 	for (j = 0; j < hp->np; j++) {
@@ -382,7 +1250,7 @@ posgradient(					/* compute position gradient */
 		yd += mag0*sinp + mag1*cosp;
 	}
 	for (i = 0; i < 3; i++)
-		gv[i] = (xd*hp->ux[i] + yd*hp->uy[i])/PI;
+		gv[i] = (xd*hp->ux[i] + yd*hp->uy[i])*(hp->nt*hp->np)/PI;
 }
 
 
@@ -390,13 +1258,13 @@ void
 dirgradient(					/* compute direction gradient */
 	FVECT  gv,
 	AMBSAMP  *da,			/* assumes standard ordering */
-	register AMBHEMI  *hp
+	AMBHEMI  *hp
 )
 {
-	register int  i, j;
+	int  i, j;
 	double  mag;
 	double  phi, xd, yd;
-	register AMBSAMP  *dp;
+	AMBSAMP  *dp;
 
 	xd = yd = 0.0;
 	for (j = 0; j < hp->np; j++) {
@@ -417,5 +1285,7 @@ dirgradient(					/* compute direction gradient */
 		yd += mag * tsin(phi);
 	}
 	for (i = 0; i < 3; i++)
-		gv[i] = (xd*hp->ux[i] + yd*hp->uy[i])/(hp->nt*hp->np);
+		gv[i] = xd*hp->ux[i] + yd*hp->uy[i];
 }
+
+#endif	/* ! NEWAMB */