--- ray/src/rt/ambcomp.c	2014/04/16 20:32:00	2.26
+++ ray/src/rt/ambcomp.c	2014/04/24 23:15:10	2.34
@@ -1,9 +1,13 @@
 #ifndef lint
-static const char	RCSid[] = "$Id: ambcomp.c,v 2.26 2014/04/16 20:32:00 greg Exp $";
+static const char	RCSid[] = "$Id: ambcomp.c,v 2.34 2014/04/24 23:15:10 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.
+ *
  *  Declarations of external symbols in ambient.h
  */
 
@@ -19,18 +23,23 @@ extern void		SDsquare2disk(double ds[2], double seedx,
 
 typedef struct {
 	RAY	*rp;		/* originating ray sample */
-	FVECT	ux, uy;		/* tangent axis directions */
+	FVECT	ux, uy;		/* tangent axis unit vectors */
 	int	ns;		/* number of samples per axis */
 	COLOR	acoef;		/* division contribution coefficient */
 	struct s_ambsamp {
 		COLOR	v;		/* hemisphere sample value */
-		float	p[3];		/* intersection point */
+		FVECT	p;		/* intersection point */
 	} sa[1];		/* sample array (extends struct) */
 }  AMBHEMI;		/* ambient sample hemisphere */
 
 #define ambsamp(h,i,j)	(h)->sa[(i)*(h)->ns + (j)]
 
+typedef struct {
+	FVECT	r_i, r_i1, e_i, rI2_eJ2;
+	double	nf, I1, I2;
+} FFTRI;		/* vectors and coefficients for Hessian calculation */
 
+
 static AMBHEMI *
 inithemi(			/* initialize sampling hemisphere */
 	COLOR	ac,
@@ -46,7 +55,7 @@ inithemi(			/* initialize sampling hemisphere */
 			wt > (d = 0.8*intens(ac)*r->rweight/(ambdiv*minweight)))
 		wt = d;			/* avoid ray termination */
 	n = sqrt(ambdiv * wt) + 0.5;
-	i = 1 + 4*(ambacc > FTINY);	/* minimum number of samples */
+	i = 1 + 5*(ambacc > FTINY);	/* minimum number of samples */
 	if (n < i)
 		n = i;
 					/* allocate sampling array */
@@ -60,50 +69,49 @@ inithemi(			/* initialize sampling hemisphere */
 	copycolor(hp->acoef, ac);
 	d = 1.0/(n*n);
 	scalecolor(hp->acoef, d);
-					/* make tangent axes */
-	hp->uy[0] = hp->uy[1] = hp->uy[2] = 0.0;
+					/* make tangent plane axes */
+	hp->uy[0] = 0.1 - 0.2*frandom();
+	hp->uy[1] = 0.1 - 0.2*frandom();
+	hp->uy[2] = 0.1 - 0.2*frandom();
 	for (i = 0; i < 3; i++)
-		if (r->rn[i] < 0.6 && r->rn[i] > -0.6)
+		if (r->ron[i] < 0.6 && r->ron[i] > -0.6)
 			break;
 	if (i >= 3)
 		error(CONSISTENCY, "bad ray direction in inithemi()");
 	hp->uy[i] = 1.0;
-	VCROSS(hp->ux, hp->uy, r->rn);
+	VCROSS(hp->ux, hp->uy, r->ron);
 	normalize(hp->ux);
-	VCROSS(hp->uy, r->rn, hp->ux);
+	VCROSS(hp->uy, r->ron, hp->ux);
 					/* we're ready to sample */
 	return(hp);
 }
 
 
-static int
+static struct s_ambsamp *
 ambsample(				/* sample an ambient direction */
 	AMBHEMI	*hp,
 	int	i,
-	int	j,
+	int	j
 )
 {
 	struct s_ambsamp	*ap = &ambsamp(hp,i,j);
 	RAY			ar;
-	int			hlist[3];
-	double			spt[2], dz;
+	double			spt[2], zd;
 	int			ii;
 					/* ambient coefficient for weight */
 	if (ambacc > FTINY)
 		setcolor(ar.rcoef, AVGREFL, AVGREFL, AVGREFL);
 	else
 		copycolor(ar.rcoef, hp->acoef);
-	if (rayorigin(&ar, AMBIENT, hp->rp, ar.rcoef) < 0) {
-		setcolor(ap->v, 0., 0., 0.);
-		ap->r = 0.;
-		return(0);		/* no sample taken */
-	}
+	if (rayorigin(&ar, AMBIENT, hp->rp, ar.rcoef) < 0)
+		goto badsample;
 	if (ambacc > FTINY) {
 		multcolor(ar.rcoef, hp->acoef);
 		scalecolor(ar.rcoef, 1./AVGREFL);
 	}
 					/* generate hemispherical sample */
-	SDsquare2disk(spt, (i+frandom())/hp->ns, (j+frandom())/hp->ns);
+	SDsquare2disk(spt,	(i+.1+.8*frandom())/hp->ns,
+				(j+.1+.8*frandom())/hp->ns );
 	zd = sqrt(1. - spt[0]*spt[0] - spt[1]*spt[1]);
 	for (ii = 3; ii--; )
 		ar.rdir[ii] =	spt[0]*hp->ux[ii] +
@@ -113,51 +121,392 @@ ambsample(				/* sample an ambient direction */
 	dimlist[ndims++] = i*hp->ns + j + 90171;
 	rayvalue(&ar);			/* evaluate ray */
 	ndims--;
+					/* limit vertex distance */
+	if (ar.rt > 10.0*thescene.cusize)
+		ar.rt = 10.0*thescene.cusize;
+	else if (ar.rt <= FTINY)	/* should never happen! */
+		goto badsample;
+	VSUM(ap->p, ar.rorg, ar.rdir, ar.rt);
 	multcolor(ar.rcol, ar.rcoef);	/* apply coefficient */
 	copycolor(ap->v, ar.rcol);
-	if (ar.rt > 20.0*maxarad)	/* limit vertex distance */
-		ar.rt = 20.0*maxarad;
-	VSUM(ap->p, ar.rorg, ar.rdir, ar.rt);
-	return(1);
+	return(ap);
+badsample:
+	setcolor(ap->v, 0., 0., 0.);
+	VCOPY(ap->p, hp->rp->rop);
+	return(NULL);
 }
 
 
+/* Compute vectors and coefficients for Hessian/gradient calcs */
 static void
+comp_fftri(FFTRI *ftp, FVECT ap0, FVECT ap1, FVECT rop)
+{
+	FVECT	vcp;
+	double	dot_e, dot_er, rdot_r, rdot_r1, J2;
+	int	i;
+
+	VSUB(ftp->r_i, ap0, rop);
+	VSUB(ftp->r_i1, ap1, rop);
+	VSUB(ftp->e_i, ap1, ap0);
+	VCROSS(vcp, ftp->e_i, ftp->r_i);
+	ftp->nf = 1.0/DOT(vcp,vcp);
+	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( ftp->nf );
+	ftp->I2 = ( DOT(ftp->e_i, ftp->r_i1)*rdot_r1 - dot_er*rdot_r +
+			dot_e*ftp->I1 )*0.5*ftp->nf;
+	J2 =  ( 0.5*(rdot_r - rdot_r1) - dot_er*ftp->I2 ) / dot_e;
+	for (i = 3; i--; )
+		ftp->rI2_eJ2[i] = ftp->I2*ftp->r_i[i] + J2*ftp->e_i[i];
+}
+
+
+/* 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	vcp;
+	FVECT	m1[3], m2[3], m3[3], m4[3];
+	double	d1, d2, d3, d4;
+	double	I3, J3, K3;
+	int	i, j;
+					/* 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 = 0.25*ftp->nf*( DOT(ftp->e_i, ftp->r_i1)*d2*d2 - d4*d1*d1 +
+				3.0/d3*ftp->I2 );
+	J3 = 0.25*d3*(d1*d1 - d2*d2) - d4*d3*I3;
+	K3 = d3*(ftp->I2 - I3/d1 - 2.0*d4*J3);
+					/* intermediate matrices */
+	VCROSS(vcp, nrm, ftp->e_i);
+	compose_matrix(m1, vcp, 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);
+	VCROSS(vcp, ftp->r_i, ftp->e_i);
+	d1 = DOT(nrm, vcp);
+	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], COLORV 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	vcp;
+	double	f1;
+	int	i;
+
+	VCROSS(vcp, ftp->r_i, ftp->r_i1);
+	f1 = 2.0*DOT(nrm, vcp);
+	VCROSS(vcp, nrm, ftp->e_i);
+	for (i = 3; i--; )
+		grad[i] = (-0.5/PI)*( ftp->I1*vcp[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, COLORV v)
+{
+	int	i;
+
+	for (i = 3; i--; )
+		grad[i] += v*( egrad1[i] + egrad2[i] + egrad3[i] );
+}
+
+
+/* Return brightness of furthest ambient sample */
+static COLORV
+back_ambval(struct s_ambsamp *ap1, struct s_ambsamp *ap2,
+		struct s_ambsamp *ap3, FVECT orig)
+{
+	COLORV	vback;
+	FVECT	vec;
+	double	d2, d2best;
+
+	VSUB(vec, ap1->p, orig);
+	d2best = DOT(vec,vec);
+	vback = colval(ap1->v,CIEY);
+	VSUB(vec, ap2->p, orig);
+	d2 = DOT(vec,vec);
+	if (d2 > d2best) {
+		d2best = d2;
+		vback = colval(ap2->v,CIEY);
+	}
+	VSUB(vec, ap3->p, orig);
+	d2 = DOT(vec,vec);
+	if (d2 > d2best)
+		return(colval(ap3->v,CIEY));
+	return(vback);
+}
+
+
+/* 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 eigenvalues */
+	if ( quadratic(evalue, 1.0, -hess2[0][0]-hess2[1][1],
+			hess2[0][0]*hess2[1][1]-hess2[0][1]*hess2[1][0]) != 2 ||
+			(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 */
-	float	pg[2]			/* returned */
+	float	ra[2],			/* returned (optional) */
+	float	pg[2]			/* returned (optional) */
 )
 {
-	if (ra != NULL) {		/* compute Hessian-derived radii */
-	} else {			/* else copy original tangent axes */
-		VCOPY(uv[0], hp->ux);
-		VCOPY(uv[1], hp->uy);
-	}
-	if (pg == NULL)			/* no position gradient requested? */
+	static char	memerrmsg[] = "out of memory in ambHessian()";
+	FVECT		(*hessrow)[3] = NULL;
+	FVECT		*gradrow = NULL;
+	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));
+	}
+					/* compute first row of edges */
+	for (j = 0; j < hp->ns-1; j++) {
+		comp_fftri(&fftr, ambsamp(hp,0,j).p,
+				ambsamp(hp,0,j+1).p, hp->rp->rop);
+		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, ambsamp(hp,i,0).p,
+			ambsamp(hp,i+1,0).p, hp->rp->rop);
+	    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;
+		COLORV	backg;
+		backg = back_ambval(&ambsamp(hp,i,j), &ambsamp(hp,i,j+1),
+					&ambsamp(hp,i+1,j), hp->rp->rop);
+					/* diagonal (inner) edge */
+		comp_fftri(&fftr, ambsamp(hp,i,j+1).p,
+				ambsamp(hp,i+1,j).p, hp->rp->rop);
+		if (hessrow != NULL) {
+		    comp_hessian(hessdia, &fftr, hp->rp->ron);
+		    rev_hessian(hesscol);
+		    add2hessian(hessian, hessrow[j], hessdia, hesscol, backg);
+		}
+		if (gradient != 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, ambsamp(hp,i+1,j+1).p,
+				ambsamp(hp,i+1,j).p, hp->rp->rop);
+		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(&ambsamp(hp,i,j+1), &ambsamp(hp,i+1,j+1),
+					&ambsamp(hp,i+1,j), hp->rp->rop);
+		comp_fftri(&fftr, ambsamp(hp,i,j+1).p, ambsamp(hp,i+1,j+1).p,
+				hp->rp->rop);
+		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);
+	
+	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])
+{
+	struct s_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);
+}
+
+
 int
 doambient(				/* compute ambient component */
 	COLOR	rcol,			/* input/output color */
 	RAY	*r,
 	double	wt,
-	FVECT	uv[2],			/* returned */
-	float	ra[2],			/* returned */
-	float	pg[2],			/* returned */
-	float	dg[2]			/* returned */
+	FVECT	uv[2],			/* returned (optional) */
+	float	ra[2],			/* returned (optional) */
+	float	pg[2],			/* returned (optional) */
+	float	dg[2]			/* returned (optional) */
 )
 {
+	AMBHEMI			*hp = inithemi(rcol, r, wt);
 	int			cnt = 0;
 	FVECT			my_uv[2];
-	AMBHEMI			*hp;
 	double			d, acol[3];
 	struct s_ambsamp	*ap;
 	int			i, j;
-					/* initialize */
-	if ((hp = inithemi(rcol, r, wt)) == NULL)
+					/* check/initialize */
+	if (hp == NULL)
 		return(0);
 	if (uv != NULL)
 		memset(uv, 0, sizeof(FVECT)*2);
@@ -169,10 +518,9 @@ doambient(				/* compute ambient component */
 		dg[0] = dg[1] = 0.0;
 					/* sample the hemisphere */
 	acol[0] = acol[1] = acol[2] = 0.0;
-	for (i = hemi.ns; i--; )
-		for (j = hemi.ns; j--; )
-			if (ambsample(hp, i, j)) {
-				ap = &ambsamp(hp,i,j);
+	for (i = hp->ns; i--; )
+		for (j = hp->ns; j--; )
+			if ((ap = ambsample(hp, i, j)) != NULL) {
 				addcolor(acol, ap->v);
 				++cnt;
 			}
@@ -181,32 +529,52 @@ doambient(				/* compute ambient component */
 		free(hp);
 		return(0);		/* no valid samples */
 	}
-	d = 1.0 / cnt;			/* final indirect irradiance/PI */
-	acol[0] *= d; acol[1] *= d; acol[2] *= d;
-	copycolor(rcol, acol);
+	copycolor(rcol, acol);		/* final indirect irradiance/PI */
 	if (cnt < hp->ns*hp->ns ||	/* incomplete sampling? */
 			(ra == NULL) & (pg == NULL) & (dg == NULL)) {
 		free(hp);
 		return(-1);		/* no radius or gradient calc. */
 	}
-	d = 0.01 * bright(rcol);	/* add in 1% before Hessian comp. */
-	if (d < FTINY) d = FTINY;
-	ap = hp->sa;			/* using Y channel from here on... */
+	if (bright(acol) > FTINY)	/* normalize Y values */
+		d = cnt/bright(acol);
+	else
+		d = 0.0;
+	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;
+		colval(ap->v,CIEY) = bright(ap->v)*d + 0.01;
 
 	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) {		/* adjust/clamp radii */
-		d = pow(wt, -0.25);
-		if ((ra[0] *= d) > maxarad)
-			ra[0] = maxarad;
+
+	if (ra != NULL) {		/* scale/clamp radii */
+		if (ra[0] < minarad) {
+			ra[0] = minarad;
+			if (ra[1] < minarad)
+				ra[1] = minarad;
+					/* cap gradient if necessary */
+			if (pg != NULL) {
+				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;
+				}
+			}
+		}
+		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;
+		}
 	}
 	free(hp);			/* clean up and return */
 	return(1);