A Low Distortion Map Between Disk and Square

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1 A Low Distortion Ma Between Disk and Square Peter Shirley Comuter Science Deartment 3190 MEB University of Utah Salt Lake City, UT 8411 Kenneth Chiu Comuter Science Deartment Lindley Hall Indiana University Bloomington, IN Astract We resent a ma etween squares and disks that associates concentric squares with concentric circles. This ma reserves adjacency and fractional area, and has roven useful in many samling alications where corresondences must e maintained etween the two shaes. We also rovide code to comute the ma that minimizes ranching and is roust for all inuts. Finally, we extend the ma to the hemishere. Though this ma has een used efore in ulications, details of its comutation have never reviously een ulished. Background Many grahics alications ma oints on the unit square S = [0 1] to oints on the unit disk D = f(x y) j x + y 1g. Samling disk-shaed camera lenses is one such alication. Though each alication can have its own unique requirements, exerience has shown that a good, general-urose ma should have the following roerties. Preserve fractional area. Let R M e a region in set M. The fractional area of R is defined as a(r)=a(m ), where the function a denotes area. Then a ma m : S! D reserves fractional area if the fractional area of R is the same as the fractional area a(m(r))=a(d) for all R S (Figure 1). This roerty ensures that a fair set of oints on the square will ma to a fair set on the disk. For distriution ray tracers this means a jittered set of oints on the square will transform to a jittered set of oints on the disk. Bicontinuous. Amaisicontinuous if the ma and its inverse are oth continuous. Such a ma will reserve adjacency; that is, oints that are close on the disk came from oints that are close on the square, and vice versa. This roerty is useful rimarily when working on the hemishere. It is necessary, for examle, if we wish to use linear distance on the square as an estimate of angular distance on the hemishere. Low distortion. By low distortion, we mean that shaes are reasonaly well reserved. Defining distortion more formally is ossile [1], ut roaly of limited enefit ecause no single definition is clearly suitale for a wide range of alications. Figure shows the olar ma, roaly the most ovious way to ma the square to the disk: r is a

2 R m(a,) = (u(a,),v(a,)) v m(r) a u S D = m(s) Figure 1: A ma that reserves fractional area will ma regions with the area constraint that a(r)=a(s) =a(m(r))=a(d). Figure : The olar ma takes horizontal stris to concentric rings. function of x and is a function of y. This will ma vertical stris on S to rings on D. Because an outer ring will have greater area than an inner ring of equivalent width, r cannot e a linear function of x: r = x = y This ma reserves fractional area, ut does not satisfy our other roerties. As can e seen in the image, shaes are grossly distorted. Another rolem is that although (r ) = ( x y) is continuous, the inverse ma is not. For some alications we would refer a icontinuous ma. Figure 3: The concentric ma takes concentric square stris to concentric rings.

3 a= 3 = = 4 a r 4 1 a = 1 = 4 = = 7 4 Figure 4: Quantities for maing region 1. Figure 5: Left: unit square with gridlines and letter F. Middle: olar maing of gridlines and F. Right: concentric maing of gridlines and F.

4 The Concentric Ma We now resent the concentric ma, which mas concentric squares to concentric circles, as shown in Figure 3. This ma has the roerties listed aove, and is easy to comute. It has een advocated for ray tracing alications [3] and has een shown emirically to rovide lower error for stochastic camera lens samling [], ut the details of its comutation have not een reviously ulished. The algera ehind the ma is illustrated in Figure 4. The square is maed to (a ) [;1 1],andis divided into four regions y the lines a = and a = ;. For the first region, the ma is: r = a = 4 This roduces an angle [;=4 =4]. The other four regions have analogous transforms. We show in the Aendix that this ma reserves fractional area. How the olar ma and concentric ma transform the connected oints in a uniform grid is shown in Figure 5. Note that the F is still recognizale in the concentric ma, ut has een grossly distorted y the olar ma. This concentric square to concentric circle ma can e imlemented in a small iece of roust code which has een written to minimize ranching: a Vector ToUnitDisk( Vector onsquare ) real hi, r, u, v real a = *onsquare.x-1 // (a,) is now on [-1,1]ˆ real = *onsquare.y-1 if (a > -) // region 1 or if (a > ) // region 1, also a > r = a hi = (PI/4 ) * (/a) else // region, also > a r = ; hi = (PI/4) * ( - (a/)) else // region 3 or 4 if (a < ) // region 3, also a >=, a!= 0 r = -a hi = (PI/4) * (4 + (/a)) else // region 4, >= a, ut a==0 and ==0 could occur. r = - if (!= 0) hi = (PI/4) * (6 - (a/)) else hi = 0 u = r * cos( hi ) v = r * sin( hi ) return Vector( u, v ) end

5 The inverse ma is straightforward rovided the function atan() is availale. The code elow assumes atan() returns a numer in the range [; ], asitdoesinansic, Draft ANSI C++, ANSI FORTRAN, andjava. Vector FromUnitDisk(Vector ondisk) real r = sqrt(ondisk.x * ondisk.x + ondisk.y * ondisk.y) Real hi = atan(ondisk.y, ondisk.x ); if (hi < -PI/4) hi += *PI // in range [-i/4,7i/4] Real a,, x, y if (hi < PI/4) // region 1 a = r = hi * a / (PI/4) else if (hi < 3*PI/4 ) // region = r a = -(hi - PI/) * / (PI/4) else if (hi < 5*PI/4) // region 3 a = -r = (hi - PI) * a / (PI/4) else // region 4 = -r a = -(hi - 3*PI/) * / (PI/4) x = (a + 1) / y = ( + 1) / return Vector(x, y) end The concentric ma can e extended to the hemishere y rojecting the oints u from the disk to the hemishere. By controlling exactly how the oints are rojected, we can generate different distriutions, as shown in the next section. Alications The concentric ma can e used in a numer of different ways. Random oint on a disk. A random oint on the disk of radius R can e generated y assing a random oint on [0 1] through the ma and multilying oth coordinates y R. Jittered oint on a disk. Asetofn jittered oints can e generated similarly using uniform random numers etween zero and one such as generated y a call to drand48(): int s = 0; for (int i = 0; i < n; i++) for (int j = 0; j < n; j++) Vector onsquare((i + drand48()) / n, (j + drand48()) / n ) ondisk[s++] = ToUnitDisk(onSquare )

6 The jittered samling roduced y the concentric ma tends to roduce lower variance than the olar ma, ecause the jitter cells stay relatively square instead of ecoming long and thin. This means that the integrand will tend to have less variation within each jitter cell. Cosine distriution on a hemishere. Radiosity and ath tracing alications often need oints on the unit hemishere with density roortional to the z-coordinate of the oint. This is commonly referred to as a cosine distriution ecause the z-coordinate of a oint on a unit shere is the cosine etween the z-axis and the direction from the shere s center to the oint. Such a distriution is commonly generated from uniform oints on the disk y rojecting the oints along the z-axis. Assuming (u v) are the coordinates on the disk and letting r = u + v, the cosine-distriuted oint (x y z) on the hemishere oriented along the ositive z-axis is x = u y = v 1 ; r z = Uniform distriution on a hemishere. Generating a uniform distriution of the z-coordinate will create a uniform distriution on the hemishere. This can e accomlished y noting that if we have a uniform distriution on a disk, then the distriution of r is also uniform. So given a uniformly distriuted random oint (u v) on a disk, we can generate a uniformly distriuted oint on the hemishere y first generating a z coordinate from r, and then assigning x and y such that the oint is on the hemishere [4]. x = u ; r y = v ; r z = 1 ; r Phong-like distriution on a hemishere. The uniform and cosine distriutions can e generalized to a Phong-like distriution where density is roortional to a ower of the cosine, or z N : 1 ; z x = u r 1 ; z y = v r 1 ; r 1 N+1 z = Note that the two revious formulas are secial cases of this formula. Sudivision data. In any alication where a sudivision data structure must e ket on the hemishere or disk, the icontinuous and low-distortion roerties make the ma ideal. For examle, a k-d tree organizing oints on the disk can e ket on the transformed oints on the square so that conventional axis-arallel two-dimensional divisions can e used.

7 Aendix When the concentric ma is exanded out so that we ma from (a ) [;1 1] to (u v) on the unit disk, the ma is: u = a cos v = a sin The ratio of differential areas in the range and domain for the ma is given y the determinant of the Jacoian matrix. If this determinant is the constant ratio of the area of the entire domain to the entire range, then the ma reserves fractional area as is desired. This is in fact the case for the = cos sin + sin ; cos!! ; 4 sin 4 cos = 4 This is the constant =4 ecause that is the ratio of the area of the unit disk to the area of [;1 1]. By symmetry, the determinant of the Jacoian matrix is the same for the other three regions. Acknowledgements Thanks to Ronen Barzel, Eric Haines, Eric Lafortune, Bill Martin, Simon Premoze, Peter-Pike Sloan, and Brian Smits for comments on the aer. Figure 5 was generated using software y Michael Callahan and Nate Roins. This work was done with suort from NSF grant ASC References [1] Frank Canters and Hugo Decleir. The World in Persective. Bath Press, Avon, Great Britain, [] Craig Kol, Pat Hanrahan, and Don Mitchell. A realistic camera model for comuter grahics. In Ro Cook, editor, Proceedings of SIGGRAPH 95 (Anaheim, California, August 6 11, 1995), Comuter Grahics Proceedings, Annual Conference Series, ages , August [3] Peter Shirley. A ray tracing method for illumination calculation in diffuse-secular scenes. In Proceedings of Grahics Interface 90, ages 05 1, May [4] Peter Shirley. Nonuniform random oint sets via waring. In David Kirk, editor, Grahics Gems III, ages Academic Press, San Diego, 199. summary of various useful samle generation transformations.

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