efficient illumination by high dynamic range images · diffuse objects by high dynamic range...
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Eurographics Symposium on Rendering 2003Per Christensen and Daniel Cohen-Or (Editors)
Efficient Illumination by High Dynamic Range Images
Thomas Kollig1 and Alexander Keller2
1 Department of Computer Science, University of Kaiserslautern, Germany2 Department of Computer Science, University of Ulm, Germany
AbstractWe present an algorithm for determining quadrature rules for computing the direct illumination of predominantlydiffuse objects by high dynamic range images. The new method precisely reproduces fine shadow detail, is muchmore efficient as compared to Monte Carlo integration, and does not require any manual intervention.
1. Introduction
Rendering synthetic objects into real scenes requires theirillumination by real world radiance, which can be capturedas a high resolution spherical high dynamic range image2, 15.This technique has been introduced by Debevec et al.4, 3, isestablished in commercial products, and has been used inmany movies.
In this context we address the efficient approximation ofthe radiance
L(x,ωo) ≈∫
Ω+x
Lhdr(ω)V(x,ω) fr (ω,x,ωo)〈n(x),ω〉dω ,
leaving a pointx in directionωo. The spherical high dynamicrange imageLhdr is of high resolution.Ω+
x is the upper hemi-sphere around the surface normaln(x) in point x. The visi-bility V(x,ω) is one if starting in pointx no objects are seenin directionω and zero otherwise.
For highly directional bidirectional reflectance distribu-tion functions fr it is easy to evaluate the above integral byvariance reduced Monte Carlo integration:
L(x,ωo)
≈ 1N
N−1
∑i=0
Lhdr(ωi)V(x,ωi) fr (ωi ,x,ωo)〈n(x),ωi〉p(ωi)
, (1)
where the directionsωi arep-distributed proportional tofr .While this is an efficient procedure for highly specular sur-faces, i.e. narrow cones of reflection, it utterly fails for pre-dominantly diffuse surfaces due to high variance intrinsic toLhdr. Choosing the directionsωi proportional toLhdr can-not be realized in an efficient way: On the one hand adap-tive integration schemes like e.g. the VEGAS or MISERtechnique11 are inefficient due to excessive additional book
keeping data structures. On the other hand importance sam-pling on the discrete set of directional light sources givenby the pixels of the high dynamic range image still resultsin high variance as will be shown in section5. In additionapproximations7 usually do not perform any better than purerandom sampling in this case. We also want to avoid vari-ance reduction by manual stratification3. Projecting the inte-gral operator into the spherical harmonics basis12, 13 allowsfor fast rendering. However, including detailed shadows inthis model imposes considerable cost and restrictions14.
In this paper the above problem of illuminating predomi-nantly diffuse surfaces by high resolution spherical high dy-namic range images is addressed. Therefore a new method(sections2 and3) is presented that reliably captures all de-tails of the high dynamic range image without manual in-tervention. Although seemingly similar to LightGen1, ourtechnique yields an almost optimal integration scheme by re-moving some flaws of the straightforward approach. This isdemonstrated by numerical evidence. Furthermore new su-perior anti-aliasing techniques are developed (section4) thatreproduce fine shadow details much more precisely and arefaster than previous approaches.
2. Quadrature Rule Construction
The basic idea of our algorithm is to determine a quadra-ture rule(ωi ,Bi)
N−1i=0 only depending on the high dynamic
range imageLhdr. Then the direct illumination can be ap-proximated by
L(x,ωo) ≈N−1
∑i=0
Bi V(x,ωi) fr (ωi ,x,ωo)〈n(x),ωi〉 , (2)
where (ωi ,Bi) in fact corresponds to a directional lightsource from directionωi with radiosityBi .
c© The Eurographics Association 2003.
Kollig and Keller / Efficient Illumination by High Dynamic Range Images
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Figure 1: The N= 256colored points in each image indicate the directionsωi generated by Lloyd’s relaxation algorithm onthe left and our improved scheme on the right. For convenience the spherical images are displayed as 2:1 lattitude/longitudemaps. Obviously the new approach captures the light distribution much more precisely resulting in a smaller integration errorduring rendering. Both images have been tone mapped for display.
Figure 2: Images rendered using the quadrature rules generated by Lloyd’s relaxation algorithm (top row) and our improvedscheme (bottom row) as illustrated in figure1. The number of light sources is N= 32,64,128,256 (from left to right). Theshadow boundary artifacts caused by the directional light sources vanish much faster with our new scheme, clearly indicatingthe faster decay of the integration error due to the more equalized and consequently smaller weights of the quadrature rule.
For a given partition(Ωi)N−1i=0 of the set of all unit direc-
tionsΩ, the quadrature rule can be determined by
Bi :=∫
Ωi
Lhdr(ω)dω (3)
and choosing the directionsωi as mass centroids ofΩi , i.e.
ωi ∈
ω′ ∈ Ω∣∣∣∫
Ωi
(arccos(〈ω′,ω〉)
)2‖Lhdr(ω)‖dω
= infω′′∈Ω
∫Ωi
(arccos(〈ω′′,ω〉)
)2‖Lhdr(ω)‖dω
.
Thus the quadrature rule is based on the piecewise constantapproximation
Lhdr(ω)≈N−1
∑i=0
Bi
|Ωi |χΩi (ω) ,
whereχΩi is the characteristic function ofΩi ⊆ Ω. In thesequel it is shown how to construct a partition such that
• the partition(Ωi)N−1i=0 is the spherical Voronoi diagram of
the set(ωi)N−1i=0 limiting the integration error of (2) if the
visibility term V is neglected and• max0≤i<N ‖Bi‖ is as small as possible limiting the inte-
gration error of (2) made on each solid angleΩi .
Note that by the first constraint an implicit definition ofΩiandωi is formed and the second one bounds
N−1
∑i=0
(Bi −
1N
N−1
∑i=0
Bi
)2
=N−1
∑i=0
(Bi −
1N
∫Ω
Lhdr(ω)dω)2
.
3. Determining the Quadrature Rule
The algorithm for determining the quadrature rule(ωi ,Bi)
N−1i=0 is based on Lloyd’s relaxation method5 on
the sphere:
c© The Eurographics Association 2003.
Kollig and Keller / Efficient Illumination by High Dynamic Range Images
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angl
eof
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ntra
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imum
angl
eof
dece
ntra
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128 256
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imum
radi
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(wei
ght)
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..........1.28
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imum
radi
osity
(wei
ght)
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θT = arccos(0.99) θT = arccos(0.999) θT = arccos(0.9999)
Figure 3: Comparison of the quadrature rules generated by Lloyd’s relaxation algorithm on the left and our improved schemeon the right using different termination thresholdsθT . The top row shows the maximum angle of decentration. Hardly noticeablethe new scheme obtains a smaller decentration at a somewhat higher variance. However, as shown in the bottom row, the decayof the maximum radiosity is quadratically faster with the new approach. Instead of onlyO(N−1/2) almost the trivial lowerbound ofO(N−1) is achieved as indicated by the black lines.
1. Randomly select an initial set(ωi)N−1i=0 of directions.
2. Construct the Voronoi tessellation(Ωi)N−1i=0 associated to
the directions(ωi)N−1i=0 .
3. For each Voronoi regionΩi replaceωi by one of its masscentroidal directions.
4. If not terminated then go to step 2.5. Compute the weights(Bi)
N−1i=0 by (3).
The relaxation procedure is terminated if the maximummovement of all directions, i.e. the maximum angle of de-centration, is below some thresholdθT . For efficiency theVoronoi tessellation and the mass centroidal directions areapproximated by uniformly samplingΩ. Thus the actualΩiimplicitly are determined6 by the samples closest toωi . Theaverage of these closest points is the corresponding masscentroid. Upon termination exactly these closest points arealso used to approximate the quadrature weightsBi . Thistechnique in fact is very similar tok-means clustering as usedin LightGen1.
This straightforward algorithm is very sensitive to the ini-tial choice of(ωi)N−1
i=0 and becoming trapped by local min-ima is almost unavoidable. This is illustrated in the left im-age of figure1, where far too many directions are located inunimportant regions.
This disadvantage easily can be overcome by incremen-tally determining the set of lighting directions. The key isto insert a new direction nearby the directionωi′ with themaximum weight‖Bi′‖:
1. Setn = 1 and select a random directionω0.2. Construct the Voronoi tessellation(Ωi)n−1
i=0 associated to
the directions(ωi)n−1i=0 .
3. For each Voronoi regionΩi replaceωi by one of its masscentroidal directions.
4. If not terminated then go to step 2.5. Compute the weights(Bi)
n−1i=0 by (3).
6. If n< N increasen by 1 and choose a new directionωn−1nearby the directionωi′ with the maximum weight‖Bi′‖and go to step 2.
c© The Eurographics Association 2003.
Kollig and Keller / Efficient Illumination by High Dynamic Range Images
3.1. Numerical Evidence
The images in figure1 illustrate the placement ofN = 256directions. Clearly, the improved algorithm places more lightsources in the brighter regions of the high dynamic rangeimage. The resulting quadrature rules have been used to ren-der the images using (2) as shown in figure2. While bothquadrature rules are almost indistinguishable in unshadowedregions, the superiority of the improved quadrature rule be-comes obvious in shadowed regions, which are rendered al-most free of artifacts at alreadyN = 256 directions. The in-creased preprocessing time reliably is compensated by thequality gain.
This result can be explained by comparing bothquadrature rules with respect to the maximum radios-ity max0≤i<N ‖Bi‖ and the maximum angle of de-centration after termination. This angle is given bymax0≤i<N arccos〈ωi ,γi〉, whereγi is the centroidal directionof the Voronoi region corresponding toωi .
As shown in figure3, forcing more iterations by low-ering the termination thresholdθT naturally decreases themaximum angle of decentration in the original approach. Inthe improved version the maximum angle of decentration ismarginally smaller at a somewhat higher variance. At thesame time the maximum radiosity practically remains un-changed independent of the approach.
However, the improved version almost achieves the triviallower bound ofO(N−1), which is quadratically better thanthe decay of the maximum radiosity in the original scheme.As already mentioned at the end of section2, this efficientlydecreases the integration error of (2).
4. Efficient Anti-Aliasing
Although our improved scheme results in much fasterquadrature rules (see section3), using an identical quadra-ture rule for the whole image results in clearly visibleshadow boundary artifacts if the numberN of light sourcesis small (see figure2). Therefore two methods are proposedthat allow one to reduce the total number of shadow rayswhile efficiently preventing aliasing.
Note that anti-aliasing is further improved by the powerfultechnique of interleaved sampling8, 10. However, the applica-tion of interleaved sampling to both methods proposed in thesequel is straightforward and therefore omitted for the sakeof clarity.
4.1. Randomly Perturbed Quadrature Rules
At low sampling rates the shadow boundary artifacts can beturned into less objectionable noise by stratified samplingusing the partition(Ωi)
N−1i=0 . However, for maximum effi-
ciency the strataΩi are approximated by inscribed sphericalcaps defined by cones of radiusαi centered inωi . Thus it
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q q qqq q qq q qa aa
a a a aa aaa aa aa a
aa
a aaa aa aaa a aa a aa a a
aa a aa
aaa aa
aa a a
aa aaa
aa aaa a aa a aa aa
a a a aaa aa aa aa a aa a
aaa aa aaa a aaa a
Figure 4: Interleaved quadrature rule consisting of M= 4separate quadrature rules at N= 32 light sources. The col-ored points represent the directionsω j,i of the light sourcesin the 2:1 lattitude/longitude map, while each color repre-sents one quadrature rule. Our improved scheme preciselycaptures the light distribution by the total of128= 4·32di-rections as well as every separate quadrature rule does asfor example highlighted by the solid points. The image hasbeen tone mapped for display.
is easy to generate a random direction inside each cone. Inorder to decorrelate the samples this random perturbation ofthe quadrature rule has to be performed each time (2) is eval-uated. It is important to note that the variance of the originalhigh dynamic range imageLhdr restricted to the strataΩi re-mains high due to the fine image details. Therefore it has tobe reduced, which is achieved by simply using the alreadyfiltered valuesBi .
4.2. Interleaved Quadrature Rules
Due to the correlation coefficient of the integrand with re-spect to the shadow rays9 it pays off to sendN shadowrays for a single primary ray, when computing the direct il-lumination. Maximum efficiency is achieved by correlatedsampling9.
We realize this kind of efficient multidimensional sam-pling for anti-aliasing by generating a separate quadrature
rule(ω j,i ,B j,i
)N−1i=0 for each primary rayj = 0, . . . ,M − 1
and imposing the two constraints that both
• each separate quadrature rule(ω j,i ,B j,i
)N−1i=0 for j =
0, . . . ,M−1 and
• the composite quadrature rule(
ω j,i , B j,i
)M−1,N−1
j,i=0
fulfill the requirements of section2. Note that both con-straints use the same directions, however, different quadra-ture weightsB j,i andB j,i result from (3) as induced by the
Voronoi tessellations of(ω j,i)N−1
i=0 for j = 0, . . . ,M−1 and(ω j,i)M−1,N−1
j,i=0 , respectively.
Figure4 shows an example of an interleaved quadraturerule composited out ofM = 4 separate quadrature rules withN = 32 light sources each that has been generated using thefollowing algorithm:
c© The Eurographics Association 2003.
Kollig and Keller / Efficient Illumination by High Dynamic Range Images
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Figure 5: Quality of the interleaved quadrature rule after k iterations. The red lines indicate the maximum of the values of theM = 4 separate quadrature rules at N= 32 light sources, while the green lines indicate the values of the composite quadraturerule. The unavoidable loss of quality of the separate quadrature rules is kept minimal by our generation process.
1. Generate an initial quadrature rule(ω0,i ,B0,i
)N−1i=0 fol-
lowing section3.2. For j = 1,2, . . . ,M − 1 construct the quadrature rule(
ω j,i ,B j,i)N−1
i=0 by randomly perturbing the directionsω0,i .
3. For j = 0,1, . . . ,M−1 construct the Voronoi tessellation(Ω j,i)N−1
i=0 of Ω associated to the directions(ω j,i)N−1i=0 .
4. For each Voronoi regionΩ j,i replaceω j,i by one of itsmass centroidal directions.
5. Construct the Voronoi tessellation(
Ω j,i
)M−1,N−1
j,i=0of Ω
associated to the directions(ω j,i)M−1,N−1j,i=0 .
6. For each Voronoi regionΩ j,i replaceω j,i by one of itsmass centroidal directions.
7. k times iterate steps 3 through 6.
8. Compute the weights(B j,i)M−1,N−1
j,i=0 by (3).
For the initialization step 2 the radiusα of the cones of per-turbation (see also section4.1) is chosen small, e.g.α =0.01. The alternate application of Lloyd’s relaxation step tothe separate quadrature rules in steps 3 and 4 and to the com-posite quadrature rule in steps 5 and 6 achieves to satisfyboth constraints as mentioned in the beginning of this sec-tion.
Figure 5 shows the development of the quality of boththe separate quadrature rules and the composite quadraturerule. Upon initialization the maximum angle of decentra-tion is small for the separate quadrature rules and due to alack of correlation huge for the composite quadrature rule.Then the iteration process efficiently increases correlationforcing a higher but nevertheless decreasing maximum an-gle of decentration in the separate quadrature rules. In con-sequence the maximum radiosities of the separate quadraturerules must be slightly increasing. At the same time, however,the maximum radiosity of the composite quadrature rule re-mains small. In practice only aboutk = 4 iteration steps aresufficient to obtain a high quality interleaved quadrature rule.
5. Results
Figure6 compares rendering methods for computing the di-rect illumination by a high dynamic range imageLhdr. Forease of comparison the sunset 2:1 lattitude/longitude scan(see figures1, 4, and8) has been used.
The images on the left of figure6 show the results of sam-pling the hemisphere (mid-left) and importance sampling thehigh dynamic range image (lower left) identifying each pixelas one directional light source. The high variance of both es-timators even cannot be sufficiently reduced by randomizedquasi-Monte Carlo9 and consequently strong noise remainsvisible. However, as expected importance sampling the dis-crete set of light sources (lower left) is less noisy and avoidsdisturbing spike noise (see the enlargements in figure6).
The images on the right of figure6 are rendered using ournew techniques. Obviously one single quadrature rule (seesection3) exposes clear shadow boundary artifacts (upperright), which are transferred to less perceivable noise (mid-right) by randomly perturbed quadrature rules (see section4.1). Finally the interleaved quadrature rule (see section4.2)reduces the shadow artifacts to an imperceptible level (lowerright). While randomly perturbed quadrature rules may befavored for rendering still images, interleaved quadraturerules are the better choice for animations, where coherentsampling over time avoids flicker.
Although each method has used 32 samples for computingthe direct illumination of each primary ray in our new ren-dering methods up to 15% of the shadow rays do not needto be shot, because they can be culled using the surface nor-mal. In addition better memory coherence and less pseudorandom number generator calls are the reasons for up to 50%reduced rendering times.
Note that pseudo random number generation can consumeconsiderable amounts of time, which also is the reason forthe 25% increase of rendering time when randomly perturb-
c© The Eurographics Association 2003.
Kollig and Keller / Efficient Illumination by High Dynamic Range Images
preprocessingtime in s
renderingtime in s
shadow raysper pixel
a a ab b bc c cd d de e e
0 0 23 2375
658
616
331
441
345
275
256
246
245
235
(c) single quadrature rule
(a) importance sampling of the hemisphere (d) randomly perturbed quadrature rule
(b) importance sampling of the discrete set of light sources(e) interleaved quadrature rule
Figure 6: Comparison of direct illumination computed by (a) importance sampling of the hemisphere, (b) importance samplingof the discrete set of light sources, (c) a single, (d) a randomly perturbed, and (e) an interleaved quadrature rule. For anti-aliasing16 primary rays per pixel were traced. For each primary ray32 samples were used for the techniques on the left . Thesingle and the randomly perturbed quadrature rule had N= 32 light sources and the interleaved quadrature rule consisted ofM = 16separate quadrature rules with N= 32 light sources each. The diagram displays the preprocessing and rendering timeson a 650 MHz Pentium III for the above images at a resolution of640×480pixels and the number of shadow rays per pixel.
c© The Eurographics Association 2003.
Kollig and Keller / Efficient Illumination by High Dynamic Range Images
ing the single quadrature rule. Considering the improved im-age quality at reduced rendering times the preprocessing of23 seconds for a single or 75 seconds for an interleavedquadrature rule clearly pays off. On the other hand obtain-ing the same quality without interleaved quadrature ruleswould require a single quadrature rule with much more lightsources that in consequence would take much longer time togenerate.
6. Conclusion
We presented a robust algorithm for the efficient compu-tation of direct illumination of predominantly diffuse sur-faces by spherical high dynamic range images. The schemeis designed for industrial production, where high resolutionspherical high dynamic range scans are used (see figure7).
Without manual intervention our new scheme generatesquadrature rules with minimized weights resulting in a con-siderable noise reduction and improved anti-aliasing. Asshown in figure8 the light distribution is captured veryprecisely. Thus it becomes redundant to e.g. cut out lightsources manually3 in order to reduce variance. The result-ing quadrature rules consume negligible amounts of memoryand can be stored along with the high dynamic range imagessaving repeated preprocessing. The directional light sourcescan be projected back onto a reconstructed geometry2 thusallowing for even more precise shadowing.
An obvious improvement to our implementation is the ac-celeration of the computation of the mass centroids and theweights of the quadrature rules by graphics hardware. Al-though specular surfaces are simple to render, the efficientcombination with our techniques is not straightforward forgeneral reflections properties, which cannot be representedas a weighted sum of basic bidirectional distribution func-tions. This, as well as the generation of caustics, is subjectto future research. We also will focus on combining quadra-ture rules from multiple high dynamic range images and onenabling our scheme for illumination by high dynamic rangevideo.
Acknowledgements
The first author has been funded by the Stiftung Rheinland-Pfalz für Innovation. The research has been supported bySpheron VR AG15.
References
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c© The Eurographics Association 2003.
Kollig and Keller / Efficient Illumination by High Dynamic Range Images
Figure 7: Reality check: A virtual car has been placed into a real set recorded by one high resolution high dynamic rangeimage. The scanned image is used both as background shot and for our fully automatic illumination technique.
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(c) single quadrature rule withN = 256 (d) interleaved quadrature rule withM = 8 andN = 32
Figure 8: The top row example demonstrates our method for a predominantly continuous light distribution. The Grace Cathe-dral example (courtesy P. Debevec) in the bottom row impressively shows that our new method reliably captures bright regionsand discontinuities of the high dynamic range image. Note that no manual intervention like e.g. replacing the bright regions byarea light sources (which would be even more tedious in the top row example) is required. All images have been tone mappedfor display.
c© The Eurographics Association 2003.