CN105869204B - A kind of unbiased Photon Mapping method for drafting in participating media - Google Patents
A kind of unbiased Photon Mapping method for drafting in participating media Download PDFInfo
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- G—PHYSICS
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- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T15/00—3D [Three Dimensional] image rendering
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Abstract
The invention discloses the unbiased Photon Mapping method for drafting in a kind of participating media, the present invention is improved unbiased photon method for congregating from the improvement of subpath sampling technique, theoretical formula transformation, the importance sampling without bias term estimation, dielectric boundaries situation discussion etc. so that obtains the drawing result of unbiased always in the scene of participating media.The present invention is also improved the weight of multiple importance sampling, to which the unbiased Photon Mapping method in participating media to be combined with two-way ray tracking technology so that can play advantage of the two methods under different scenes simultaneously.
Description
Technical field
The present invention relates to the unbiased Photon Mapping drafting sides in medium rendering technique field more particularly to a kind of participating media
Method.
Background technology
The core concept of Photon Mapping is that local luminous energy is estimated using the density of photon, specifically refers to JENSEN,
H.W.2001.Realistic Image Synthesis Using Photon Mapping.A.K.Peters,Ltd.,
Natick,MA,USA.The locality of photon collection makes this method between challenging caustic and multiple reflections
It is especially effective to connect illumination.In addition, Photon Mapping can reuse the light path corresponding to photon, to reduce the cost of sampling.It is close
Degree estimation can be effectively reduced noise, but also final result can be given to introduce deviation, therefore photons all before simultaneously
Mapping method all has inclined.Although a series of newest progressive Photon Mapping methods can be one when photon numbers be unlimited
Cause converge to correctly as a result, specific method can refer to HACHISUKA, T., OGAKI, S., AND JENSEN,
H.W.2008.Progressive photon mapping.ACM Trans.Graph.27,5(Dec.),130:1–130:8；
KNAUS,C.,AND ZWICKER,M.2011.Progressive photon mapping:A probabilistic
approach.ACM Trans.Graph.30,3(May),25:1–25:13；HACHISUKA,T.,AND JENSEN,
H.W.2009.Stochastic progressive photon mapping.ACM Trans.Graph.28,5(Dec.),
141:1–141:8；KAPLANYAN,A.S.,AND DACHSBACHER,C.2013.Adaptive progressive photon
mapping.ACM Trans.Graph.32,2(Apr.),16:1–16:13.But these methods without exception are arbitrarily having
Interior result all has inclined in limited time, and the method invented herein is realized in medium field of drawing, and our method is from beginning
To the drawing result that can obtain unbiased eventually.
The rendering equation that Kajiya was proposed in 1986 is the mathematical description for the physics law propagated for luminous energy, the equation
It is a multiple integral, can be solved by the method for Monte Carlo, which may refer to KAJIYA,
J.T.1986.The rendering equation.SIGGRAPH Comput.Graph.20,4(Aug.),143–150.It is two-way
Ray trace is then to carry out complete light path by sampling sight and light respectively, and by the method that two light paths are connected
Sampling.This method is especially effective for the light source and camera of indoor scene and limited size, and specific method may refer to
LAFORTUNE,E.P.,AND WILLEMS,Y.D.1993.Bi-directional path tracing.In
Proceedings of CompuGraphics,vol.93,145–153；VEACH,E.,AND GUIBAS,
L.J.1994.Bidirectional estimators for light transport.In Proceedings of the
Fifth Eurographics Workshop on Rendering,Eurographics,147–162。
Multiple importance sampling is also proposed by Veach et al. later, is mainly used for tying among two-way ray trace at that time
The different connection type of various light paths is closed, referring to VEACH, E.1998.Robust Monte Carlo Methods for
Light Transport Simulation.PhD thesis,Stanford,CA,USA.AAI9837162.Photon Mapping and double
All it is to form the light path completed sampling by connecting sight and light to ray trace.For two-way ray trace, light
Son mapping is shone since sight-line end and photon are very close in the indirect light for sampling more complicated caustic and multiple reflections
It is more advantageous；However when direct illumination, two-way ray trace is then more efficient.Therefore both methods is in difference
Type light path sampling in be complementary, specifically may refer to HASAN ˇ, M., KRIV ˇ ANEK ', J., WALTER, B., AND
BALA,K.2009.Virtual spherical lights for many-light rendering of glossy
scenes.ACM Trans.Graph.28,5(Dec.),143:1–143:6；VORBA,J.2011.Bidirectional
photon mapping.In Proc.of the Central European Seminar on Computer Graphics
(CESCG 11)。
Invention content
In view of the above-mentioned deficiencies in the prior art, it is an object of the present invention to the unbiased Photon Mapping in providing a kind of participating media
Method for drafting.The present invention solves while giving full play to higher efficiency of the Photon Mapping method for drafting in medium drafting
Error problem caused by approximate evaluation in this method.The applicability on medium is done to unbiased Photon Mapping method for drafting to change
Into enabling the method to preferably to apply in medium draws scene, there is very high use value.
The purpose of the present invention is achieved through the following technical solutions：A kind of unbiased Photon Mapping in participating media is painted
Method processed, includes the following steps：
Step 1：In the scene for having participating media, light source subpath is generated according to the light source in scene, according in scene
Viewpoint position generate viewpoint subpath, specific generation method is as follows：
1.1, it is starting point from light source or viewpoint position, photon is emitted into scene.
1.2, photon continues to advance to exit direction, when it encounters the object in scene, the material that is defined according to object
And the incidence angle of photon samples to obtain the exit direction of photon, photon moves in this direction.
1.3, when photon encounters the medium in scene, due to the scattering properties of medium, photon can be after a distance
It scatters, finds out photon and arrive generation scattering phenomenon when institute forward travel distance next time on the medium.
1.4, after the forward travel distance of photon in the medium has been calculated, it is also necessary to sample side when being scattered after photon advances
To the direction can be acquired according to the phase function property and incident direction of medium.
1.5, the whole string location information that the photon is sampled is recorded, as the sample path of the photon, the i.e. sub- road of light source
DiameterOr viewpoint subpathI=1,2 ... s, j=1,2 ... t；Wherein yi
Refer to i-th of subpath vertex, z on light source subpathjRefer to j-th of subpath vertex, y on viewpoint subpath1For light source position
It sets, z1For viewpoint position, ysFor the last term point on light source subpath, ztFor the last term point on viewpoint subpath, s is light source
Subpath vertex number on path, t are the subpath vertex number on viewpoint subpath.
Step 2：According to light source subpathWith viewpoint subpathMeter
Calculate the measurement contribution margin of simultaneously recording light source subpathThe probability density function of light source subpathViewpoint subpath
Measurement contribution marginThe probability density function of viewpoint subpath
The probability density function of light source subpathRefer to the conditional probability density letter on this string of vertex on the path
Several formed Joint Distributions.First, i-th of subpath vertex y is definediProbability density function：
And intermediate variable：
Wherein, p (y1) what is indicated is to up-sample y in light source1The probability of point, pσ(y1→y2) indicate from light source position y1Place refers to
To next sub- path vertices y2When y1Locate the probability density function in solid angle, pσ(yi-1→yi→yi+1) indicate that incidence angle is
yi-1→yiThe angle of emergence is yi→yi+1When yi-1Locate the probability density function in solid angle, θi→i+1Refer to subpath vertex yiOn
Surface normal and direction vectorBetween angle, subfix σ indicates that probability density function is estimated for solid angle.
According to subpath vertex yiProbability density function p (yi), the probability density function of light source subpath can be obtained
The measurement contribution margin of light source subpath
Wherein,For light source subpath vertex yiWith yi+1Between geometry item；Indicate light
Source subpath vertex yiWith yi+1Between visual function, 0 is taken when being blocked, otherwise take 1；fr(yi-1→yi→yi+1) indicate i-th
Bidirectional reflectance distribution function at a sub- path vertices.
Similarly：The probability density function of viewpoint subpath
The measurement contribution margin of viewpoint subpath
Wherein,For viewpoint subpath vertex zjWith zj+1Between geometry item；Expression regards
Idea path vertices zjWith zj+1Between visual function；fs(zj-1→zj→zj+1) indicate the two-way of j-th subpath apex
Reflectance Distribution Function.
Step 3：All light source subpath is tracked in sampling, by they directly with viewpoint position z1It is connected, calculates viewpoint position
Set z1The direct illumination at place：
Wherein, Ω is the sample space of light source subpath,For the probability-distribution function of light source subpath, N is light source
The number of subpath, k are the ordinal number of light source subpath.
Simultaneously to y on all light source subpaths1y2…ysSubpath last term point ys, established using balance search tree method
Space querying accelerating structure.
Step 4：By all obtained light source subpath y1y2…ysWith viewpoint subpath z1z2…ztDo attended operation, i.e. handle
Light source subpath last term point ysWith viewpoint subpath last term point ztIt is connected, calculates connection path y1y2…yszt…z2z1Brightness,
It is added to viewpoint position z1On corresponding pixel, the connection pathRadiance：
Wherein, subscript VC indicates vertex attended operation,It is the expansion item of light source subpath：
It is the expansion item of viewpoint subpath：
For light source subpath last term point ysWith viewpoint subpath last term point ztThe measurement contribution margin of junction：
fr(ys-1→ys→zt) refer to incidence angle as ys-1→ysThe angle of emergence is ys→ztShi Guangyuan subpaths vertex ysPair at place
To Reflectance Distribution Function, fr(zt-1→zt→ys) refer to incidence angle as zt-1→ztThe angle of emergence is zt→ysWhen viewpoint subpath vertex zt
The bidirectional reflectance distribution function at place,Refer to light source subpath last term point ysWith viewpoint subpath last term point ztBetween
Geometry item.
Step 5：Obtained viewpoint subpath z is sampled to every1z2…ztOn subpath last term point zt, in step 3
Balance search tree is searched and ztDistance is less than all light source subpath last term point y of rs, according to ysThe corresponding sub- road of light source can be obtained
Diameter y1y2…ys, then by viewpoint subpath z1z2…ztWith light source subpath y1y2…ysMerged with subpath last term point, calculates and close
And pathBrightness, be added to viewpoint position z1On corresponding pixel, the merging path
y1y2…yszt-1…z2z1Radiance：
Wherein, subscript VM indicates vertex attended operation,It is the expansion item of light source subpath：
It is the expansion item of viewpoint subpath：
It is light source subpath last term point ysWith viewpoint subpath last term point ztIntermediate variable at merging：
Wherein,It is light source subpath last term point ysWith viewpoint subpath last term point ztMeasurement contribution at merging
Value,It is light source subpath last term point ysWith viewpoint subpath last term point ztProbability density function at merging calculates
Formula is as follows：
F hereins(ys-1→ys→zt-1) refer to incidence angle be ys-1→ysThe angle of emergence is ys→zt-1When subpath vertex ys's
Bidirectional reflectance distribution function, similarly fs(zt-2→zt-1→ys) refer to incidence angle be zt-2→zt-1The angle of emergence is zt-1→ysShi Zilu
Diameter vertex zt-1Bidirectional reflectance distribution function.Then indicate subpath vertex ysWith zt-1Between geometry item；
Wherein, r is the radius of photon collection, S (ys, r) and it is exactly with ysCentered on, radius is the Neighbourhood set of r, px(zt-2
→zt-1→ z) it is by the first two subpath vertex zt-2,zt-1Sampling obtains the probability density of position z：
For from current view point subpath vertex zt-1Constantly to light source subpath last term point ysTransmitting sampling light,
Light hits sphere neighborhood S (y for the first times, r) after get in total test light number, E [] indicate it is expected calculate.
Step 6 obtains the light intensity of all vertex connection paths in corresponding pixel points according to step 4With step 5
Obtain the light intensity that all vertex in corresponding pixel points merge pathUsing multiple importance sampling technology, obtain
Light intensity I after the weighting of corresponding pixel pointsVCM, so as to complete the drafting of entire unbiased Photon Mapping：
Wherein, subscript VC indicates that vertex attended operation, subscript VM indicate that vertex attended operation, subscript VCM indicate that vertex connects
Meet and merge joint operation, nVCFor the fullpath number that all vertex attended operations on the pixel obtain, nVMFor the picture
The fullpath number that all vertex union operations on vegetarian refreshments obtain, subscript k are the ordinal number in the path, and subscript s is the path
In the number on light source subpath vertex that contains, subscript s is the number on the viewpoint subpath vertex contained in the path.Refer to the light intensity for the fullpath that the attended operation of kth vertex obtains,It is corresponding multiple for the path
Weights of importance.Refer to the light intensity for the fullpath that kth vertex union operation obtains,For the road
The corresponding multiple weights of importance of diameter.IdirectFor the direct light intensity obtained in step 3.
The calculation formula of multiple weights of importance is as follows：
It is the probability that light source subpath and viewpoint subpath end-node do the fullpath obtained after attended operation
Density function：
It is that light source subpath and viewpoint subpath end-node do the general of the fullpath obtained after union operation
Rate density function：
Wherein, the probability density function of subpathCalculation formula is provided by step 2, for vertex
Probability density function at mergingCalculation formula is provided by step 5.
The beneficial effects of the invention are as follows：The present invention eliminates while keeping Photon Mapping method medium to draw efficient
Approximation in conventional photonic mapping method has bias term, keeps final drawing result accurate.And it solves applied to medium
Applicable sex chromosome mosaicism under inside and outside different scenes, and in contrast to conventional photonic mapping method for, method of the invention will not be by light
Son aggregation effect of radius stability.Specifically, in terms of being embodied in following four：
1, distance samples are carried out to the subpath in medium scene with Woodcook Tracking methods, then uses medium
Phase function importance sampling technology samples direction, forms a strip light path.
2, the optical characteristics of binding medium, by pair in the radiance calculation formula of complete optical path in unbiased Photon Mapping
Item is answered to carry out formula transformation.
3, by sampling the limitation carried out on angle and sampled distance to Bernoulli Jacob, to improve Bernoulli Jacob's sampling
Efficiency makes the cost of the probability calculation of unbiased reduce as far as possible.
4, a point situation discussion is done to the different integrated forms inside and outside dielectric boundaries, and rewrites multiple important under medium participates in
Property sample weight formula, completes the consistent unification of method, enables the method part suitable for whether there is or not medium and scene simultaneously
There are all kinds of situations such as participating media in region.
Description of the drawings
Fig. 1 be in the present invention about in medium scene Bernoulli trials and sampling angle, sampling direction of advance limitation
Schematic diagram, wherein the left side is to sample azimuthal limitation schematic diagram, and the right be the limitation schematic diagram of sampling forward travel distance；
Fig. 2 is the schematic diagram about the virtual boundary point in sample path in the present invention；Wherein, left side is to have participating media
Scene rendering as a result, right side is the schematic diagram of a scenario for having participating media；
Fig. 3 is the result schematic diagram of the method for drafting introduced in the present invention compared with other state-of-the-art method for drafting；
Wherein, (a) is two-way approach tracking method for drafting handling result figure, is (b) handling result that Photon Mapping handles method for drafting
Figure is (c) the handling result figure of method for drafting of the present invention, is (d) original image；
Fig. 4 is result of the method for drafting of the present invention compared with traditional Photon Mapping method is under different aggregation radiuses
Schematic diagram, wherein figure (a)-(d) be the present invention method under different aggregation radiuses handling result figure, figure (e)-(h) is traditional
Photon Mapping method under different aggregation radiuses handling result figure；And r=3.0 in (a) and (e), r=1.5 (b) and in (f),
(c) and (g) in r=0.5, (d) and (h) in r=0.2.
Specific implementation mode
Unbiased Photon Mapping method for drafting in participating media of the present invention, specific implementation steps are as follows：
Step 1：In the scene for having participating media, light source subpath is generated according to the light source in scene, according in scene
Viewpoint position generate viewpoint subpath, specific generation method is as follows：
1.1, it is starting point from light source or viewpoint position, photon is emitted into scene.
1.2,1.2, photon continues to advance to exit direction, when it encounters the object in scene, is defined according to object
Material and the incidence angle of photon sample to obtain the exit direction of photon, and photon moves in this direction.
The method of sample direction can refer to PHARR M, HUMPHREYS G.Physically based rendering:
From theory to implementation[M].[S.l.]:Morgan Kaufmann,2004。
1.3, when photon encounters the medium in scene, due to the scattering properties of medium, photon can be after a distance
It scatters, finds out photon and arrive generation scattering phenomenon when institute forward travel distance next time on the medium.
The method of sampled distance uses WOODCOCK E, MURPHY T, HEMMINGS P, et al.Techniques
used in the GEM code for Monte Carlo neutronics calculations in reactors and
other systems of complex geometry[C]//Proc.Conf.Applications of Computing
Methods to Reactor Problems:Vol 557.1965。
1.4, after the forward travel distance of photon in the medium has been calculated, it is also necessary to sample side when being scattered after photon advances
To the direction can be acquired according to the phase function property and incident direction of medium.
1.5, the whole string location information that the photon is sampled is recorded, as the sample path of the photon, the i.e. sub- road of light source
DiameterOr viewpoint subpathI=1,2 ... s, j=1,2 ... t；Wherein yi
Refer to i-th of subpath vertex, z on light source subpathjRefer to j-th of subpath vertex, y on viewpoint subpath1For light source position
It sets, z1For viewpoint position, ysFor the last term point on light source subpath, ztFor the last term point on viewpoint subpath, s is light source
Subpath vertex number on path, t are the subpath vertex number on viewpoint subpath.
Step 2：According to light source subpathWith viewpoint subpathMeter
Calculate the measurement contribution margin of simultaneously recording light source subpathThe probability density function of light source subpathViewpoint subpath
Measurement contribution marginThe probability density function of viewpoint subpath
The probability density function of light source subpathRefer to the conditional probability density letter on this string of vertex on the path
Several formed Joint Distributions.First, i-th of subpath vertex y is definediProbability density function：
And intermediate variable：
Wherein, p (y1) what is indicated is to up-sample y in light source1The probability of point, pσ(y1→y2) indicate from light source position y1Place refers to
To next sub- path vertices y2When y1Locate the probability density function in solid angle, pσ(yi-1→yi→yi+1) indicate that incidence angle is
yi-1→yiThe angle of emergence is yi→yi+1When yi-1Locate the probability density function in solid angle, θi→i+1Refer to subpath vertex yiOn
Surface normal and direction vectorBetween angle, subfix σ indicates that probability density function is estimated for solid angle.
According to subpath vertex yiProbability density function p (yi), the probability density function of light source subpath can be obtained
The measurement contribution margin of light source subpath
Wherein,For light source subpath vertex yiWith yi+1Between geometry item；Indicate light
Source subpath vertex yiWith yi+1Between visual function, 0 is taken when being blocked, otherwise take 1；fr(yi-1→yi→yi+1) indicate i-th
Bidirectional reflectance distribution function at a sub- path vertices is specifically defined and can refer to PHARR M, HUMPHREYS G.Physically
based rendering:From theory to implementation[M].[S.l.]:Morgan Kaufmann,2004。
Similarly：The probability density function of viewpoint subpath
The measurement contribution margin of viewpoint subpath
Wherein,For viewpoint subpath vertex zjWith zj+1Between geometry item；Expression regards
Idea path vertices zjWith zj+1Between visual function；fs(zj-1→zj→zj+1) indicate the two-way of j-th subpath apex
Reflectance Distribution Function.
Step 3：All light source subpath is tracked in sampling, by they directly with viewpoint position z1It is connected, calculates viewpoint position
Set z1The direct illumination at place：
Wherein, Ω is the sample space of light source subpath,For the probability-distribution function of light source subpath, N is light source
The number of subpath, k are the ordinal number of light source subpath.
Simultaneously to y on all light source subpaths1y2…ysSubpath last term point ys, established using balance search tree method
Space querying accelerating structure.
Step 4：(vertex attended operation) is by all obtained light source subpath y1y2…ysWith viewpoint subpath z1z2…zt
Attended operation is done, i.e., light source subpath last term point ysWith viewpoint subpath last term point ztIt is connected, calculates connection path y1y2…
yszt…z2z1Brightness, be added to viewpoint position z1On corresponding pixel, the connection path
Radiance：
Wherein, subscript VC indicates vertex attended operation (Vertex Connection),It is the exhibition of light source subpath
Open item：
It is the expansion item of viewpoint subpath：
For light source subpath last term point ysWith viewpoint subpath last term point ztThe measurement contribution margin of junction：
fr(ys-1→ys→zt) refer to incidence angle as ys-1→ysThe angle of emergence is ys→ztShi Guangyuan subpaths vertex ysPair at place
To Reflectance Distribution Function, fr(zt-1→zt→ys) refer to incidence angle as zt-1→ztThe angle of emergence is zt→ysWhen viewpoint subpath vertex zt
The bidirectional reflectance distribution function at place,Refer to light source subpath last term point ysWith viewpoint subpath last term point ztBetween
Geometry item.
In the scene of participating media, Wo Menyou：
Wherein,WithRespectively by xi-1It is directed toward xiIncident direction with by xiIt is directed toward xi+1Exit direction,
Have when there was only isotropic medium in scene for the phase function of participating media
For the visual function in step 2 geometry item formula,For from current vertex xiLocate to next vertex xi+1Between
Medium Optics thickness.In addition and the form of optical thickness is indicated with symbol tau：
Wherein σtFor the extinction coefficient of medium here, d is vertex xiTo xi+1The distance between,For from xiIt is directed toward xi+1's
Unit vector, t are from vertex xiTo xi+1The distance of advance.
Step 5：(the vertex union operation of unbiased) we sample obtained viewpoint subpath z to every herein1z2…ztOn
Subpath last term point zt, searched with the balance search tree in step 3 and ztDistance is less than all light source subpath last terms of r
Point ys, according to ysCorresponding light source subpath y can be obtained1y2…ys, then by viewpoint subpath z1z2…ztWith light source subpath
y1y2…ysMerged with subpath last term point, calculates and merge path y1y2…yszt-1…z2z1Brightness, be added to viewpoint position z1
On corresponding pixel, the merging path y1y2…yszt-1…z2z1Radiance：
Wherein, subscript VM indicates vertex attended operation (Vertex Merging), is the expansion item of light source subpath：
It is the expansion item of viewpoint subpath：
It is light source subpath last term point ysWith viewpoint subpath last term point ztIntermediate variable at merging：
Wherein,It is light source subpath last term point ysWith viewpoint subpath last term point ztMeasurement contribution at merging
Value,It is light source subpath last term point ysWith viewpoint subpath last term point ztProbability density function at merging.To ask
SolutionWe need to calculate separatelyWithWithout bias：
5.1、Unbiased in item medium calculates
In traditional Photon Mapping method,Item is not the light source subpath last term point y after mergingsWith the sub- road of viewpoint
Diameter vertex zt-1It is connected, but approximate evaluation in the following way：
Wherein ys(zt) refer to using vertex ysCover vertex zt.And fr(ys-1→ys(zt)→zt-1) refer to incidence angle be ys-1
→ysThe angle of emergence is zt→zt-1Shi Dingdian ysBidirectional reflectance distribution function, it means that in ysOn BSDF analyses be using close
As zt→zt-1To substitute ys→zt-1.And subsequent geometry itemWith bidirectional reflectance distribution function fr(zt-2→
zt-1→zt) calculated for convenience all using approximate item, calculation formula is provided by step 2.And in fact, we use strictly
The formula form of unbiased：
F hereins(ys-1→ys→zt-1) refer to incidence angle be ys-1→ysThe angle of emergence is ys→zt-1When subpath vertex ys's
Bidirectional reflectance distribution function, similarly fs(zt-2→zt-1→ys) refer to incidence angle be zt-2→zt-1The angle of emergence is zt-1→ysShi Zilu
Diameter vertex zt-1Bidirectional reflectance distribution function.Then indicate subpath vertex ysWith zt-1Between geometry item, specifically
Formula provided in step 2.
Therefore in our method, the sub- mapping method of polarisation becomes unbiased traditional, it is necessary first in use
Unbiased formula is stated to replace original approximate formula.
5.2、Unbiased in item medium calculates
In traditional Photon Mapping method,Item is block to be used to filter using traditional photon density estimation mode
Device function：
Wherein px(zt-2→zt-1→zt) it is by the first two vertex zt-2,zt-1Sampling obtains vertex ztProbability density, r is
The radius of photon collection.And stringent definition should calculate ztAll photon probability density value p in collect range r neighborhoodsx's
With it is as follows using the form of integral：
Wherein r is the radius of photon collection, S (ys, r) and it is exactly with ysCentered on, radius is the Neighbourhood set of r, px(zt-2→
zt-1→ z) it is by the first two subpath vertex zt-2,zt-1Sampling obtains the probability density of position z.
We can not direct solutionValue, but we can be reciprocal with unbiased esti-mator probability integral
Value：
For from current view point subpath vertex zt-1Constantly to light source subpath last term point ysTransmitting sampling light
(the same step 1) of sampling process in the direction and forward travel distance of sampling light, light hit sphere neighborhood S (y for the first times, r) after in total
The number for the test light got, E [] indicate it is expected to calculate.We term it Bernoulli trials for above-mentioned sampling experiment.
And an apparent problem of the solution be exactly this unknown Bernoulli trials number may be very big.Due to
It needs to be sampled out in entire scene space with vertex ys'Centered on r be radius neighborhood probability, it is evident that such probability is non-
Often small, therefore Bernoulli trials number will be very big, then the calculating cost of unbiased esti-mator very big (such as can may pass through herein
100 samplings could terminate, and primary photon mapping method then only needs 1 approximate calculation).
Then in the scene of not medium, the range of sampling can be reduced by azimuthai boundary, to reduce primary
Nu Li tests total degree, as shown in fig. 1 on the left-hand side.And in the scene for having medium, since photon samples in dielectric space (not
Only on a surface of an), Bernoulli trials total degree still can not effectively only be reduced by azimuth limitation.Therefore we carry
Go out and limited jointly by azimuthai boundary and forward travel distance boundary, the test light that experiment is got is made to sample the institute on the right side of Fig. 1
The area of space shown.
We are defined forward travel distance using the theory deduction in Woodcock Tracking simultaneously.For given
Equally distributed random number r ∈ [0,1) according to sampled probability density functionForward travel distance, which can be obtained, is：
Wherein σtFor the extinction coefficient of medium.Later we can write out apart from boundary R and corresponding random number range
rboundBetween relationship：
Therefore we generate random number r progress scope limitations using by initial distance, make
The forward travel distance of test light can be allowed to be limited in range [R1,R2] in, to be effectively improved our method in medium
To the unbiased esti-mator efficiency of probability integral inverse in scene.
And in our realization, R1With R2It is to add or subtract by the distance between initial vertex and representative points
Aggregation radius r is gone to be acquired to calculate.In addition to this, it would be desirable to which surface vertices or medium top are distinguished to two connection vertex
The a variety of situation discussion of point：
1) it when starting point vertex is surface vertices, needs to carry out azimuth about using the bidirectional reflectance distribution function in material
Otherwise beam uses the phase function on medium；
2) when representative points are surface vertices, as long as then ensureing apart from lower limit R1, upper limit R is more than for sampled distance2Institute
There is something special all directly beats on the surface, therefore our needs will be apart from upper limit R2Remove, corresponding is exactly to allow the upper limit of random number rEqual to 1.
Then after we take importance sampling to accelerate, the form of calculation of probability density function inverse is as follows：
Wherein Nb'Indicate time that Bernoulli trials is tested in total after first time success after taking importance sampling to accelerate
Number.It indicates to limit the product of probability score value in distance range in our importance samplings.σt(z) it indicates at the z of position
Medium extinction coefficient.Ωb'It indicates to limit the locational space in distance range.Due to before we to being limited on its sampled distance
System, thenThe analytical form of probability item as after limited samples distance.Then have in uniform dielectric：
Therefore, there are the limitation on above-mentioned azimuth and forward travel distance, the expectation number of final Bernoulli trials big
Width reduces, and greatly increases probability integral inverseUnbiased esti-mator efficiency.
Step 6：There is the light intensity for all vertex connection paths that step 4 obtains in corresponding pixel pointsWith step 5
Obtain the light intensity that all vertex in corresponding pixel points merge pathAfterwards, it would be desirable to be adopted using multiple importance
Sample (Multiple Importance Sampling, referred to as MIS) technology, to obtain the light after the weighting of corresponding pixel points
Strong IVCM, so as to complete the drafting flow of entire unbiased Photon Mapping：
Wherein subscript VC indicates that vertex attended operation (Vertex Connection), subscript VM indicate vertex attended operation
(Vertex Merging), subscript VCM indicate vertex connection and merge joint operation (Vertex Connection and
Merging),nVCFor the fullpath number that all vertex attended operations on the pixel obtain, nVMFor on the pixel
The fullpath number that all vertex union operations obtain, subscript k are the ordinal number in the path, and subscript s is to contain in the path
The number on light source subpath vertex, subscript s are the number on the viewpoint subpath vertex contained in the path；Refer to
The light intensity for the fullpath that k vertex attended operation obtains,For the corresponding multiple weights of importance in the path；Refer to the light intensity for the fullpath that kth vertex union operation obtains,It is corresponding more for the path
Weight weights of importance；IdirectFor the direct light intensity obtained in step 3.
The calculation formula of multiple weights of importance is as follows：
It is the probability that light source subpath and viewpoint subpath end-node do the fullpath obtained after attended operation
Density function：
It is that light source subpath and viewpoint subpath end-node do the general of the fullpath obtained after union operation
Rate density function：
Wherein, the probability density function of subpathCalculation formula is provided by step 2, for vertex
Probability density function at mergingCalculation formula is provided by step 5.
Significantly, since the addition of medium, the calculation formula of multiple importance sampling weight can do certain tune
It is whole.Consistency is maintained in order to make to pass through the corresponding formula of fullpath inside and outside medium, you can estimate with direct use simultaneously
Contribution rate and probability density function, which are calculated, no longer needs to consider that ray is the different form in medium or outside medium
Expression, we add virtual boundary point and are marked.Then when calculating multiple importance sampling weight weight, it should be noted that
Vertex connects and union operation weight ratio ηVCMVariation.Understand the η on surfaceVCMForm is as follows：
And in participating media in the case of for, we use following form：
Wherein nVCIndicate the light path sum of vertex attended operation, nVMIndicate that the light path sum of vertex union operation, r are top
The combined radius of neighbourhood of point.Then we can realize vertex connection in medium scene and merge multiple under two kinds of strategies
Importance sampling scheme.
Embodiment
Inventor is equipped with the double-core central processing unit of Intel i5 3.00GHz and the computer of 16GB memories at one
On realize method as described above, and realize other method for drafting being closely related with our methods, generate in figure
Rendering result.
Experimental result (Fig. 3) shows to be substantially better than traditional pair in medium rendering by our improved integrated processes
To path tracing and Photon Mapping method.In contrast to two-way approach method for tracing, the convergence of our methods is with obvious effects to get well.Phase
Than in conventional photonic mapping method, our methods have apparent while keeping method is efficient in the unbiasedness of result
Advantage.Fig. 4 considers from photon aggregation radius factor later, and rendering performance of our method in medium drafting is relative to tradition
Photon Mapping method for, aggregation radius may be significantly smaller the influence of our method performance.
Claims (1)
1. the unbiased Photon Mapping method for drafting in a kind of participating media, which is characterized in that include the following steps：
Step 1：In the scene for having participating media, light source subpath is generated according to the light source in scene, according to regarding in scene
Point position generates viewpoint subpath, and specific generation method is as follows：
1.1, it is starting point from light source or viewpoint position, photon is emitted into scene；
1.2, photon continues to advance to exit direction, when it encounters the object in scene, the material that is defined according to object and
The incidence angle of photon samples to obtain the exit direction of photon, and photon moves in this direction；
1.3, when photon encounters the medium in scene, due to the scattering properties of medium, photon can occur after a distance
Scattering finds out photon and arrives generation scattering phenomenon when institute forward travel distance next time on the medium；
1.4, after the forward travel distance of photon in the medium has been calculated, it is also necessary to direction when being scattered after photon advances is sampled,
The direction can be acquired according to the phase function property and incident direction of medium；
1.5, the whole string location information that the photon is sampled is recorded, as the sample path of the photon, i.e. light source subpathOr viewpoint subpathWherein yiIt is
Refer to i-th of subpath vertex on light source subpath, zjRefer to j-th of subpath vertex, y on viewpoint subpath1For light source position,
z1For viewpoint position, ysFor the last term point on light source subpath, ztFor the last term point on viewpoint subpath, s is the light source subpath
On subpath vertex number, t be viewpoint subpath on subpath vertex number；
Step 2：According to light source subpathWith viewpoint subpathIt calculates simultaneously
The measurement contribution margin of recording light source subpathThe probability density function of light source subpathThe survey of viewpoint subpath
Measure contribution marginThe probability density function of viewpoint subpath
The probability density function of light source subpathRefer to the conditional probability density function institute on this string of vertex on the path
The Joint Distribution of composition；First, i-th of subpath vertex y is definediProbability density function：
And intermediate variable：
Wherein, p (y1) what is indicated is to up-sample y in light source1The probability of point, pσ(y0→y1) indicate from light source position y0Under place is directed toward
One sub- path vertices y1When y1Locate the probability density function in solid angle, pσ(yi-1→yi→yi+1) expression incidence angle be yi-1→
yiThe angle of emergence is yi→yi+1When yiLocate the probability density function in solid angle, θi→i+1Refer to subpath vertex yiOn surface normal
Amount and direction vectorBetween angle, subfix σ indicates that probability density function is estimated for solid angle；
According to subpath vertex yiProbability density function p (yi), the probability density function of light source subpath can be obtained
The measurement contribution margin of light source subpath
Wherein,For light source subpath vertex yiWith yi+1Between geometry item；Indicate the sub- road of light source
Diameter vertex yiWith yi+1Between visual function, 0 is taken when being blocked, otherwise take 1；fr(yi-1→yi→yi+1) indicate i-th of sub- road
The bidirectional reflectance distribution function of diameter apex；
Similarly：The probability density function of viewpoint subpath
The measurement contribution margin of viewpoint subpath
Wherein,For viewpoint subpath vertex zjWith zj+1Between geometry item；Indicate the sub- road of viewpoint
Diameter vertex zjWith zj+1Between visual function；fs(zj-1→zj→zj+1) indicate that the bidirectional reflectance of j-th of subpath apex divides
Cloth function；
Step 3：All light source subpath is tracked in sampling, by they directly with viewpoint position z1It is connected, calculates viewpoint position z1Place
Direct illumination：
Wherein, Ω is the sample space of light source subpath,For the probability-distribution function of light source subpath, N is the sub- road of light source
The number of diameter, k are the ordinal number of light source subpath；
Simultaneously to y on all light source subpaths1y2…ysSubpath last term point ys, space is established using balance search tree method and is looked into
Ask accelerating structure；
Step 4：By all obtained light source subpath y1y2…ysWith viewpoint subpath z1z2…ztAttended operation is done, i.e., light source
Subpath last term point ysWith viewpoint subpath last term point ztIt is connected, calculates connection path y1y2…yszt…z2z1Brightness, add up
To viewpoint position z1On corresponding pixel, the connection pathRadiance：
Wherein, subscript VC indicates vertex attended operation,It is the expansion item of light source subpath：
It is the expansion item of viewpoint subpath：
For light source subpath last term point ysWith viewpoint subpath last term point ztThe measurement contribution margin of junction：
fr(ys-1→ys→zt) refer to incidence angle as ys-1→ysThe angle of emergence is ys→ztShi Guangyuan subpaths vertex ysThe bidirectional reflectance at place
Distribution function, fr(zt-1→zt→ys) refer to incidence angle as zt-1→ztThe angle of emergence is zt→ysWhen viewpoint subpath vertex ztPair at place
To Reflectance Distribution Function,Refer to light source subpath last term point ysWith viewpoint subpath last term point ztBetween geometry item；
Step 5：Obtained viewpoint subpath z is sampled to every1z2…ztOn subpath last term point zt, with the balance in step 3
Search tree is searched and ztDistance is less than all light source subpath last term point y of rs, according to ysCorresponding light source subpath can be obtained
y1y2…ys, then by viewpoint subpath z1z2…ztWith light source subpath y1y2…ysMerged with subpath last term point, calculates and merge
PathBrightness, be added to viewpoint position z1On corresponding pixel, the merging path
y1y2…yszt-1…z2z1Radiance：
Wherein, subscript VM indicates vertex union operation,It is the expansion item of light source subpath：
It is the expansion item of viewpoint subpath：
It is light source subpath last term point ysWith viewpoint subpath last term point ztIntermediate variable at merging：
Wherein,It is light source subpath last term point ysWith viewpoint subpath last term point ztMeasurement contribution margin at merging,It is light source subpath last term point ysWith viewpoint subpath last term point ztProbability density function at merging, calculation formula
It is as follows：
F hereins(ys-1→ys→zt-1) refer to incidence angle be ys-1→ysThe angle of emergence is ys→zt-1When subpath vertex ysIt is two-way
Reflectance Distribution Function, similarly fs(zt-2→zt-1→ys) refer to incidence angle be zt-2→zt-1The angle of emergence is zt-1→ysWhen subpath top
Point zt-1Bidirectional reflectance distribution function；Then indicate subpath vertex ysWith zt-1Between geometry item；
Wherein, r is the radius of photon collection, S (ys, r) and it is exactly with ysCentered on, radius is the Neighbourhood set of r, px(zt-2→zt-1
→ z) it is by the first two subpath vertex zt-2,zt-1Sampling obtains the probability density of position z：
For from current view point subpath vertex zt-1Constantly to light source subpath last term point ysTransmitting sampling light, light
Sphere neighborhood S (y are hit for the first times, r) after get in total test light number, E [] indicate it is expected calculate；
Step 6：The light intensity of all vertex connection paths in corresponding pixel points is obtained according to step 4It is obtained with step 5
All vertex in corresponding pixel points merge the light intensity in pathUsing multiple importance sampling technology, corresponded to
Light intensity I after the weighting of pixelVCM, so as to complete the drafting of entire unbiased Photon Mapping：
Wherein, subscript VC indicates vertex attended operation, and subscript VM indicates vertex union operation, subscript VCM indicate vertex connection and
Merge joint operation, nVCFor the fullpath number that all vertex attended operations on the pixel obtain, nVMFor the pixel
On the obtained fullpath number of all vertex union operations, subscript k is the ordinal number in the path, and subscript s is to contain in the path
The number on some light source subpaths vertex, subscript s are the number on the viewpoint subpath vertex contained in the path；It is
Refer to the light intensity for the fullpath that kth vertex attended operation obtains,For the corresponding multiple weights of importance in the path；Refer to the light intensity for the fullpath that kth vertex union operation obtains,It is corresponding more for the path
Weight weights of importance, IdirectFor the direct light intensity obtained in step 3；
The calculation formula of multiple weights of importance is as follows：
It is the probability density that light source subpath and viewpoint subpath end-node do the fullpath obtained after attended operation
Function：
Be light source subpath and viewpoint subpath end-node do the fullpath obtained after union operation probability it is close
Spend function：
Wherein, the probability density function of subpathCalculation formula is provided by step 2, and vertex is merged
The probability density function at placeCalculation formula is provided by step 5.
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