CN102496171A - Hierarchical replacement method for (t, s) sequences - Google Patents

Abstract

The invention provides a computer image rendering method. According to the method, the position of sampling points is determined by hierarchical replacement operation based on (t, s) sequences. Compared with a Latin hypercube sampling method, the computer image rendering method has the advantages that: the consumption of internal memory during running of a program can be remarkably reduced; the quantity of the sampling points is not required to be predetermined; and the application range is remarkably expanded.

Description

A kind of being used for (t, s) the hierarchical method of replacing of sequence

Technical field

The present invention relates to a kind of computer picture rendering intent, particularly a kind of rendering intent that utilizes quasi-random numbers (quasi-random numbers) to confirm sampling point position.

Background technology

Play up the field at computer picture, for simulating various effect of shadow, for example, the depth of field, motion blur, global illumination etc., need carry out evaluation to the integration with following form:

$\underset{I}{\∫}...\underset{I}{\∫}f(x^{\left(1\right)},x^{\left(2\right)},...,x^{\left(S\right)}){\mathrm{dx}}^{\left(1\right)}{\mathrm{dx}}^{\left(2\right)}...{\mathrm{dx}}^{\left(S\right)}---\left(1\right)$

Wherein I is the unit interval, function f () by the contextual data decision of the effect that will simulate and input.Formula (1) generally can't accurately be found the solution, and in the render process of reality, always is to use the method for numerical evaluation to ask its approximate value:

$\underset{I}{\∫}...\underset{I}{\∫}f(x^{\left(1\right)},x^{\left(2\right)},...,x^{\left(S\right)}){\mathrm{dx}}^{\left(1\right)}{\mathrm{dx}}^{\left(2\right)}...{\mathrm{dx}}^{\left(S\right)}\≈\frac{1}{N}\underset{i=0}{\overset{N-1}{\mathrm{\Σ}}}f(x_i^{\left(1\right)},x_i^{\left(2\right)},...,x_i^{\left(S\right)})---\left(2\right)$

The method of confirming sampling point position

in the formula (2) can be divided into two types; One type of method that is to use random number, the another kind of method that is to use quasi-random numbers.Quasi-random numbers is a kind of distribution pattern with satisfactory texture, and the efficient higher than random number can be provided in the Integral Estimation of formula (2).

(t, s) sequence is a kind of main quasi-random numbers type, generation b system (t, s) method of the j of i some dimension component is following in the sequence:

$x_i^{\left(j\right)}=(b^{-1},b^{-2},...b^{-M})C^{\left(j\right)}\left(\begin{array}{c}{b}_0\left(i\right)\\ b_1\left(i\right)\\ \·\\ \·\\ \·\\ b_{M-1}\left(i\right)\end{array}\right)---\left(3\right)$

Wherein For (t, s) j of sequence supports one's family into matrix (generator matrices), its coefficient

Be Galois field F _bMiddle element.b _r(i) the r bit digital of expression integer i under the b system is represented, promptly

M is selected maximal accuracy under the b system is represented.

(t, s) example of sequence comprises: Sobol sequence, Faure sequence, Niederreiter sequence and Niederreiter-Xing sequence etc.It is different that (t, s) difference of sequence is that the building method of its generator matrix is different.

Although quasi-random numbers can provide the efficient higher than random number, along with the increase of dimension, the quality of quasi-random numbers can descend gradually, and random number can not.For fully utilizing quasi-random numbers and random number advantage separately; Art B.Owen; " Latin Supercube Sampling for Very High-Dimensional Simulations ", ACM Transactions on Modeling and Computer Simulation, vol.8; No.1; Jan.1998, pp.71-102. provide the method for the super cube sampling of a kind of Latin (Latin Supercube Sampling), the method is characterized in that through quasi-random numbers being carried out the higher-dimension component that random permutation generates sampled point.

Because what adopt is common replacement operator, realizes that the computer program of the super cube sampling of Latin need consume and the proportional memory size of the quantity of sampled point when operation, thereby be unwell to the task of playing up of a large amount of sampled points of needs.On the other hand, this common replacement operator needs to confirm in advance the quantity of sampled point, thereby can't be used for asymptotic expression and play up (progressive rendering).

Summary of the invention

The invention provides a kind of computer picture rendering intent, this method is based on (t, s) sequence are confirmed the position of sampled point through the hierarchical replacement operator.

Realize the scheme of technical purpose of the present invention, may further comprise the steps:

The integration that needs evaluation during (A) for rendering image

$\underset{I}{\∫}...\underset{I}{\∫}f(x^{\left(1\right)},x^{\left(2\right)},...,x^{\left(S\right)}){\mathrm{dx}}^{\left(1\right)}{\mathrm{dx}}^{\left(2\right)}...{\mathrm{dx}}^{\left(S\right)}$

Use the method for numerical evaluation to ask its approximate value

$\underset{I}{\∫}...\underset{I}{\∫}f(x^{\left(1\right)},x^{\left(2\right)},...,x^{\left(S\right)}){\mathrm{dx}}^{\left(1\right)}{\mathrm{dx}}^{\left(2\right)}...{\mathrm{dx}}^{\left(S\right)}\≈\frac{1}{N}\underset{i=0}{\overset{N-1}{\mathrm{\Σ}}}f(x_i^{\left(1\right)},x_i^{\left(2\right)},...,x_i^{\left(S\right)});$

(B) confirm the method for the value of variable described in the step (A)

; 0≤i≤N-1 wherein may further comprise the steps:

Will $(x_i^{\left(1\right)},x_i^{\left(2\right)},...,x_i^{\left(S\right)})$ Be grouped into $(x_i^{(p_0+1)},...x_i^{\left(p_1\right)},x_i^{(p_1+1)},...,x_i^{\left(p_2\right)},...,x_i^{(p_q+1)},...,x_i^{\left(p_{q+1}\right)}),$ P wherein ₀=0, p _Q+1=s, 0＜p _V+1-p _v≤s _Max, 0≤v≤q, s _MaxBe selected largest;

confirms that the method for its value may further comprise the steps for set of variables:

$x_i^{\left(j\right)}=h_j\left(i\right),$ 1≤j≤p ₁；

H wherein _j(i) :=(b ^-1, b ^-2..., b ^-M) C ^(j)(b ₀(i), b ₁(i) ..., b _M-1(i)) ^T,

Be that (t, s) j of sequence supports one's family into matrix to a b system, b _r(i) the r bit digital of expression integer i under the b system is represented, promptly

M is selected maximal accuracy under the b system is represented;

For set of variables 1≤v≤q wherein, confirm that the method for its value may further comprise the steps:

$x_i^{(p_v+\mathrm{\Δ})}=h_{\mathrm{\Δ}}\left(i^{\left(v\right)}\right),$ 1≤v≤q，1≤Δ≤p _v+1-p _v；

i(v)＝π _v(i)，1≤v≤q；

π wherein _v() is the hierarchical replacement operator, to an integer

Wherein

$b_{M-1}^{\left(v\right)}\left(x\right)=(b_{M-1}\left(x\right)+{\mathrm{\ξ}}_v),\mathrm{mod}b$

$b_{M-2}^{\left(v\right)}\left(x\right)=(b_{M-2}\left(x\right)+{\mathrm{\ξ}}_{v,b_{M-1}\left(x\right)}),\mathrm{mod}b$

$b_{M-3}^{\left(v\right)}\left(x\right)=(b_{M-3}\left(x\right)+{\mathrm{\ξ}}_{v,b_{M-1}\left(x\right),b_{M-2}\left(x\right)}),\mathrm{mod}b$

$b_0^{\left(v\right)}\left(x\right)\stackrel{\stackrel{\stackrel{\·}{\·}}{\·}}{=}(b_0\left(x\right)+{\mathrm{\ξ}}_{v,b_{M-1}\left(x\right),b_{M-2}\left(x\right),...,b_1\left(x\right)}),\mathrm{mod}b$

ξ wherein _v, ξ _{V, 0}, ξ _{V, 1}..., ξ _{V, b-1}, ξ _{V, 0,0}..., ξ _{V, b-1, b-1}..., ξ _{V, b-1}..., b-1,1≤v≤q is separate random number.

The present invention has the following advantages:

(t, s) sequence are confirmed the position of sampled point to compare the memory consumption when the present invention can significantly reduce program run with the super cube sampling of Latin through the hierarchical replacement operator owing to the present invention is based on.

(t, s) sequence are confirmed the position of sampled point through the hierarchical replacement operator, need not confirm the quantity of sampled point in advance, sample with the super cube of Latin and compare, and the scope of application of the present invention significantly strengthens owing to the present invention is based on.

Embodiment

Below in conjunction with embodiment the present invention is elaborated.

Table 1 has been listed the C++ example code that calculates 2 system Niederreiter sequences:

Table 1

In code shown in the table 1, maximal accuracy is made as 32.The reason of setting a maximal accuracy is, in numerical evaluation, and both can not the also unnecessary calculating of carrying out infinite precision.

In code shown in the table 1, largest is made as 12 dimensions.The effect of setting largest is, on the one hand, calculate (t, s) data of the generator matrix that need use of sequence need be stored in the computer-readable medium in advance, and any equipment in the reality all can not the unlimited many data of storage; On the other hand, though the quality of quasi-random numbers is superior to random number, along with the increase of dimension, the quality of quasi-random numbers can descend gradually, thus in reality the infinite dimensional quasi-random numbers of also unnecessary use.

(choosing of largest will decide according to the maximal accuracy of its mass parameter (qualty parameters) t and setting for t, s) sequence, and its principle is that the mass parameter of selected largest should not surpass the maximal accuracy of setting to different.

Table 2 has been listed 2 system Sobol sequences, Niederreiter sequence and the mass parameter of Niederreiter-Xing sequence in different dimensions.The little expression quality of this more parameters is high more, and minimum is 0.

Table 2

Dimension	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	20
																	Sobol	0	0	1	3	5	8	11	15	19	23	27	31	35	40	45	71
Niederreiter	0	0	1	3	5	8	11	14	18	22	26	30	34	38	43	68
																	Niederreiter-Xing	0	0	1	1	2	3	4	5	6	8	9	10	11	13	15	21

Can find out from last table; For 2 system Sobol sequences and Niederreiter sequence, if maximal accuracy is made as 32, then largest should be elected 12 dimensions as; Because the Sobol sequence of 12 dimensions and the mass parameter of Niederreiter sequence are respectively 31 and 30; Do not surpass 32, and the Sobol sequence of 13 dimensions and the mass parameter of Niederreiter sequence are respectively 35 and 34, have surpassed 32.

Use (t, s) sequence need be set a largest, when the dimension of the required sampled point of rendering image surpasses this setting value; Need be to (t; S) sequence number of sequence is carried out a hierarchical replacement operator, and the sequence number after the displacement can be used for, and former (t, s) sequence generates new component.Table 3 has been listed the C++ example code of realizing this hierarchical replacement operator:

Table 3

Can find out from code shown in the table 3; Different with the common replacement operator that the super cube sampling of Latin is adopted; Realize that its memory consumption of computer program of hierarchical replacement operator and the quantity of sampled point have nothing to do; Compare with the super cube sampling of Latin, the memory consumption during program run significantly reduces.

Table 4 has been listed based on 2 system Niederreiter sequences, confirms the C++ example code of sampling point position through the hierarchical replacement operator, has wherein used at code shown in table 1 and the table 3:

Table 4

Table 5 has been listed the C++ example code of code shown in the use table 4 in render process:

Table 5

Can find out from code shown in table 4 and the table 5; Different with the common replacement operator that the super cube sampling of Latin is adopted; The hierarchical replacement operator need not be confirmed the quantity of sampled point in advance; The position that only need provide corresponding sequence number can generate required sampled point, this point for gradual play up particularly important.In gradual playing up, renderer feeds back current rendering result in real time and gives the user, when stops playing up task by user's decision, thereby can't confirm the quantity of sampled point in advance, can only rely on the sequence number of sampled point to generate the position of sampled point.The hierarchical replacement operator can satisfy gradual this requirement of playing up fully.

In code shown in the table 4, to each divide into groups all to have used identical (t, s) sequence of all dimensions of sampled point component; I.e. 2 system Niederreiter sequences; Remove to use identical (t, s) sequence also can be used radix identical but (t, s) sequence that type is different outward; For example, 2 system Niederreiter-Xing sequences.Though the quality of Niederreiter-Xing sequence is superior to Sobol sequence and Niederreiter sequence; But there is a restriction in the use; Promptly must confirm total sampling dimension in advance; Thereby suitable play up as image in a kind of blanket method, if but the dimension of current group is known and do not have the more sampling of higher-dimension, then available Niederreiter-Xing sequence replacement Niederreiter sequence is to obtain better rendering effect.

It is a kind of that (t, s) expansion of sequence is referred to as randomization (t, s) (randomized (t, s) sequences) its role is to estimate the error size of integral approach value to sequence.Though needn't go to estimate the error size of integral approach value during rendering image; But this method of randomization is used in the flaw of alleviating in the playing up of animation in the image; The pictures different frame uses different random numbers that sampled point is carried out randomization in promptly animation being played up, to desalinate the flaw pattern that possibly exist.

Table 6 has been listed the C++ example code that on the basis of code shown in the table 4, has increased the randomization step:

Table 6

Claims (4)

1. a computer picture rendering intent is characterized by, and may further comprise the steps:

The integration that needs evaluation during (A) for rendering image

$\underset{I}{\∫}...\underset{I}{\∫}f(x^{\left(1\right)},x^{\left(2\right)},...,x^{\left(S\right)}){\mathrm{dx}}^{\left(1\right)}{\mathrm{dx}}^{\left(2\right)}...{\mathrm{dx}}^{\left(S\right)}$

Use the method for numerical evaluation to ask its approximate value

$\underset{I}{\∫}...\underset{I}{\∫}f(x^{\left(1\right)},x^{\left(2\right)},...,x^{\left(S\right)}){\mathrm{dx}}^{\left(1\right)}{\mathrm{dx}}^{\left(2\right)}...{\mathrm{dx}}^{\left(S\right)}\≈\frac{1}{N}\underset{i=0}{\overset{N-1}{\mathrm{\Σ}}}f(x_i^{\left(1\right)},x_i^{\left(2\right)},...,x_i^{\left(S\right)});$

(B) confirm the method for the value of variable described in the step (A) ; 0≤i≤N-1 wherein may further comprise the steps:

Will $(x_i^{\left(1\right)},x_i^{\left(2\right)},...,x_i^{\left(S\right)})$ Be grouped into $(x_i^{(p_0+1)},...x_i^{\left(p_1\right)},x_i^{(p_1+1)},...,x_i^{\left(p_2\right)},...,x_i^{(p_q+1)},...,x_i^{\left(p_{q+1}\right)}),$ P wherein ₀=0, p _Q+1=s, 0＜p _V+1-p _v≤smax, 0≤v≤q, smax are selected largest;

confirms that the method for its value may further comprise the steps for set of variables:

$x_i^{\left(j\right)}=h_j\left(i\right),$ 1≤j≤p ₁；

H wherein _j(i) :=(b ^-1, b ^-2..., b ^-M) C ^(j)(b ₀(i), b ₁(i) ..., b _M-1(i)) ^T,

Be that (t, s) j of sequence supports one's family into matrix to a b system, b _r(i) the r bit digital of expression integer i under the b system is represented, promptly

M is selected maximal accuracy under the b system is represented;

For set of variables

1≤v≤q wherein, confirm that the method for its value may further comprise the steps:

$x_i^{(p_v+\mathrm{\Δ})}=h_{\mathrm{\Δ}}\left(i^{\left(v\right)}\right),$ 1≤v≤q，1≤Δ≤p _v+1-p _v；

i(v)＝πv(i)，1≤v≤q；

π wherein _v() is the hierarchical replacement operator, to an integer

Wherein

$b_{M-1}^{\left(v\right)}\left(x\right)=(b_{M-1}\left(x\right)+{\mathrm{\ξ}}_v),\mathrm{mod}b$

$b_{M-2}^{\left(v\right)}\left(x\right)=(b_{M-2}\left(x\right)+{\mathrm{\ξ}}_{v,b_{M-1}\left(x\right)}),\mathrm{mod}b$

$b_{M-3}^{\left(v\right)}\left(x\right)=(b_{M-3}\left(x\right)+{\mathrm{\ξ}}_{v,b_{M-1}\left(x\right),b_{M-2}\left(x\right)}),\mathrm{mod}b$

$b_0^{\left(v\right)}\left(x\right)\stackrel{\stackrel{\stackrel{\·}{\·}}{\·}}{=}(b_0\left(x\right)+{\mathrm{\ξ}}_{v,b_{M-1}\left(x\right),b_{M-2}\left(x\right),...,b_1\left(x\right)}),\mathrm{mod}b$

ξ wherein _v, ξ _{V, 0}, ξ _{V, 1}..., ξ _{V, b-1}, ξ _{V, 0,0}..., ξ _{V, b-1, b-1}..., ξ _{V, b-1 ..., b-1}, 1≤v≤q is separate random number.

2. like the said method of claim 1., also comprise, each of all dimensions of sampled point component divided into groups to use identical (t, s) identical but (t, s) sequence that type is different of sequence or radix.

3. like the said method of claim 1., wherein, said (t, s) sequence comprises Sobol sequence, Faure sequence, Niederreiter sequence and Niederreiter-Xing sequence.

4. like the said method of claim 1., wherein, said (t, s) sequence also comprises randomized (t, s) sequence.