1. A computer program product for use with a computer graphics system, the computer graphics system comprising a computer and a display module, the computer being operable to generate a display-controlling electrical output based on numerical evaluation of an integral of a function over an s-dimensional integration domain, the display-controlling electrical output being transmittable by the computer to the display module to cause the display module to display a human-perceptible image responsive to the numerical evaluation of the integral, the computer program product comprising a computer readable physical medium having recorded thereon:
A. first computer-executable instructions, recorded on the computer-readable physical medium and executable by the computer to enable the computer to generate a selected number of sample points over the integration domain, the sample points being generated such that there is at least one sample point in each of a plurality of strata distributed over the integration domain, the strata being defined by a rank-1 lattice;
B. second computer-executable instructions, recorded on the computer-readable physical medium and executable by the computer to enable the computer to, for respective ones of the sample points, generate a value for the function at the respective sample point; and
C. third computer-executable instructions, recorded on the computer-readable physical medium and executable by the computer to enable the computer to use the function values generated at the respective sample points in generating an estimate for the value of the integral and to generate therefrom the display-controlling electrical output based on the estimate, and to enable the computer to cause the display module to display a human-perceptible image based on the display-controlling electrical output.
2. A computer program product as defined in claim 1 in which the integration domain comprises the \u201cs\u201d-dimensional unit cube 0,1)s.
3. A computer program product as defined in claim 2 in which, for each stratum Aj, the first computer-executable instructions comprise instructions to enable the computer to generate the \u201cj-th\u201d sample point Rj(x) in accordance with the bijection
R
j
\u2062
:
\u2062
0
,
1
)
s
\u2192
A
j
x
\u21a6
(
j
n
\xb7
g
+
Bx
)
\u2062
\u2062
mod
\u2062
0
,
1
)
s
where \u201cn\u201d is the selected number of sample points, g is a generator vector for the rank-1 lattice and B is a matrix that defines a linear transformation that represents a rotation andor shear.
4. A computer program product as defined in claim 3 in which the first computer-executable instructions comprise instructions executable to enable the computer to generate the \u201cj-th\u201d sample point such that \u201cx\u201d is a vector.
5. A computer program product as defined in claim 4 in which the first computer-executable instructions comprise instructions executable to enable the computer to generate the \u201cj-th\u201d sample point such that the vector \u201cx\u201d is the same for all sample points.
6. A computer program product as defined in claim 4 in which, for each value of index \u201cj,\u201d the first computer-executable instructions comprise instructions to enable the computer to generate the \u201cj-th\u201d sample point such that the vector \u201cx\u201d is the \u201cj-th\u201d element of a sequence of vectors
Pn=(\u03be0, . . . , \u03ben\u22121).
7. A computer program product as defined in claim 6 in which the elements of the sequence are generated using a random sequence generation methodology.
8. A computer program product as defined in claim 6 in which at least one vector in the sequence is an \u201cs\u201d-dimensional vector.
9. A computer program product as defined in claim 2 in which the sample point generator module is configured to enable the computer to generate the \u201cj-th\u201d sample point such that the \u201cs-\u201ddimensional unit cube 0,1)s comprises an \u201cs1\u201d-dimensional portion and an \u201cs2\u201d-dimensional portion, where each sample point is defined by
(\u03bei,Rj(\u03b6i))
where \u03bei is an element of an \u201cs\u201d-dimensional sequence, \u03b6i is an element of an \u201cs2\u201d-dimensional sequence, and Rj is defined by the bijection
R
j
\u2062
:
\u2062
0
,
1
)
s
2
\u2192
A
j
x
\u21a6
(
j
n
\xb7
g
+
Bx
)
\u2062
\u2062
mod
\u2062
0
,
1
)
s
2
where \u201cn\u201d is the selected number of sample points, g is a generator vector for the rank-1 lattice, and B is a matrix that defines a linear transformation that represents a rotation andor shear.
10. A computer program product as defined in claim 9 in which the first computer-executable instructions comprise instructions executable to enable the computer to generate the \u201cj-th\u201d sample point such that the elements of the \u201cs1\u201d-dimensional sequence are generated using a random sequence generation methodology.
11. A computer program product as defined in claim 9 in which the first computer-executable instructions comprise instructions executable to enable the computer to generate the \u201cj-th\u201d sample point such that the elements of the \u201cs2\u201d-dimensional sequence are generated using a random sequence generation methodology.
12. A computer program product for use with a computer graphics system, the computer graphics system comprising a computer and a display module, the computer being operable to generate a pixel value for a pixel in an image, and to generate a display-controlling electrical output based on the pixel value, the pixel being representative of a point in a scene, the scene comprising at least one object and at least one light source, the computer being operable to generate the pixel value by an evaluation of an integral of a selected function over an s-dimensional integration domain, the display-controlling electrical output being transmittable by the computer to the display module to cause the display module to display a human-perceptible image based on the pixel value, the computer program product comprising a computer readable physical medium having recorded thereon:
A. first computer-executable instructions, recorded on the computer-readable physical medium and executable by the computer to enable the computer to generate a selected number of sample points over the integration domain, the sample points being generated such that there is at least one sample point in each of a plurality of strata distributed over the integration domain, the strata being defined by a rank-1 lattice;
B. second computer-executable instructions, recorded on the computer-readable physical medium and executable by the computer to enable the computer to, for respective ones of the sample points, generate a value for the function at the respective sample point; and
C. third computer-executable instructions, recorded on the computer-readable physical medium and executable by the computer to enable the computer to use the function values generated at the respective sample points in generating an estimate for the value of the integral in relation to the at least one object and at least one light source, the estimate corresponding to the pixel value for the image, to enable the computer to generate the display-controlling electrical output based on the pixel value, and to enable the computer to cause the display module to display a human-perceptible image based on the display-controlling electrical output.
13. A computer program product as defined in claim 12 in which the integration domain comprises the \u201cs\u201d-dimensional unit cube 0,1)s and for each stratum Aj, the sample point generator module is configured to enable the computer to generate the \u201cj-th\u201d sample point Rj(x) in accordance with the bijection
R
j
\u2062
:
\u2062
0
,
1
)
s
\u2192
A
j
x
\u21a6
(
j
n
\xb7
g
+
Bx
)
\u2062
\u2062
mod
\u2062
0
,
1
)
s
where \u201cn\u201d is the selected number of sample points, g is a generator vector for the rank-1 lattice, and B is a matrix that defines a linear transformation that represents a rotation andor shear.
14. A computer program product as defined in claim 13 in which the first computer-executable instructions comprise instructions to enable the computer to use the same vector \u201cx\u201d for at least some of the sample points.
15. A computer program product as defined in claim 14 in which at least part of the selected function represents a path tracing methodology in which the second computer-executable instructions comprise instructions executable to enable the computer to generate the function value associated with a point in the scene in relation to a plurality of simulated traces projected from the point, the directions of the traces being determined by strata defined on at least a portion of a sphere centered on the point induced by at least a portion of the rank-1 lattice.
16. A computer program product as defined in claim 13 in which, for each value of index \u201cj,\u201d the first computer-executable instructions comprise instructions executable to enable the computer to generate the \u201cj-th\u201d sample point such that the vector \u201cx\u201d is the \u201cj-th\u201d element of a sequence of vectors
Pn=(\u03be0, . . . , \u03ben\u22121).
17. A computer program product as defined in claim 16 in which the elements of the sequence are generated using a random sequence generation methodology.
18. A computer program product as defined in claim 17 in which at least part of the selected function represents a photon map methodology in which the second computer-executable instructions comprise instructions executable to enable the computer to generate the function values by using sample points generated by the computer using least some of the vectors from the sequence to control a random walk of traces from the light source through at least a portion of the scene.
19. A computer program product as defined in claim 13 in which the first computer-executable instructions comprise instructions executable to enable the computer to generate the \u201cj-th\u201d sample point such that the \u201cs-\u201ddimensional unit cube 0,1)s comprises an \u201cs1\u201d-dimensional portion and an \u201cs2\u201d-dimensional portion, where each sample point is defined by
(\u03be1,Rj(\u03b6i)),
where \u03bei is an element of an \u201cs1\u201d-dimensional sequence, \u03b6i is an element of an \u201cs2\u201d-dimensional sequence, and Rj is defined by the bijection
R
j
\u2062
:
\u2062
0
,
1
)
s
2
\u2192
A
j
x
\u21a6
(
j
n
\xb7
g
+
Bx
)
\u2062
\u2062
mod
\u2062
0
,
1
)
s
2
where \u201cn\u201d is the selected number of sample points, g is a generator vector for the rank-1 lattice, and B is a matrix that defines a linear transformation that represents a rotation andor shear.
20. A computer program product as defined in claim 19 in which the first computer-executable instructions comprise instructions executable to enable the computer to generate the \u201cj-th\u201d sample point such that the elements of the \u201cs1\u201d-dimensional sequence are generated using a random sequence generation methodology.
21. A computer program product as defined in claim 19 in which the first computer-executable instructions comprise instructions executable to enable the computer to generate the \u201cj-th\u201d sample point such that the elements of the \u201cs2\u201d-dimensional sequence are generated using a random sequence generation methodology.
22. A computer program product as defined in claim 19 in which at least part of the selected function represents a shadow ray methodology, in which a plurality of shadow rays comprising simulated traces are traced from a point in the scene towards respective points on the light source, the locations of the points on the light source being determined by Rj, the second computer-executable instructions comprising instructions executable to generate the function value in relation to whether respective ones of the shadow rays intersect at least one of said objects in the scene.
23. A computer program product as defined in claim 19 in which at least a part of the selected function represents a distribution path tracing methodology, in which the second computer-executable instructions comprise instructions executable to enable the computer to generate the function value associated with a point in the scene in relation to a plurality of simulated traces projected from the point, the directions of the traces being determined by strata defined on at least a portion of a sphere centered on the point induced by Rj.
The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.
1. A method for collecting valuable metal from an ITO scrap including the steps of subjecting the ITO scrap to electrolysis as an anode so as to deposit indium on a cathode, and collecting the indium.
2. The method for collecting valuable metal from an ITO scrap including the steps of subjecting the ITO scrap to electrolysis as an anode in an electrolytic bath partitioned with a diaphragm or an ion-exchange membrane so as to obtain anolyte containing indium, subsequently extracting the anolyte temporarily, eliminating tin contained in the anolyte, placing the anolyte from which the tin was eliminated in a cathode again, performing electrolysis thereto, and selectively collecting metallic indium.
3. A method for collecting valuable metal from an ITO scrap including the steps of providing an ITO electrolytic bath and an indium collecting bath, subjecting the ITO scrap to electrolysis as an anode in the electrolytic bath, obtaining anolyte containing indium by the electrolysis, placing the anolyte in the collecting bath and collecting indium from the anolyte containing indium in the collecting bath.