Image generation using low-discrepancy sequences
View Patent ↗A computer graphics system generates a pixel value for a pixel in an image and displays a human-perceptible image on an LCD CRT or other display device based on an electrical output generated in response to the pixel value, the pixel being representative of a point in a scene as recorded on an image plane of a simulated camera, the computer graphics system being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot from respective ones of a plurality of subpixels in the pixel each subpixel having coordinates in the image plane. The computer graphics system comprises a sample point generator and a function evaluator.
1. In a computer graphics system comprising a computer operable to (1) generate a pixel value for a pixel in an image, (2) generate an electrical output signal responsive to a plurality of pixel values, and (3) display the image on a display device in response to the electrical output signal, wherein the pixel value is representative of a point in a scene as recorded on an image plane of a simulated camera, the computer being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot from respective ones of a plurality of subpixels in the pixel, each subpixel having coordinated (s s , s y ) in the image plane, the improvement comprising:
A. a sample point generator, in the computer, configured to map subpixel coordinates (s s ,s y ) onto strata coordinates
(j,k):=(s x mod 2 n ,s y mod 2 n ),
from which a ray is to be shot in accordance with
x
i
=
(
s
x
+
σ
(
k
)
2
n
,
s
y
+
σ
(
j
)
2
n
)
,
where “i” is an instance number for the ray generated as i=j2 n +σ(k), were integer permutation
σ( k ):=2 n Φ b ( k ) for 0≦k<2 n
for selected “n,” where φ b (x) is a radical inverse function given by
Φ
b
:
N
0
→
I
x
=
∑
j
=
0
∞
a
j
(
x
)
b
j
↦
∑
j
=
0
∞
a
j
(
x
)
b
-
j
-
1
,
where
(
a
j
)
j
=
0
∞
,
is the representation of “x” in a selected integer base “b”; and
B. a function evaluator in the computer, in communication with the sample point generator, configured to generate at least one value representing an evaluation of a selected function using the strata coordinates generated by the sample point generator, the value generated by the function evaluator corresponding to the pixel value, so as to enable the generation of the pixel value, the electrical output signal and the display of the image.
2. In a computer graphics system comprising a computer operable to (1) generate a pixel value for a pixel in an image, (2) generate an electrical output signal responsive to a plurality of pixel values, and (3) display the image on a display device in response to the electrical output signal, wherein the pixel value is representative of a point in a scene as recorded on an image plane of a simulated camera, the computer being configured to generate the pixel value for an image using a selected ray-tracing methodology in which simulated rays are shot from respective ones of a plurality of subpixels in the pixel, each subpixel having coordinates (s x , s y ) in the image plane, the method comprising:
A. generating, in the computer, sample points by mapping subpixel coordinates (s x , s y ) onto strata coordinates
(j,k):=(s x med 2 n ,s y mod 2 n ),
from which a ray is to be shot in accordance with
x
i
=
(
s
x
+
σ
(
k
)
2
n
,
s
y
+
σ
(
j
)
2
n
)
,
where “i” is an instance number for the ray generated as i=j2 n +σ(k), where integer permutation
σ(k):=2 n Φ b (k) for 0≦k<2 n
for selected “n,” where φ b (x) is a radical inverse function given by
Φ
b
:
N
0
→
I
x
=
∑
j
=
0
∞
a
j
(
x
)
b
j
↦
∑
j
=
0
∞
a
j
(
x
)
b
-
j
-
1
where
(
a
j
)
j
=
0
∞
,
is the representation of “x” in a selected integer base “b”, and
B. generating in the computer, at least one function evaluation value, representing an evaluation of a selected function, using the strata coordinates generated by the generating of sample points, the function evaluation value corresponding to the pixel value, so as to enable the generation of the pixel value, the electrical output signal and the display of the image.