![](bg3.png)
Figure 1: Spatial filter kernel shaped by the positional dif-
ferential vectors, D
u
x
x
x and D
v
x
x
x.
Sporring et al. evaluates the full differentials for
a parameterized ray. This allows for an extension of
parameters such that the derivatives of a ray can be
considered with respect to time. From Sporring et
al.’s equations for transfer, reflection and refraction,
we observe that non-zero time-dependent element dif-
ferentials (eg. D
t
q
q
q) propagate through these interac-
tions to the differentials of the interacting photon. We
exploit this behavior such that a footprint from a pho-
ton differential traveling in a dynamic scene not only
describes the spatial coherence of the ray but also the
temporal coherence of the ray.
When a photon differential hits a surface, its po-
sitional differential vectors are projected onto the sur-
face’s tangent plane at the intersection point. The spa-
tial footprint of the photon differential is the area on
the tangent plane of a parallelogram spanned by the
positional differential vectors. The spatial footprint
can be used to shape an anisotropic filter kernel as il-
lustrated in Figure 1.
The time dependent positional differential vector,
D
t
x
x
x, tells us either how the photon’s footprint is go-
ing to behave over consecutive frames, or how the
footprint has behaved in former frames. In the for-
mer case, the direction of D
t
x
x
x predicts the direction
on the surface that the footprint will move, and the
magnitude of the vector predicts how far the footprint
is likely to move. Basically, the magnitude and the
direction of D
t
x
x
x depends on the estimation method
used to calculate the time derivatives of an element,
which again depends on the geometry representation.
In the present method, we simply use finite differ-
ences and triangle meshes. Except for the last frame,
in which we use backward differences, we estimate
the time dependent differentials using forward differ-
ences. When we want to predict how a footprint is go-
ing to behave, having intersected a moving element,
we estimate the element’s positional time derivatives
by
D
t
q
q
q
f
= e(q
q
q
f +1
−q
q
q
f
), (5)
where D
t
q
q
q
f
is the derivative of the vertex q
f
with re-
(b)
Figure 2: Temporal filter kernel shaped by a spatial kernels
translation along the time dependent differential vector.
spect to time at frame step f , and e is the exposure.
The exposure is a parameter for how much we trust
our prediction. Generally, it works as a smoothing pa-
rameter for the time dependent footprint that decides
how much motion blur we induce. Its unit is given
in frames as it depends on the movement of the scene
elements between frames. The exposure is related to
the exposure time by the frame rate such that the ex-
posure time is equal to the exposure divided by the
frame rate.
The time dependent footprint constitutes an inte-
gration of the spatial footprint over the time depen-
dent differential vector such that the spatial footprint
is elongated along the vector. We achieve this by
translating the spatial footprint along the time depen-
dent differential vector. As in the spatial case, the time
dependent footprint describes a filter kernel. In Fig-
ure 2(a), D
t
x
x
x
pd
is the time dependent differential vec-
tor, x
x
x
pd
is the center of the spatial kernel, and x
x
x is the
estimation point, for which the kernel weight is esti-
mated.
The kernel is translated along D
t
x
x
x
pd
to the point,
x
x
x
0
pd
, on the line segment, (x
x
x
pd
→x
x
x
pd
+D
t
x
x
x
pd
), where
x
x
x
0
pd
is the point on the segment having the shortest
distance to the estimation point, x
x
x. Using x
x
x
0
pd
as cen-
ter for the spatial kernel, the resulting time dependent
kernel will achieve an elongated shape as illustrated
in Figure 2(b).
The irradiance of the time dependent photon dif-
ferential is estimated as
E
pd
= Φ
pd
/A
pd
, (6)
where Φ
pd
is the radiant flux carried by the photon,
and A
pd
is the surface area, to which the radiant flux
Figure 1: Spatial filter kernel shaped by the positional dif-
ferential vectors, D
u
x
x
x and D
v
x
x
x.
al.’s equations for transfer, reflection and refraction,
we observe that non-zero time-dependent element dif-
ferentials (eg. D
t
q
q
q) propagate through these interac-
tions to the differentials of the interacting photon. We
exploit this behavior such that a footprint from a pho-
ton differential traveling in a dynamic scene not only
describes the spatial coherence of the ray but also the
temporal coherence of the ray.
When a photon differential hits a surface, its po-
sitional differential vectors are projected onto the sur-
face’s tangent plane at the intersection point. The spa-
tial footprint of the photon differential is the area on
the tangent plane of a parallelogram spanned by the
positional differential vectors. The spatial footprint
can be used to shape an anisotropic filter kernel as il-
lustrated in Figure 1.
The time dependent positional differential vector,
D
t
x
x
x, tells us either how the photon’s footprint is go-
ing to behave over consecutive frames, or how the
footprint has behaved in former frames. In the for-
mer case, the direction of D
t
x
x
x predicts the direction
on the surface that the footprint will move, and the
magnitude of the vector predicts how far the footprint
is likely to move. Basically, the magnitude and the
direction of D
t
x
x
x depends on the estimation method
used to calculate the time derivatives of an element,
which again depends on the geometry representation.
In the present method, we simply use finite differ-
ences and triangle meshes. Except for the last frame,
in which we use backward differences, we estimate
the time dependent differentials using forward differ-
ences. When we want to predict how a footprint is go-
ing to behave, having intersected a moving element,
we estimate the element’s positional time derivatives
by
D
t
q
q
q
f
= e(q
q
q
f +1
−q
q
q
f
), (5)
where D
t
q
q
q
f
is the derivative of the vertex q
f
with re-
spect to time at frame step f , and e is the exposure.
The exposure is a parameter for how much we trust
our prediction. Generally, it works as a smoothing pa-
rameter for the time dependent footprint that decides
(a)
(b)
Figure 2: Temporal filter kernel shaped by a spatial kernels
translation along the time dependent differential vector.
how much motion blur we induce. Its unit is given
in frames as it depends on the movement of the scene
elements between frames. The exposure is related to
the exposure time by the frame rate such that the ex-
posure time is equal to the exposure divided by the
frame rate.
The time dependent footprint constitutes an inte-
gration of the spatial footprint over the time depen-
dent differential vector such that the spatial footprint
is elongated along the vector. We achieve this by
translating the spatial footprint along the time depen-
dent differential vector. As in the spatial case, the time
dependent footprint describes a filter kernel. In Fig-
ure 2(a), D
t
x
x
x
pd
is the time dependent differential vec-
tor, x
x
x
pd
is the center of the spatial kernel, and x
x
x is the
estimation point, for which the kernel weight is esti-
mated.
The kernel is translated along D
t
x
x
x
pd
to the point,
x
x
x
0
pd
, on the line segment, (x
x
x
pd
→x
x
x
pd
+D
t
x
x
x
pd
), where
x
x
x
0
pd
is the point on the segment having the shortest
distance to the estimation point, x
x
x. Using x
x
x
0
pd
as cen-
ter for the spatial kernel, the resulting time dependent
kernel will achieve an elongated shape as illustrated
in Figure 2(b).
The irradiance of the time dependent photon dif-
ferential is estimated as
E
pd
= Φ
pd
/A
pd
, (6)
where Φ
pd
is the radiant flux carried by the photon,
and A
pd
is the surface area, to which the radiant flux
is incident. For the time dependent photon differen-
tial, this area is the area of the time dependent kernel.
Referring to Figure 3 this area is calculated as
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