Shape from camera motion for unknown material reflectance
A computer vision method that includes deriving a relationship of spatial and temporal image derivatives of an object to bidirectional reflectance distribution function (BRDF) derivatives under camera motion, and deriving with a processor a quasilinear partial differential equation for solving surfaced depth for orthographic projections using the relationship of spatial and temporal image derivatives without requiring knowledge of the BRDF. The method may further recover surface depth for an object with unknown BRDF under perspective projection.
1. A computer vision method comprising:
deriving, with a processor, a relationship of spatial and temporal image derivatives of an object to bidirectional reflectance distribution function (BRDF) derivatives under camera motion that provides an image sequence;
deriving a quasilinear partial differential equation for solving surfaced depth for orthographic projections using the relationship of spatial and temporal image derivatives to bidirectional reflectance distribution function (BRDF) derivatives;
solving the surface depth from the quasilinear partial differential equation; and
reconstructing an image of the object from the solving of the surface depth for orthographic projections,
wherein the BRDF comprises a negative incoming light direction, ω i , and outgoing direction, ω r , both defined with respect to the surface normal, n, and returns the ratio of reflected radiance exiting along ω r to the irradiance incident on the surface from direction ω i , each direction ω is itself parameterized by azimuth angle φ and zenith angle θ, therefore the BRDF as a whole is 4-dimensional and the BRDF has units sr −1 , with steradian (sr) being a unit of solid angle, therefore the BRDF as a whole is 4-dimensional and the BRDF has units sr −1 , with steradian (sr) being a unit of solid angle,
wherein the BRDF comprises
f
r
(
ω
i
,
ω
r
)
=
ⅆ
L
r
(
ω
r
)
ⅆ
E
i
(
ω
i
)
=
ⅆ
L
r
(
ω
r
)
L
i
(
ω
i
)
cos
θ
i
ⅆ
ω
i
where L is radiance, or power per unit solid-angle-in-the-direction-of-a-ray per unit projected-area-perpendicular-to-the-ray, E is irradiance, or power per unit surface area, Ø i is angle between ω i and surface normal, n and index i indicates incident light, whereas index rr indicates reflected light.
2. The method of claim 1 , wherein the quasilinear partial differential equation is solved using a method of characteristics.
3. The method of claim 1 , wherein a shape of the object for the image is recovered from camera motion under conditions of said bidirectional reflectance distribution function (BRDF) that is unknown.
4. The method of claim 1 further comprising a non-transitory computer readable medium comprising a computer readable program for executing the method of computer vision, wherein the computer readable program is executed on a computer.
5. A system for computer vision comprising:
a spatial and temporal image derivative module for deriving a relationship of spatial and temporal image derivatives from an object to unknown bidirectional reflectance distribution function (BRDF) derivatives under camera motion to provide an image sequence using a processor;
an orthographic module for deriving with a processor a quasilinear partial differential equation for solving surfaced depth for orthographic projections using the relationship of spatial and temporal image derivatives to bidirectional reflectance distribution function (BRDF) derivatives, solving the surface depth from the quasilinear partial differential equation, and reconstructing an image of the object from the solving of the surface depth for orthographic projections; and
a perspective module for estimating depth from the spatial and temporal image derivatives, deriving a constraint on surface gradient of the object, combining the depth and the constraint of the surface gradient to yield a solution for surface depth for perspective projections as a sparse linear system, and reconstructing an image of the object from the solution for surface depth for perspective projections,
wherein the BRDF comprises a negative incoming light direction, ω i , and outgoing direction, ω r , both defined with respect to the surface normal, n, and returns the ratio of reflected radiance exiting along ω r to the irradiance incident on the surface from direction ω i , each direction ω is itself parameterized by azimuth angle φ and zenith angle θ, therefore the BRDF as a whole is 4-dimensional and the BRDF has unit sr −1 , with steradian (sr) being unit of solid angle, therefore the BRDF as a whole is 4-dimensional and the BRDF has units sr −1 , with steradian (sr) being a unit of solid angle,
wherein the BRDF comprises
f
r
(
ω
i
,
ω
r
)
=
ⅆ
L
r
(
ω
r
)
ⅆ
E
i
(
ω
i
)
=
ⅆ
L
r
(
ω
r
)
L
i
(
ω
i
)
cos
θ
i
ⅆ
ω
i
where L is radiance, or power per unit solid-angle-in-the-direction-of-a-ray per unit projected-area-perpendicular-to-the-ray, E is irradiance, or power per unit surface area, Ø i is angle between ω i and surface normal, n and index i indicates incident light, whereas index rr indicates reflected light.
6. The method of claim 1 , wherein the BRDF comprises a four-dimensional function that defines how light is reflected at an opaque surface.