Graphics Reference
In-Depth Information
Real space
Fourier space
f ( x , y )
f ( wx , wy )
2D Fourier
transform
θ
θ
Integral
projection
Slicing
1D Fourier
transform
Figure 5.25 The Fourier slice theorem. Projecting in real space corresponds to slicing in Fourier space.
A rotation by θ applied before the projection is matched by the same rotation in Fourier
space before the slice is taken.
integral transform of a function is a slice of the Fourier transform of the function.
In contrast to integral projections, which involve multiple integration, slices are
almost trivial to compute—there lies the benefit of the Fourier slice theorem.
Integral projections and slices can be generalized to include a linear trans-
formation of the variables, which is also known as a change of basis . This does
not affect the validity of the Fourier slice theorem because the Fourier transform
is itself linear, meaning it “passes through” linear transformations. Figure 5.25
illustrates the Fourier slice theorem in this more general context: the projection
and slice are both applied after the coordinates are rotated by
. The refocusing
transformation implied in Equation (5.9) is in fact a linear transformation of the
light field parameters:
θ
1
u
1
v
v
1
α
1
α
1
α
1
α
(
x
,
y
,
u
,
v
)
+
x
,
+
y
,
u
,
.
(5.11)
Consequently, the process of refocusing an image from an in-camera light field
can be formulated as the integral projection (Equation (5.10)) of the light field
after a linear transformation of the parameters (Equation (5.11)). By the Fourier
slice theorem, the Fourier transform of a refocused image is a 2D slice of the 4D
Fourier transform of the in-camera light field. This, combined with the particular
transformations involved, is the Fourier slice photography theorem.
Analysis of sampling and image quality is mathematically simpler in the
Fourier domain. The mathematical framework of the “Fourier Slice Photogra-
 
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