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Page 101

.. photographic imaging in the fourier domain 

complexity compares favorably to the O(n4) approach of existing algorithms, which are es-
sentially different approximations of numerical integration in the d spatial domain.

5.3 Photographic Imaging in the Fourier Domain


Figure .: Photographic Imag-
ing Operator.

Chapter  introduced the imaging integral in Equa-
tion ., which relates light fields and photographs
focused at different depths. Our first step here is to
codify Equation . in the operator notation that will
be used throughout this chapter.

Operator notation provides a higher level of
mathematical abstraction, allowing the theorems
derived below to express the relationship between
transformations that we are interested in (e.g. image
formation) rather than being tied up in the underly-
ing functions being acted upon (e.g. light fields and
photographs). Throughout this chapter, calligraphic
letters, such as A, are reserved for operators. If f is a
function in the domain of A, then A [ f ] denotes the
application of A to f .

Photographic Imaging Operator Let Pα be the operator that transforms an in-
camera light field parameterized by a separation of depth F into the photograph
formed on film at depth (α · F):

Pα [LF] (x′, y′) =


∫ ∫


1− 1



, v

1− 1




, u, v

du dv. (.)

This operator is what is implemented by a digital refocusing code that accepts light fields
and computes refocused photographs (Figure .). As noted in Chapter , the operator can
be thought of as shearing the d space, and then projecting down to d.

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 chapter . signal processing framework

As described in the chapter overview, the key to analyzing the imaging operator is the
Fourier Slice Theorem. The classical version of the Fourier Slice Theorem [Deans ]
states that a d slice of a d function’s Fourier spectrum is the Fourier transform of an or-
thographic integral projection of the d function. The slicing line is perpendicular to the
projection lines, as illustrated in Figure .. Conceptually, the theorem works because the
value at the origin of frequency space gives the dc value (integrated value) of the signal, and
rotations do not fundamentally change this fact. From this perspective, it makes sense that
the theorem generalizes to higher dimensions. It also makes sense that the theorem works
for shearing operations as well as rotations, because shearing a space is equivalent to rotating
and dilating the space.

These observations mean that we can expect that photographic imaging, which we have
observed is a shear followed by projection, should be proportional to a dilated d slice of
the light field’s d Fourier transform. With this intuition in mind, Sections .. and ..
are simply the mathematical derivations in specifying this slice precisely, culminating in
Equations . and ..

5.3.1 Generalization of the Fourier Slice Theorem

Let us first digress to study a generalization of the theorem to higher dimensions and pro-
jections, so that we can apply it in our d space. A closely related generalization is given by
the partial Radon transform [Liang and Munson ], which handles orthographic pro-
jections from N dimensions down to M dimensions. The generalization here formulates a
broader class of projections and slices of a function as canonical projection or slicing follow-
ing an appropriate change of basis (e.g. an N-dimensional rotation or shear). This approach
is embodied in the following operator definitions.

Integral Projection Let INM be the canonical projection operator that reduces
an N-dimensional function down to M-dimensions by integrating out the last
N − M dimensions: INM [ f ] (x1, . . . , xM) =

f (x1, . . . , xN) dxM+1 . . . dxN .

Slicing Let SNM be the canonical slicing operator that reduces an N-dimensional
function down to an M dimensional one by zero-ing out the last N − M dimen-
sions: SNM [ f ] (x1, . . . , xM) = f (x1, . . . , xM, 0, . . . , 0).

Page 202

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