4D Frequency Analysis of Computational Cameras for Depth of Field

4D Frequency Analysis of Computational Cameras for Depth of Field Extension
Anat Levin1,2
Samuel W. Hasinoff1 Paul Green1 Frédo Durand1
1 MIT CSAIL 2 Weizmann Institute
Standard lens image
Our lattice-focal lens: input
William T. Freeman1
Lattice-focal lens: all-focused output
Figure 1: Left: Image from a standard lens showing limited depth of field, with only the rightmost subject in focus. Center: Input from our
lattice-focal lens. The defocus kernel of this lens is designed to preserve high frequencies over a wide depth range. Right: An all-focused
image processed from the lattice-focal lens input. Since the defocus kernel preserves high frequencies, we achieve a good restoration over the
full depth range.
Abstract
Depth of field (DOF), the range of scene depths that appear sharp
in a photograph, poses a fundamental tradeoff in photography—
wide apertures are important to reduce imaging noise, but they also
increase defocus blur. Recent advances in computational imaging
modify the acquisition process to extend the DOF through deconvolution. Because deconvolution quality is a tight function of the
frequency power spectrum of the defocus kernel, designs with high
spectra are desirable. In this paper we study how to design effective
extended-DOF systems, and show an upper bound on the maximal
power spectrum that can be achieved. We analyze defocus kernels
in the 4D light field space and show that in the frequency domain,
only a low-dimensional 3D manifold contributes to focus. Thus,
to maximize the defocus spectrum, imaging systems should concentrate their limited energy on this manifold. We review several
computational imaging systems and show either that they spend energy outside the focal manifold or do not achieve a high spectrum
over the DOF. Guided by this analysis we introduce the lattice-focal
lens, which concentrates energy at the low-dimensional focal manifold and achieves a higher power spectrum than previous designs.
We have built a prototype lattice-focal lens and present extended
depth of field results.
response that can be achieved. In this paper, we use a standard
computational photography tool, the light field, e.g., [Levoy and
Hanrahan 1996; Ng 2005; Levin et al. 2008a], to address these issues. Using arguments of conservation of energy and taking into
account the finite size of the aperture, we present bounds on the
power spectrum of all defocus kernels.
Furthermore, a dimensionality gap has been observed between the
4D light field and the space of 2D images over the 1D set of depths
[Gu et al. 1997; Ng 2005]. In the frequency domain, only a 3D
manifold contributes to standard photographs, which corresponds
to focal optical conditions. Given the above bounds, we show that
it is desirable to avoid spending power in the other afocal regions
of the light field spectrum. We review existing camera designs and
find that some spend significant power in these afocal regions, while
others do not achieve a high spectrum over the depth range.
Keywords: Computational camera, depth of field, light field,
Fourier analysis.
Our analysis leads to the development of the lattice-focal lens—a
novel design which allows for improved image reconstruction. It
is designed to concentrate energy at the focal manifold of the light
field spectrum, and achieves defocus kernels with high spectra. The
design is a simple arrangement of lens patches with different focal
powers, but the patches’ size and powers are carefully derived. The
defocus kernels of a lattice-focal lens are high over a wide depth
range, but they are not depth invariant. This both requires and enables coarse depth estimation. We have constructed a prototype and
demonstrate encouraging extended depth of field results.
1
1.1
Introduction
Depth of field, the depth range over which objects in a photograph
appear acceptably sharp, presents an important tradeoff. Lenses
gather more light than a pinhole, which is critical to reduce noise,
but this comes at the expense of defocus outside the focal plane.
While some defocus can be removed computationally using deconvolution, the results depend heavily on the information preserved
by the blur, as characterized by the frequency power spectrum
of the defocus kernel. Recent advances in computational imaging [Dowski and Cathey 1995; Levin et al. 2007; Veeraraghavan
et al. 2007; Hausler 1972; Nagahara et al. 2008] modify the image
acquisition process to enable extended depth of field through such
a deconvolution approach.
Computational imaging systems can dramatically extend depth of
field, but little is known about the maximal frequency magnitude
Depth of field evaluation
To facilitate equal comparison across designs all systems are allocated a fixed time budget and maximal aperture width, and hence
can collect an equal amount of photons. All systems are expected
to cover an equal depth range d ∈ [dmin , dmax ].
Similar to previous work, we focus on Lambertian scenes and assume locally constant depth. The observed image B of an object at depth d is then described as a convolution B = φd ⊗ I + N,
where I is the ideally sharp image, N is the imaging noise, and
φd is the defocus kernel, commonly referred to as the point spread
function (PSF). The defocus PSF φd is often analyzed in terms of
its Fourier transform φ̂d , known as the optical transfer function
(OTF). In the frequency domain, convolution is a multiplication
ˆ ω ) + N̂(ω ) where hats denote Fourier transforms.
B̂(ω ) = φ̂d (ω )I(
In a nutshell, deblurring divides every spatial frequency by the ker-
While the tools derived here apply to many computational cameras,
our focus is on designs capturing only a single input image. In
appendix B we present one possible extension to multiple measurement strategies like the focal stack and the plenoptic camera.
1.2
Related work
Depth of field is traditionally increased by reducing the aperture,
but this unfortunately lowers the light collected and increases noise.
Alternatively, a focal stack [Horn 1968; Hasinoff and Kutulakos
2008] captures a sequence of images with narrow depth of field
but varying focus, which can be merged for extended depth of field
[Ogden et al. 1985; Agarwala et al. 2004]. Our new lattice-focal
lens can be thought of as capturing all the images from a special
focal stack, shifted and summed together in a single photo.
New designs have achieved improved frequency response together
with a depth invariant PSFs, allowing for deconvolution without
depth estimation. Wavefront coding achieves this with a cubic optical element [Dowski and Cathey 1995]. Others use a log asphere
[George and Chi 2003] and focus sweep approaches modify the focus configuration continuously during the exposure [Hausler 1972;
Nagahara et al. 2008].
In contrast, coded aperture approaches [Veeraraghavan et al. 2007;
Levin et al. 2007] make the defocus blur more discriminative to
depth variations. Having identified the defocus diameter, blur can
be partially removed via deconvolution. One disadvantage of this
design is that some light rays are blocked. A more serious problem is that the lens is still focused only at one particular depth and
objects located away from the focus depth are still heavily blurred.
Other designs [Ben-Eliezer et al. 2005] divide the aperture into subsquares consisting of standard lenses, similar to our lattice-focal
lens. But while these methods involve redundant focal lengths, our
analysis lets us optimize the combination of different focal powers
for improved depth of field.
We build on previous analysis of cameras and defocus in light field
space [Ng 2005; Adams and Levoy 2007; Levin et al. 2008a]. A
related representation in the Fourier optics literature is the Ambiguity function [Rihaczek 1969; Papoulis 1974; Brenner et al. 1983;
FitzGerrell et al. 1997], allowing a simultaneous analysis of defocus over a continuous depth range.
2
Background on defocus in light field space
Our main analysis is based on geometric optics and the light field,
but appendix C provides complementary derivations using wave optics. We first review how the light field can be used to analyze cameras [Ng 2005; Levin et al. 2008a]. It is a 4D function ℓ(x, y, u, v)
Wavefont Coding Lens
2D Scene
x−plane
x−plane
u−plane
Light Field and
Integration Surface c(u)
In this paper, we focus on the stability of the deblurring process to
noise and evaluate imaging systems according to the spectra they
achieve over a specified depth range. We note, however, that many
approaches such as coded apertures and our new lattice-focal lens
involve a depth-dependent PSF φd and require a challenging depth
identification stage. On the positive side, such systems output a
coarse depth map of the scene in addition to the all-focused image.
In contrast, designs like wavefront coding and focus sweep have an
important advantage: their blur kernel is invariant to depth.
Standard Lens
u−plane
(aperture plane)
(aperture plane)
sensor plane
sensor plane
u
u
x
x
ω
ω
Fourier Lens Spectrum
nel spectrum, so the information preserved at a spatial frequency ω
depends strongly on the kernel spectrum. If |φ̂d (ω )| is low, noise is
amplified and image reconstruction is degraded. To capture scenes
with a given depth range d ∈ [dmin , dmax ], we want PSFs φd whose
modulation transfer function (MTF) |φ̂d | is as high as possible for
every spatial frequency ω , over the full depth range. Noise is absent
from the equations in the rest of this paper, because whatever noise
is introduced by the sensor gets amplified as a monotonic function
of |φ̂d (ω )|.
ω
ω
Figure 2: Integration surfaces in flatland. Top: Ray mapping diagrams. Middle: The corresponding light field and integration surface c(u). Bottom: The lens spectrum k̂. The blue/red slices represent OTF-slices of the blue/red objects respectively. The vertical
yellow slices represent ωx0 slices discussed in Sec. 3. Left: Standard lens focused at the blue object. Right: Wavefront coding.
u, v
x, y
ωx,y
Ω
φ (x, y)
φ̂ (ωx , ωy )
k(x, y, u, v)
k̂(ωx , ωy , ωu , ωv )
A
εA
α (ωx,y ), β (ωx,y )
aperture plane coordinates
spatial coordinates (at focus plane)
spatial frequencies
max spatial frequency
point spread function (PSF)
optical transfer function (OTF)
4D lens kernel
4D lens spectrum
aperture width
hole/subsquare width
bounded multiplicative factors (Eqs. (43,11))
Table 1: Notation.
describing radiance for all rays in a scene, where a ray is parameterized by its intersections with two parallel planes, the uv-plane
and the xy-plane [Levoy and Hanrahan 1996]. Figure 2 shows a 2D
flatland scene and its corresponding 2D light field. We assume the
camera aperture is positioned on the uv-plane, and xy is a plane in
the scene (e.g., the focal plane of a standard lens). x, y are spatial
coordinates and the u, v coordinates denote the viewpoint direction.
An important property is that the light rays emerging from a given
physical point correspond to a 2D plane in 4D of the form
x = su + (1 − s)px , y = sv + (1 − s)py ,
(1)
whose slope s encodes the object’s depth:
s = (d − do )/d ,
(2)
where d is the object depth and do the distance between the uv, xy
planes. The offsets px and py characterize the location of the scene
point within the plane at depth d.
Each sensor element gathers light over its 2D area and the 2D aperture. This is a 4D integral over a set of rays, and under first order
s=1
k(x0 − x, y0 − y, −u, −v)ℓ(x, y, u, v) dxdydudv .
s=0
0
(3)
For most designs, the 4D kernel is effectively non-zero only at a 2D
integration surface because the pixel area is small compared to the
aperture. That is, the 4D kernel is of the form
k(x, y, u, v) = δ (x − cx (u, v), y − cy (u, v))R(u/A)R(v/A) ,
(4)
where R is a rect function, δ denotes a Dirac delta, and c(u, v) →
(x, y) is a 2D → 2D surface describing the ray mapping at the lens’s
aperture, which we assume to be square and of size A × A. The
surface c is shown in black in the middle row of Figure 2.
For example, a standard lens focuses rays emerging from a point
at the focus depth and the integration surface c is linear c(u, v) =
(su, sv). The integration slope s corresponds to the slope of the focusing distance (Fig. 2, left). When integrating a light field with the
same slope (blue object in Fig. 2), all rays contributing to a sensor
element come from the same 3D point. In contrast, when the object
is misfocused (e.g., red/green objects), values from multiple scene
points get averaged, causing defocus. Wavefront coding [Dowski
and Cathey 1995] involves a cubic lens. Since refraction is a function of the surface normal, the kernel is a parabolic surface [Levin
et al. 2008b; Zhang and Levoy 2009] (Fig. 2, right) defined by
c(u, v) = (au2 , av2 ) .
−0.5Ω
(5)
Finally, the kernel of the focus sweep is not a 2D surface but the
integral of standard lens kernels with different slopes/depths.
Consider a Lambertian scene with locally constant depth. If the local scene depth, or slope, is known, the noise-free defocused image
B̃ can be expressed as a convolution of an ideal sharp image I with
a PSF φs : B̃ = φs ⊗ I. As demonstrated in [Levin et al. 2008c], for a
given slope s this PSF is fully determined by projecting the 4D lens
kernel k along the slope s:
−Ω
ωy
ωx
−Ω
−0.5Ω
0
0.5Ω
Ω
slope color coding
s=0.5
0.5Ω
s=−0.5
B̃(x0 , y0 ) =
ZZZZ
Ω
s=−1
optics (paraxial optics), it can be modeled as a convolution [Ng
2005; Levin et al. 2008a]. A shift-invariant kernel k(x, y, u, v) determines which rays are summed for each element, as governed by the
lens. Before applying imaging noise, the value recorded at a sensor
element is then:
Figure 3: Layout of the 4D lens spectrum, highlighting the focal
manifold. Each subplot represents a ωx0 ,y0 -slice, k̂ωx0 ,y0 (ωu , ωv ).
The outer axes vary the spatial frequency ωx0 ,y0 , i.e., the slicing
position. The inner axes of each subplot, i.e., of each slice, vary
ωu,v . The entries of k̂ along each focal segment are color coded, so
that the 2D set of points sharing the same color corresponds to an
OTF with a given depth/slope (e.g., the red points define an OTF for
the slope s = −1). This illustrates the dimensionality gap: the set of
entries contributing to an OTF at any physical depth occupies only
a 1D segment in each 2D ωx0 ,y0 -slice. In the flatland case (Fig. 2),
each ωx0 ,y0 -slice corresponds to a vertical column.
Below we refer to slices of this form as OTF-slices, because they
directly provide the OTF, describing the frequency response due to
defocus at a given depth. OTF-slices in flatland are illustrated in
the last row of Figure 2 (dashed red/blue). These are slanted slices
whose slope is orthogonal to the object slope in the primal light field
domain. Low spectrum values in k̂ leads to low magnitudes in the
OTF for the corresponding depth. In particular, for a standard lens,
only the OTF-slice corresponding to the focusing distance (dashed
blue, Fig. 2 left) has high values.
All systems in this paper are allocated a fixed exposure time, w.l.o.g. 1. The aperture size is A × A.
∆ denotes a pixel width back-projected onto the focal xy-plane.
In the frequency domain we deal with the range [−Ω, Ω], where
Ω = 1/(2∆). ωx,y , ωu,v are shortcuts for the 2D vectors (ωx , ωy ),
(ωu , ωv ). Table 1 summarizes notations.
Notations and assumptions:
φs (x, y) =
ZZ
k(x, y, u + sx, v + sy)dudv .
(6)
That is, we simply integrate over all rays (x, y, u + sx, v + sy) corresponding to a given point in the xy-plane (see Eq. 1).
For example, we have seen that the 4D kernel k for a standard lens is
planar. If the slope s of an object and the orientation of this planar
k coincide, the object is in focus and the projected PSF φs is an
impulse. For a different slope the projected PSF is a box filter, and
the width of this box depends on the difference between the slopes
of the object and that of k. For wavefront coding, the parabolic 4D
kernel has an equal projection in all directions, explaining why the
resulting PSF is invariant to object depth [Levin et al. 2008b; Zhang
and Levoy 2009].
Now that we have expressed defocus as a convolution, we can
analyze it in the frequency domain. Let k̂(ωx , ωy , ωu , ωv ) denote
the 4D lens spectrum, the Fourier transform of the 4D lens kernel
k(x, y, u, v). Figure 2 visualizes lenses spectra k̂ in flatland for a
standard and wavefront coding lenses. As the PSF φs is obtained
from k by projection (Eq. (6)), by the Fourier slice theorem, the
OTF (optical transfer function) φ̂s is a slice of the 4D lens spectrum
k̂ in the orthogonal direction [Ng 2005; Levin et al. 2008c]:
φ̂s (ωx , ωy ) = k̂(ωx , ωy , −sωx , −sωy ) .
(7)
We seek to capture a fixed depth range [dmin , dmax ]. To simplify the
light field parameterization, we select the location of the xy plane
according to the harmonic mean do = d2dminmin+ddmax
, corresponding to
max
the point at which one would focus a standard lens to equalize defocus diameter at both ends of the depth range, e.g., [Hasinoff and
Kutulakos 2008]. This maps the depth range to the symmetric slope
2(dmax −dmin )
range [−S/2, S/2], where S = dmax
+dmin (Eq. (2)). Under this parameterization the defocus diameter (on the xy-plane) of slope s can
be expressed simply as A|s|.
We also assume that scene radiance is fairly constant over the narrow solid angle subtended by the camera aperture. This assumption
is violated by highly specular objects or at occlusion boundaries.
3 Frequency analysis of depth of field
We now analyze the requirements, strategies, and limits of depth
of field extension. We show that a key factor for depth of field
optimization is the presence of a dimensionality gap in the 4D light
field: only a manifold of the 4D spectrum, which we call focal,
contributes to focusing at physical depths. Furthermore, we show
that the energy in a 4D lens spectrum is bounded. This suggests that
to optimize depth of field, most energy should be concentrated on
the focal manifold. We discuss existing lens designs and show that
many of them spend energy outside the focal manifold. In Sec. 4 we
propose a novel design which significantly reduces this problem.
3.1
The dimensionality gap
As described above, scene depth corresponds to slope s in the light
field. It has, however, been observed that the 4D light field has
a dimensionality gap, in that most slopes do not correspond to a
physical depth [Gu et al. 1997; Ng 2005]. Indeed, the set of all 2D
planes x = su u + px , y = sv v + py described by their slope su , sv and
offset px , py is 4D. In contrast, the set corresponding to real depth,
i.e., where s = su = sv , is only 3D, as described by Eq. (1). This
makes sense because scene points are 3D. The dimensionality gap
is a property of the 4D light field, and does not exist for the 2D
light field in flatland. The other slopes where su 6= sv are afocal
and represent rays from astigmatic refractive or reflective surfaces,
which are surfaces with anisotropic curvature [Adams and Levoy
2007], e.g., the reflection from a cylindrical mirror. Since we consider scenes which are sufficiently Lambertian over the aperture,
afocal light field orientations hold no interesting information.
The dimensionality gap is particularly clear in the Fourier domain [Ng 2005]. Consider the 4D lens spectrum k̂, and examine
the 2D slices k̂ωx0 ,y0 (ωu , ωv ), in which the the spatial frequencies
ωx0 , ωy0 are held constant (Fig. 3). We call these ωx0 ,y0 -slices. In
flatland, ωx0 ,y0 -slices are vertical slices (yellow in Fig. 2). Following Eq. (7), we note that the set of entries in each k̂ωx0 ,y0 participating in the OTF for any depth is restricted to a 1D line:
k̂ωx0 ,y0 (−sωx0 , −sωy0 ) ,
As an example, Figure 4(b-e) shows the 2D families of 2D ωx0 ,y0 slices for a variety of cameras. A standard lens has a high response
for an isolated point in each slice, corresponding to the focusing
distance. In contrast, wavefront coding (Fig. 4(e)) has a broader
response that spans more of the focal segment, but also over the
afocal region. While the spectrum of the focus sweep (Fig. 4(d)) is
on the focal segment, its magnitude is lower magnitude than that of
a standard lens.
Upper bound on the defocus MTF
In this section we derive a bound on the defocus MTF. As introduced earlier, we pose depth of field extension as maximizing the
MTFs |φ̂s (ωx,y )| over all slopes s ∈ [−S/2, S/2] and over all spatial
frequencies ωx,y . Since the OTFs are slices from the 4D lens spectrum k̂ (Eq. (7)), this is equivalent to maximizing the spectrum on
the focal segments of k̂.
We first derive the available energy budget, using a direct extension
of the 1D case [FitzGerrell et al. 1997; Levin et al. 2008c].
Claim 1 For an aperture of size A × A and exposure length 1, the
total energy in each ωx0 ,y0 -slice is bounded by A2 :
ZZ
|k̂ωx0 ,y0 (ωu , ωv )|2 d ωu d ωv ≤ A2 .
As in the 1D space-time case [Levin et al. 2008c], optimal worstcase performance can be realized by spreading the energy budget
uniformly over the range of slopes. The key difference in this paper
is the dimensionality gap. As shown in Figure 3, the OTFs φ̂s cover
only a 1D line segment, and most entries in an ωx0 ,y0 -slice k̂ωx0 ,y0
do not contribute to any OTF. Therefore, the energy budget should
be spread evenly over the 1D focal segment only.
Given a power budget for each ωx0 ,y0 -slice, the upper bound for
the defocus MTF concentrates this budget on the 1D focal segment
only. Distributing energy over the focal manifold requires caution,
however, because the segment effectively has non-zero thickness
due to its finite support in the primal domain. If a 1D focal segment
had zero thickness, its spectrum values could be made infinite while
still obeying the norm constraints of Claim 1. As we show below,
since the primal support of k is finite (k admits no light outside the
aperture), the spectrum must be finite as well, so the 1D focal segment must have non-zero thickness. Slices from this ideal spectrum
are visualized in Figure 4(a).
Claim 2 The worst-case defocus MTF for the range [−S/2, S/2] is
bounded. For every spatial frequency ωx,y :
(8)
for which ωu = −sωx0 , ωv = −sωy0 . For a fixed slope range s ∈
[−S/2, S/2] the set of entries participating in any OTF φ̂s is a 1D
segment. These segments, which we refer to as focal segments,
are highlighted in Figure 3. The rest of the spectrum is afocal. This
property is especially important, because it implies that most entries
of k̂ do not contribute to an OTF at any depth.
3.2
The proof, provided in appendix A, follows from the finite amount
of light passing through a bounded aperture over a fixed exposure.
As a consequence of Parseval’s theorem, this energy budget then
applies to every ωx0 ,y0 -slice k̂ωx0 ,y0 . While Claim 1 involves geometric optics, similar bounds can be obtained with Fourier optics
using slices of the ambiguity function [Rihaczek 1969; FitzGerrell
et al. 1997]. In appendix C we derive an analogous bound under
Fourier optics, with a small difference—the budget is no longer
equal across spatial frequencies, but decreases with the diffractionlimited MTF.
(9)
min
s∈[−S/2,S/2]
|φ̂s (ωx , ωy )|2 ≤
β (ωx,y )A3
,
S|ωx,y |
(10)
where the factor
β (ωx,y ) =
|ωx,y |
min(|ωx |, |ωy |)
1−
max(|ωx |, |ωy |)
3 · max(|ωx |, |ωy |)
(11)
√
is in the range [ 5125 , 1] ≈ [0.93, 1].
Proof: For each ωx0 ,y0 -slice k̂ωx0 ,y0 the 1D focal segment is of length
S|ωx0 ,y0 |. We first show that the focal segment norm is bounded by
A3 , and then the worst-case optimal strategy is to spread the budget
evenly over the segment.
To simplify notations, we consider the case ωy0 = 0 since the general proof is similar after a basis change. For this case, the 1D focal
segment is a horizontal line of the form k̂ωx0 ,y0 (ωu , 0), shown in the
central row of Figure 3. For a fixed value of ωx0 , this line is the
Fourier transform of:
ZZZ
k(x, y, u, v)e−2iπ (ωx0 x+0y+0v) dxdydv .
(12)
By showing that the total power of Eq. (12) is bounded by A3 , Parseval’s theorem gives us the same bound for the focal segment.
Since the exposure time is assumed to be 1, we collect unit energy
through every u, v point lying within the clear aperture1 :
ZZ
1
|u| ≤ A/2, |v| ≤ A/2
k(x, y, u, v)dxdy =
. (13)
0
otherwise
1 If an amplitude mask is placed at the aperture (e.g., a coded aperture)
the energy will be reduced and the upper bound still holds.
Camera type
Squared MTF
a. Upper bound
|φ̂s (ωx,y )|2 ≤
b. Standard lens
|φ̂s (ωx,y )|2 = A4 sinc2 (A(s − s0 )ωx )sinc2 (A(s − s0 )ωy )
c. Coded aperture
E[|φ̂s (ωx,y )|2 ] ≈
A3
S|ωx,y |
ε 2 A4
2
sinc2 (ε A(s − s0 )ωx )sinc2 (ε A(s − s0)ωy )
d. Focus sweep
|φ̂s (ωx,y )|2 ≈
A2 α (ωx,y )2
S2 |ωx,y |2
e. Wavefront coding
|φ̂s (ωx,y )|2 ≈
A2
S2 |ωx ||ωy |
f. Lattice-focal
E[|φ̂s (ωx,y )|2 ] ≈
A8/3 β (ωx,y )
S4/3 Ω1/3 |ωx,y |
Table 2: Squared MTFs of computational imaging designs. See
Table 1 for notation. The optimal spectrum bound falls off linearly
as a function of spatial frequency, yet existing designs such as the
focus sweep and wavefront coding fall off quadratically and do not
utilize the full budget. The new lattice-focal lens derived in this
paper achieves a higher spectrum, closer to the upper bound.
A phase change to the integral in Eq. (13) does not increase its
magnitude, therefore, for every spatial frequency ωx0 ,y0 ,
ZZ
−2iπ (ωx0 x+ωy0 y)
(14)
dxdy ≤ 1 .
k(x, y, u, v)e
Using Eq. (14) and the fact that the aperture is width A along on the
v-axis, we obtain:
2
ZZZ
−2iπωx0 x+0y+0v
≤ A2 .
(15)
dxdydv
k(x,
y,
u,
v)e
On the u-axis, the aperture has width A as well. By integrating
Eq. (15) over u we see the power is bounded by A3 :
2
Z ZZZ
−2iπ (ωx0 x+ωy0 y)
du ≤ A3 .
k(x,
y,
u,
v)e
(16)
dxdydv
Since the left-hand side of Eq. (15) is the power spectrum of
k̂ωx0 ,y0 (ωu , 0), by applying Parseval’s theorem we see that the total power over the focal segment is bounded by A3 as well:
Z
|k̂ωx0 ,y0 (ωu , 0)|2 d ωu ≤ A3 .
(17)
Since the focal segment norm is bounded by A3 , and since we aim
to maximize the worst-case magnitude, the best we can do is to
spread the budget uniformly over the length S|ωx0 ,y0 | focal segment,
which bounds the worst MTF power by A3 /S|ωx0 |. In the general
case, Eq. (16) is bounded by β (ωx,y )A3 rather than A3 , and Eq. (10)
follows.
3.3
Analysis of existing designs
We analyze the spectra of existing imaging designs with particular
attention paid to the spectrum on the focal manifold since it is the
portion of the spectrum that contributes to focus at physical depths.
Figure 4 visualizes ωx0 ,y0 -slices through a 4D lens spectrum |k̂| for
recent imaging systems. Figure 5 shows the corresponding MTFs
(OTF-slices) at a few depths. A low spectrum value at a point on
the focal segment leads to low spectrum content at the OTF of the
corresponding depth. Examining Figures 4 and 5, we see that some
designs spend a significant portion of the budget on afocal regions.
The MTFs for the previous designs shown in Figure 5 are lower than
the upper bound. We have analytically computed spectra for these
designs. The derivation is provided in appendix A and summarized
in Table 2. We observe that no existing spectrum reaches the upper
bound. Below we review the results in Table 2b-e and provide some
intuitive arguments. In the next section we introduce a new design
whose spectrum is higher than all known designs, but still does not
fully meet the bound.
Standard lens: For a standard lens focused at depth s0 we see
in Figure 4(b) high frequency content near the isolated points
k̂ωx0 ,y0 (−s0 ωx0 , −s0 ωy0 ), which correspond to the in-focus depth
φ̂s0 . The spectrum falls off rapidly away from these points, with
a sinc whose width is inversely proportional to the aperture. When
the deviation between the focus slope and the object slope |s0 − s|
is large, this sinc severely attenuates high frequencies.
The coded aperture [Levin et al. 2007; Veeraraghavan et al. 2007] incorporates a pattern blocking light rays.
The integration surface is linear, like that of a standard lens, but has
holes at the blocked areas. Compared to the sinc of a standard aperture, the coded aperture camera has a broader spectrum (Fig. 4(c)),
but is still far from the bound. To see why, assume w.l.o.g. that
the lens is focused at s0 = 0. The primal integration surface lies
on the x = 0, y = 0 plane and k̂ is constant over all ωx,y . Indeed,
all ωx0 ,y0 -slices in Figure 4(c) are equal. Since the union of focal
segment orientations from all ωx0 ,y0 -slices covers the plane, to guarantee worst-case performance, the coded aperture spectrum should
be spread over the entire 2D plane of each ωx0 ,y0 -slice. This implies
significant energy away from focal segments.
Coded aperture:
For a focus sweep camera [Hausler 1972; Nagahara et al. 2008], the focus distance is varied continuously during
exposure and the 4D lens spectrum is the average of standard lenses
spectra over a range of slopes s0 (Figs. 4(d) and 5(d)). In contrast to
the isolated points covered by a static lens, this spreads energy over
the entire focal segment, since the focus varies during exposure.
This design does not spend budget away from the focal segment of
interest. However, as discussed in appendix A, since the lens kernel
describing a focus sweep camera is not a Dirac delta, phase cancellation occurs between different focus settings and the magnitude is
lower than the upper bound (Fig. 4(a)).
Focus sweep:
The integration surface of a wavefront
coding lens [Dowski and Cathey 1995] is a separable 2D
parabola [Levin et al. 2008b; Zhang and Levoy 2009]. The spectrum is a separable extension of that of the 1D parabola [Levin et al.
2008c]. However, while the 1D parabola achieves an optimal worstcase spectrum, this is no longer the case for a 2D parabola in 4D,
and the wavefront coding spectrum (Table 2e, Figs. 4(e) and 5(e))
is lower than the bound. The ωx0 ,y0 -slices in Figure 4(e) reveal
why. Due to the separability, energy is spread uniformly within the
minimal rectangle bounding the focal segment. For another perspective, consider the wavefront coding integration surface in the
primal domain, which is a separable parabola c(u, v) = (au2 , av2 ).
A local planar approximation to that surface around an aperture
point u0 , v0 is of the form c(u, v) = (su u, sv v), for su = ∂∂cux = 2au0 ,
Wavefront coding:
∂c
sv = ∂ vy = 2av0 . For u0 6= v0 the lens is locally astigmatic, and
as discussed in Sec. 3.1, this is an afocal surface. Thus, the only
focal part of the wavefront coding lens is the narrow strip along its
diagonal, where u0 = v0 .
Still, the wavefront coding spectrum is superior to that of coded
apertures at low-to-mid frequencies. It spreads budget only within
the minimal rectangle bounding the focal segment, but not up to the
maximal cutoff spatial frequency. The wavefront coding spectrum
and that of a focus sweep are equal if |ωx | = |ωy |. However, the
wavefront coding spectrum is significantly improved for |ωx | → 0
(b) standard lens, focused at s0 = 0.5
(a) upper bound
(c) coded aperture, focused at s0 = 0
Ω
Ω
Ω
0.5Ω
0.5Ω
0.5Ω
0
0
0
−0.5Ω
−0.5Ω
−0.5Ω
−Ω
ωy
ωx
−Ω
−Ω
−0.5Ω
0
0.5Ω
Ω
ωy
ωx
−Ω
−Ω
(d) focus sweep
−0.5Ω
0
0.5Ω
Ω
ωy ωx
(e) wave front coding
Ω
Ω
0.5Ω
0.5Ω
0.5Ω
0
0
0
−0.5Ω
−0.5Ω
−0.5Ω
−Ω
ωy
ωx
−Ω
−Ω
−0.5Ω
0
0.5Ω
Ω
ωy
ωx
−0.5Ω
0
0.5Ω
Ω
0.5Ω
Ω
(f) lattice-focal
Ω
−Ω
−Ω
−Ω
−0.5Ω
0
0.5Ω
Ω
ωy ωx
−Ω
−0.5Ω
0
Figure 4: 4D lens spectrum for different optical designs. Each subplot is an ωx0 ,y0 -slice as described in Figure 3. In the flatland case of
Figure 2, these ωx0 ,y0 -slices correspond to vertical columns. An ideal design (a) should account for the dimensionality gap and spend energy
only on the focal segments. Yet, this bound is not reached by any existing design. A standard lens (b) devotes energy only to a point in
each subplot. A coded aperture (c) is more broadband, but its spectrum is constant over all ωx0 ,y0 -slices, so it cannot cover only the focal
segment in each ωx0 ,y0 -slice. The focus sweep camera (d) covers only the focal segments, but has reduced energy due to phase cancellations
and does not achieve the bound. A wavefront coding lens (e) is separable in the ωu , ωv directions and spends significant energy on afocal
areas. Our new lattice-focal lens (f) is an improvement over existing designs, and spreads energy budget over the focal segments. Note that
all subplots show the numerical simulation of particular design instances, with parameters for each design tuned to the depth range (see
Sec. 5.1), approximating the analytic spectra in Table 2. The intensity scale is constant for all subplots.
s
(a)
upper
bound
(b)
standard
s0 = 0.5
(c)
coded
s0 = 0
(d)
focus
sweep
(e)
wavefront
coding
(f)
latticefocal
1
0.5
or |ωy | → 0, because the rectangle becomes compact, as shown in
the central row and column of Figure 4(e).
In appendix B we also analyze the plenoptic camera and the focal stack imaging models. Note that despite all the sinc patterns
mentioned so far, the derivation in this section and the simulations
in Figures 4 and 5 model pure geometric optics. Diffraction and
wave optics effects are also discussed in appendix C. In most cases
Fourier optics models lead to small adjustments to the spectra in
Table 2, and the spectra are scaled by the diffraction-limited OTF.
0
−0.5
−1
Figure 5: Spectra of OTF-slices for different optical designs over
a set of depths. The subplots represent the MTF of a given imaging
system for slope s, |φ̂s (ωx , ωy )|, where the subplot axes are ωx,y .
These OTF-slices are the 2D analog of the slanted red and blue
slices in Figure 2. Our new lattice-focal lens design best approximates the ideal spectrum upper bound. Note that all subplots show
the numerical simulation of particular design instances, with parameters for each design tuned to the depth range (see Sec. 5.1),
approximating the analytic spectra in Table 2.
Having reviewed several previous computational imaging approaches to extending depth of field, we conclude that none of them
spends the energy budget in an optimal way. In a standard lens the
entire aperture area is focal, but light is focused only from a single depth. A wavefront coding lens attempts to cover a full depth
range, but at the expense that most aperture area is afocal. In the
next section we propose a new lens design, the lattice-focal lens,
with the best attributes of both—all aperture area is focal, yet it
focuses light from multiple depths. This lets our new design get
closer to the upper bound compared to existing imaging systems.
4 The lattice-focal lens
Motivated by the previous discussion, we propose a new design,
which we call the lattice-focal lens. The spectrum it achieves is
higher than previous designs but still lower than the upper bound.
In this design, the aperture is divided into 1/ε 2 subsquares of
Lattice-Focus
x−plane
u
x
u−plane
(aperture plane)
sensor plane
Figure 6: Left: Ray mapping for a lattice-focal lens in flatland. The
aperture is divided into three color-coded sections, each focused on
a different depth. Right: In the 2D light field the integration surface
is a set of slanted segments, shown with corresponding colors.
(a) Lattice-focal lens
(a) Lattice-focal lens
(b) PSFs
Figure 7: (a) Toy lattice-focal lens design with only 4 subsquares.
(b) The PSFs φs in the primal domain, at two different depths. Each
subsquare (color-coded) corresponds to a box in the PSF. The width
of each box is a function of the deviation between the subsquare
focal depth and the object depth.
size ε A × ε A each (for 0 < ε < 1). Each subsquare is a focal element cropped from a standard lens focused at some slope
s j ∈ [−S/2, S/2]. That is, the integration surface is defined as:
c(u, v) = (s j u, s j v) for (u, v) ∈ W j ,
(18)
where W j denotes the area of the j-th subsquare. Figure 6 visualizes the integration surface of a lattice-focal lens, composed of
linear surfaces with different slopes (compare with Figure 2, left).
Figure 7 illustrates a toy four-element lattice-focal lens and its PSF
for two different depths. In the primal domain, the PSF is a superposition of scaled and shifted boxes corresponding to the various
aperture subsquares. For this example, one of the subsquares is focused at the correct depth for each scene depth, so the PSF consists
of an impulse plus three defocused boxes. The box width is a function of the deviation between the lens focal depth and the object
depth.
The OTF φ̂s (ωx , ωy ) of a lattice-focal lens is a sum of sincs corresponding to the different subsquares:
∑ ε 2 A2 e2π i(γ j,x ωx +γ j,y ωy ) sinc ε Aωx (s j − s) sinc ε Aωy(s j − s) .
j
(19)
For a subsquare centered at aperture point (u j , v j ), (γ j,x , γ j,y ) =
(u j (s j − s), v j (s j − s)) denotes the phase shift of the j-th subsquare,
corresponding to its translated center. The 4D spectrum of a single
aperture subsquare is a sinc around one point in the focal segment:
k̂ωx0 ,y0 (−s j ωx0 , −s j ωx0 ). However since each subsquare is focused
at a different slope s j the summed spectra cover the focal segment
(Figure 4(f)). In contrast to the spectrum for wavefront coding, the
lattice-focal spectrum does not spend much budget away from the
focal manifold. This follows from the fact that the subsquare slopes
in Eq. (18) are set to be equal in u and v, therefore the entire aperture
area is focal.
The lattice-focal design resembles the focus sweep in that both
distribute focus over the DOF—focus sweep over time, and the
lattice-focal design over aperture area. The crucial difference is
that since each lattice-focal subsquare is smaller than the full aper-
(b) Discrete focus sweep
Figure 8: Focus sweep vs. the lattice-focal lens. (a) Lattice-focal
lens whose aperture is divided into 3 differently-focused bins. (b)
Discrete focus sweep, dividing the integration time into 3 bins, each
focusing on a different depth (note that an actual focus sweep camera varies focus continuously). Depth ranges with defocus diameter
below a threshold are colored. While in both cases each bin lets in
1/3 of the energy, the sub-apertures for the lattice-focal lens are
narrower than the full aperture used by the focus sweep, hence the
effective DOF for each of the lattice-focal bins is larger.
ture, its effective DOF is larger than the DOF for the full aperture (Figure 8). As shown in Fig. 4(d,f) and Fig. 5(d,f), the
lattice-focal lens achieves significantly higher spectra than focus
sweep. Mathematically, by discretizing the exposure time into N
bins, each bin of the focus sweep (focused at slope s j ) contributes
A2 sinc(A(s − s )ω )sinc(A(s − s )ω ) to the OTF. By contrast, by
j y
j x
N
dividing the aperture into N bins, each bin of the lattice-focal lens
2
contributes AN sinc(AN −1/2 (s − s j )ωx )sinc(AN −1/2 (s − s j )ωy ). In
both cases each bin collects 1/N of the total energy (and the sincs’
height is A2 /N), but the lattice-focal sinc is wider. While coincidental phase alignments may narrow the sincs, these alignments
occur in isolation and do not persist across all depths and all spatial
frequencies. Therefore, the lattice-focal lens has a higher spectrum
when integrating over s j .
The ωx0 ,y0 -slices in Figure 4(f), and the OTF-slices in Figure 5(f)
suggest that the lattice-focal lens achieves a higher spectrum compared to previous designs. In the rest of this section we develop an
analytic, average-case approximation for the lattice-focal spectrum,
which enables order-of-magnitude comparison to other designs. We
then discuss the effect of window size ε and show it is a critical parameter of the construction, and implies a major difference between
our design and previous multi-focus designs [George and Chi 2003;
Ben-Eliezer et al. 2005].
The spectrum of a particular lattice focal lens can be computed numerically (Eq. (19)), and
Figures 4 and 5 plot such a numerical evaluation. However, to
allow an asymptotic order-of-magnitude comparison between lens
designs we compute the expected spectrum over random choices of
the slopes s j and subsquare centers (u j , v j ) in Eq. (18) (note that to
simplify the proof, the subsquares in a generic random lattice-focal
are allowed to overlap and to leave gaps in the aperture area). Given
sufficiently many subsquares, the law of large numbers applies and
a sample lattice-focal lens resembles the expected spectrum. While
this analysis confers insight, the expected spectrum should not be
confused with the spectrum of a particular lattice-focal lens. The
spectrum of any particular lattice-focal instance is not equal to the
expected one.
Spectrum of the lattice-focal lens:
Claim 3 Consider a lattice-focal lens whose subsquare slopes
in Eq. (18) are sampled uniformly from the range [−S/2, S/2],
and subsquares centers sampled uniformly over the aperture area
j
Proof: Let s denote a particular scene depth of interest and let φ̂s
denote the OTF of the j-th subsquare focused at slope s j , so that
j
the lattice-focal OTF is φ̂s = ∑ j φ̂s . For a subsquare size of ε A ×
ε A, the aperture area is covered by m = 1/ε 2 subsquares. Since
j
the m random variables φ̂s are drawn independently from the same
distribution, it follows that
j
j
E[|φ̂s |2 ] = mE[|φ̂s |2 ] + m(m − 1)|E[φ̂s ]|2 .
(21)
The second term in Eq. (21) is positive, and one can show it is
small relative to the first term. For simplicity we make the conj
servative approximation E[|φ̂s |2 ] ≈ mE[|φ̂s |2 ], and show how to
j 2
compute E[|φ̂s | ] below. Note that the exact lattice-focal spectrum
(Eq. (19), and the right-hand side of Eq. (21)) involves interference
from the phase of each subsquare. An advantage of our approximation mE[|φ̂sj |2 ] is that it bypasses the need to model phase precisely.
Recall that the PSF from each subsquare is a box filter and the OTF
is a sinc. If the j-th subsquare is focused at s j ,
j
|φ̂s (ωx,y )|2 = ε 4 A4 sinc2 (ε Aωx (s − s j ))sinc2 (ε Aωy (s − s j )) . (22)
Since the subsquare slopes are drawn uniformly from [−S/2, S/2],
the expected spectrum is obtained by averaging Eq. (22) over s j .
j
E[|φ̂s |2 ] =
ε 4 A4
S
Z S/2
−S/2
sinc2 ε Aωx (s j − s) sinc2 ε Aωy (s j − s) ds j .
(23)
To compute this integral we make use of the following identity: for
a 2D vector r = (r1 , r2 ),
Z ∞
−∞
sinc2 (r1t)sinc2 (r2t)dt =
β (|r|)
.
|r|
(24)
If −S/2 < s < S/2 and S is large, we can assume that the integration
boundaries of Eq. (23) are sufficiently large2 , and asymptotically
approximate Eq. (23) with the unbounded integration of Eq. (24):
ε 4 A4
j
E[|φ̂s |2 ] =
S
ε 4 A4
=
S
≈
Z S/2
−S/2
sinc2
Z S/2+s
−S/2+s
ε Aωx (s j − s) sinc2 ε Aωy (s j − s) ds j
sinc2 ε Aωx s j sinc2 ε Aωy s j ds j
ε 3 A3 β (ωx,y )
.
S|ωx,y |
(25)
Eq. (20) now follows from Eq. (25), after multiplying by the number of subsquares, m = ε12 .
2 Note that the approximation in Eq. (25) is reasonable for |ω |,|ω | >
x
y
(Sε A)−1 . The approximation is crude at the low frequencies but becomes
accurate at higher frequencies, for which the MTF approaches the desired
fall off. Furthermore, note that at the exact integration boundaries (s =
±S/2) one gets only half of the contrast. Thus, in practice, one should set S
a bit higher than the actual depth range to be covered.
defocus diameter
defocus diameter
where β is defined in Eq. (11).
Defocus diameter
defocus diameter
(20)
(a) Undersampled
ε > ε∗
ε A3
β (ωx,y ) ,
S|ωx,y |
(b) Optimal
ε = ε∗
E[|φ̂s (ωx , ωy )|2 ] ≈
Expected spect. Particular spect.
(c) Redundant
ε < ε∗
[−A/2, A/2] × [−A/2, A/2]. For |ωx |, |ωy | > (ε SA)−1 , the expected
power spectrum asymptotically approaches
2
1.5
1
0.5
0
100
120
100
120
100
120
140
160
180
140
160
180
140
160
180
depth
2
1.5
1
0.5
0
depth
2
1.5
1
0.5
0
depth
Figure 9: The lattice-focal lens with varying window sizes. Left:
ωx0 ,y0 -slice at ωx = 0.9Ω, ωy = −0.9Ω, through the expected spectrum. Middle: ωx0 ,y0 -slice from a particular lattice-focal lens instance. Right: The defocus diameter over the depth of field. The
expected spectrum improves when the windows number is reduced,
but every particular lattice-focal lens becomes undersampled and
does not cover the full depth range.
According to Claim 3, the expected
power spectrum of a lattice-focal lens increases with window size
ε (Fig. 9). For larger subsquares the sinc blur around the central
focal segment is narrower, so more energy is concentrated on the
focal segment. However, it is clear that we cannot make ε arbitrarily large. When the number of subsquares is small, the expected
power spectrum is high, but there are not enough samples to cover
the full focal segment (Figure 9(a)). On the other hand, when the
number of subsquares is too large, every subsquare has wide support around the main focal segment, leading to lower energy on the
focal segment (Fig. 9(c)).
Optimal subsquare size:
Posed another way, each subsquare is focused at a different point
in the depth range, and provides reasonable coverage over the subrange of depths for which it achieves a defocus diameter of less
than 1 pixel (Fig. 9, rightmost column). The subsquares’ arrangement is undersampled if the minimum defocus diameter for some
depth range is above 1 pixel, and redundant when the subsquares’
effective depth coverage overlap. In the optimal arrangement each
depth is covered by exactly one subsquare.
We derive the minimal number of windows providing full coverage
of the depth of field, resulting in an optimal ε ∗ .
Claim 4 The maximal subsquare size which allows full spectrum
coverage is
ε ∗ = (ASΩ)−1/3 .
(26)
Proof: If the spacing between spatial samples is ∆ the maximal frequency we need to be concerned with is ΩS/2 = S/(4∆).
For window size ε we obtain 1/ε 2 subsquares. If the slopes of
the subsquares are equally spaced over the range [−S/2, S/2], the
spacing between samples in the frequency domain is τ = ΩSε 2 .
Using subsquares of width ε A, we convolve the samples with
sinc(ε Aωx )sinc(ε Aωy ). For full coverage, we thus require ε A ≤
1/τ , implying:
SΩε 2 ≤
1
εA
⇒
ε ≤ (ASΩ)−1/3 .
(27)
If we plug the optimal ε ∗ from Eq. (26) into Eq. (20) we conclude
that the expected power spectrum of a lattice-focal lens with optimal window size is:
E[|φ̂s (ωx , ωy )|2 ] ≈
A8/3
S4/3 Ω1/3 |ωx,y |
β (ωx,y ) .
The lattice-focal lens with an optimal window size achieves the highest power spectrum (i.e., closest to the upper bound) among all computational imaging designs
listed in Table 2. While the squared MTFs for wavefront coding
and focus sweep fall off quadratically as a function of ωx,y , for
the lattice-focal lens the squared MTF only falls off linearly. Furthermore, while the squared MTFs for wavefront coding and focus
sweep scale with A2 , for the lattice-focal lens the squared MTF
scales with A8/3 . Still, there exists a gap of (ASΩ)1/3 between the
power spectrum of the lattice-focal lens and the upper bound. It
should be noted that the advantage of the lattice-focal lens is asymptotic and is most effective for large depth of field ranges. When the
depth range of interest is small the difference is less noticeable, as
demonstrated below.
From the above discussion, the aperture area should be divided more or less equally into
elements focused at different depths. However, beyond equal area
we also want the aperture regions focused at each depth to be
grouped together. Eq. (20) indicates that the expected power spectrum is higher if we use few wide windows, rather than many small
ones. This can shed some light on earlier multi-focus designs. For
example, [George and Chi 2003] use annular focus rings, and [BenEliezer et al. 2005] use multiplexed subsquares, but multiple nonadjacent subsquares are assigned the same focal length. In both
cases, the support of the aperture area focused at each depth is not
at all compact, leading to sub-optimal MTFs.
Compact support in other designs:
Experiments
We first perform a synthetic comparison between extended depth of
field approaches. We then describe a prototype construction of the
lattice-focal lens and demonstrate real extended-DOF images.
5.1
Small depth range (S = 0.1)
Wavefront coding
Lattice-focal
(28)
Discussion of lens spectra:
5
Large depth range (S = 2)
Wavefront coding
Lattice-focal
Simulation
We start with a synthetic simulation using spatially-invariant first
order (paraxial) optics. The OTFs in this simulation are computed
numerically with precision, and do not rely on the approximate formulas in Table 2 .
Our simulation uses A = 1000∆ and considers two depth of field
ranges given by S = 2 and S = 0.1. Assuming a planar scene,
we synthetically convolved an image with the PSF of each design
adding i.i.d. Gaussian noise with standard deviation η = 0.004.
Non-blind deconvolution was performed using Wiener filtering and
the results are visualized in Figures 10 and 11. We set the free parameters of each design to best match the depth range—for example, we adjust the parabola width a (in Eq. (5)), and select the optimal subsquare size of the lattice-focal lens. The standard and coded
lenses were focused at the middle of the depth range, at s0 = 0.
In Figure 10 we simulate the effect of varying the depth of the object. Using cameras tuned for depth range S = 2, we positioned
the planar object at s = 0 (Fig. 10, top row) and s = −0.9 (Fig. 10,
bottom row). As expected, higher spectra improve the visual quality of the deconvolution. Standard and coded lenses obtain excellent reconstructions when the object is positioned at the focus slope
s = 0, but away from the focus depth the image deconvolution cannot recover much information. Focus sweep, wavefront coding and
the lattice-focal lens achieve uniform reconstruction quality across
depth. The best reconstruction is obtained by our lattice-focal PSF,
Figure 12: ωx0 ,y0 -slice (at ωx0 = 0.9Ω, ωy0 = −0.9Ω) for two depth
ranges defined by slope bounds S = 2 (left) and S = 0.1 (right). For
the smaller range, the difference between the focal segment and the
full bounding square is lower, and the spectra for wavefront coding
and the lattice-focal lens are more similar.
followed by wavefront coding, then focus sweep. Note that since
we use a square aperture, several imaging systems have more horizontal and vertical frequency content. This leads to horizontal and
vertical structure in the reconstructions of Figure 10, particularly
noticeable in the standard lens and the wavefront coding results.
In Figure 11 we simulate the effect of varying the depth range. The
planar object was positioned at s = −0.5, and the camera parameters were adjusted to cover a narrow depth range S = 0.1 (Fig. 11,
top row) and a wider range S = 2 (Fig. 11, second row). When the
focus sweep, wavefront coding and lattice-focal lens are adjusted
to a narrower depth range their performance significantly improves,
since they now distribute the same budget over a narrower range.
The difference between the designs becomes more critical when the
depth range is large. Figure 12 visualizes a ωx0 ,y0 -slice for both S
values. For S = 0.1, the length of the focal segment is so short
that there is little difference between the segment and its bounding
square. Thus, with a smaller depth range the wavefront coding lens
incurs less of penalty for spending its budget on afocal regions.
Assume that the
camera has sensor resolution ∆0 = 0.007mm, and that we use an
f = 85mm focal length lens focused at depth do = 70cm. This
depth also specifies the location of the xy light field plane. The DOF
is defined by the range [dmin , dmax ] corresponding to slopes ±S/2.
From Eq. (2), the depth range can be expressed as do /(1 ± S/2),
yielding a DOF of [35, ∞]cm for S = 2 and [66.2, 74.3]cm for
S = 0.1. The pixel size in the light field is ∆ = ∆0 /M, where
M = f /(do − f ) = 0.13 is the magnification. We set the effective
aperture size A to 1000∆ = 1000∆0 /M = 50.6mm, which corresponds to f /1.68. The subsquares number and focal lengths are
selected such that for each point in the depth range, there is exactly one subsquare achieving defocus diameter of less than one
pixel. The subsquare number is given by Eq. (26), in this simulation m = 100 aperture subsquares with S = 2, and m = 16 subsquares with S = 0.1. To set the focal lengths of each subsquare we
select m equally spaced slopes s j in the range [−S/2, S/2]. A slope
s j is mapped to a physical depth d j according to Eq. (2). To make
the j-th subsquare focus at depth d j we select its focal length f j according to the Gaussian lens formula: 1/ f j = 1/d j + 1/ds (where
ds denotes the sensor-to-lens distance).
Mapping slope ranges to physical distances:
5.2
Implementation
To demonstrate our design we have
built a prototype lattice-focal lens. Our construction provides a
proof of concept showing that a lattice-focal lens can be implemented in practice and lead to reasonably good results, however
it is not an optimized or fully-characterized system.
Hardware construction:
As shown in Figure 13, our lattice-focal lens mounts to a main
lens using the standard threaded interface for a lens filter. The subsquares of the lattice-focal lens were cut from BK7 spherical planoconvex lens elements using a computer-controlled saw. The squares
are of size 5.5 × 5.5mm and thickness 3mm. By attaching our
Standard Lens
Coded aperture
Focus sweep
Wavefront coding
Lattice-focal
Figure 10: Synthetic comparison of image reconstruction at different object depths Top row: object depth s = 0, Bottom row: object depth
s = −0.9 Standard and coded lenses produce high quality reconstruction for an object at the focus depth, but a very poor one away from
the focus depth. Focus sweep, wavefront coding and the lattice focal lens perform equally across depth. The highest quality reconstruction
produced by our lattice-focal lens.
Standard Lens
Coded aperture
Focus sweep
Wavefront coding
Lattice-focal
Figure 11: Synthetic comparison of image reconstruction when camera parameters are adjusted for different depth ranges. Top row: narrow
depth range bounded by S = 0.1, Bottom row: wider range bounded by S = 2. Most designs improve when they attempt to cover a narrower
range. The difference between the designs is more drastic at large depth ranges.
lattice-focal lens to a high-quality main lens (Canon 85mm f1.2L),
we reduce aberrations. Since most of the focusing is achieved by
the main lens, our new elements require low focal powers, and correspond to very low-curvature surfaces with limited aberrations (in
our prototype, the subsquare focal lengths varied from 1m to 10m).
In theory the lattice-focal element should be placed in the plane of
the main lens aperture or at one of its images, e.g., the entrance or
exit pupils. To avoid disassembling the main lens to access these
planes, we note that a sufficiently narrow stop in front of the main
lens redefines a new aperture plane. This lets us attach our latticefocal lens at the front, where the stop required to define a new aperture still let us use 60% of the lens diameter.
The minimal subsquare size is limited by diffraction. Since a
normal lens starts being diffraction-limited around an f /12 aperture [Goodman 1968], we can fit about 100 subsquares within an
f /1.2 aperture. To simplify the construction, however, our prototype included only 12 subsquares. The DOF this allowed us to
cover was small and, as discussed in Sec. 5.1, in this range the
lattice-focal lens advantage over wavefront coding is limited. Still,
our prototype demonstrates the effectiveness of our approach.
Given a fixed budget of m subsquares of a given width, we can
invert the arguments in Sec. 4 and determine the DOF it can
cover in the optimal way. As discussed at the end of Sec. 5.1
and illustrated in Figure 9(b), for every point in the optimal
DOF, there is exactly one subsquare achieving defocus diameter of less than 1 pixel. This constraint also determines the
focal length for each of these subsquares. For our prototype
we focused the main lens at 180cm and chose subsquare focal
lengths covering a depth range of [60, 180]cm. Given the limited
availability of commercial plano-convex elements, our subsquares’
coverage was not perfectly uniform, and we used focal lengths of
10000,5000,4000,3000,2500,2000,1750,1500,1300,1200,1000mm,
plus one flat subsquare (infinity focal length). However, for a
custom-manufactured lens this would not be a limitation.
To calibrate the lattice-focal lens, we used a planar
white noise scene and captured a stack of 30 images for different
depths of the scene. Given a blurred and sharp pair of images Bd , Id
at depth d, we solved for the kernel φd minimizing |φd ⊗ Id − Bd |.
We show the recovered PSF at 3 depths in Figure 13. As discussed
in Sec. 4, the PSF is a superposition of boxes of varying sizes, but
Calibration:
Input
Contrast-adjusted/
Deconvolved
Standard lens, f /16
Standard lens, f /4
Lattice-focal lens
150cm
90cm
Figure 14: Comparison between a lattice-focal lens and a standard lens, both for a narrow aperture ( f /16) and for the same aperture size
as our lattice-focal lens prototype ( f /4). All photos were captured with equal exposure time, so the f /16 image is very noisy. The standard
f /4 image is focused at the white book, but elsewhere produces a defocused image. The lattice-focal output is sharper over the entire scene.
the wrong PSF leads to convolution error, we can locally score the
explanation provided by PSF φd around pixel i as:
Ei (d) = |Bi − B̃d,i |2 + λ ρ (gx,i (Id )) + ρ (gy,i (Id ) ,
(30)
where B̃d = φd ⊗ Id 3 . We regularize the local depth scores using
a Markov random field (MRF), then generate an all-focus image
using the Photomontage algorithm of Agarwala et al. [2004].
In Figure 14 we compare the reconstruction using our
lattice-focal lens with a standard lens focused at the middle of the
depth range (i.e., the white book). Using a narrow aperture ( f /16),
the standard lens produces a very noisy image, since we held exposure time constant over all conditions. Using the same aperture
size as our prototype ( f /4), the standard lens resolves a sharp image of the white book, but the rest of the scene is defocused. For
the purpose of comparison, we specified the depth layers manually
and deconvolved both the standard and lattice-focal images with
PSFs corresponding to the true depth. Because the spectrum of
the lattice-focal lens is higher than a standard lens across the depth
range, greater detail can be resolved after deconvolution.
180cm
Results:
Figure 13: Our prototype lattice-focal lens and PSFs calibrated at
three depths. The prototype attaches to the main lens like a standard lens filter. The PSFs are a sum of box filters from the different
subsquares, where the exact box width is a function of the deviation
between the subsquare focal depth and the object depth.
the exact arrangement of boxes varies with depth. For comparison,
we did the same calibration using a standard lens as well.
Given the calibrated per-depth PSFs, we deblur an image using sparse deconvolution [Levin et al. 2007]. This
algorithm computes the latent image Id as
Id = arg min |φd ⊗ I − B|2 + λ ∑ ρ (gx,i (I)) + ρ (gy,i (I)) , (29)
Depth estimation:
I
i
where gx,i , gy,i denote horizontal and vertical derivatives of the i-th
pixel, ρ is a robust function, and λ is a weighting coefficient.
Since the PSF varies over depth, rough depth estimation is required
for deblurring. If an image region is deconvolved with a PSF corresponding to the incorrect depth, the result will include ringing
artifacts. To estimate depth, we start by deconvolving the entire
image with the stack of all depth-varying PSFs, and obtain a stack
of candidate deconvolved images {Id }. Since deconvolution with
Figure 15 shows all-focus images and depth maps captured using
our lattice-focal lens. More results are available online4 . Since
the MRF of Agarwala et al. [2004] seeks invisible seams, the layer
transitions usually happen at low-texture regions and not at the actual contours. Despite the MRF’s preference for piecewise-constant
depth structures we handle continuous depth variations, as shown in
the rightmost column of Figure 15.
The results in Figure 15 were obtained fully automatically. However, depth estimation can fail, especially next to occlusion boundaries, which present a general problem for all computational
extended-DOF systems [Dowski and Cathey 1995; Nagahara et al.
2008; Levin et al. 2007; Veeraraghavan et al. 2007]. While a principled solution to this problem is beyond the scope of this paper, most
artifacts can be eliminated with simple manual layer refinement.
3 Note that despite the discussion in [Levin et al. 2009], we employ a
MAPx,k approach that scores a depth d based on the best Id explanation
alone. The reason this approach works here is that a delta explanation is absent from the search space, and there is a roughly equal volume of solutions
around all PSFs φd .
4 www.wisdom.weizmann.ac.il/˜levina/papers/lattice
Standard lens
Lattice-focal lens
Figure 15: Partially defocused images from a standard lens, compared with an all-focused image and depth map produced by the lattice-focal
lens.
Figure 16: Synthetic refocusing using the coarse depth map estimated with the lattice-focal lens.
Relying on depth estimation to decode an image from a lattice-focal
lens is a disadvantage compared to depth-invariant solutions, but it
also allows coarse depth recovery. In Figure 16 we used the rough
depth map to synthetically refocus a scene post exposure.
6
Discussion
This paper analyzes extended depth of field systems in light field
space. We show that while effective extended DOF systems seek
high spectrum content, the maximal possible spectrum is bounded.
The dimensionality gap between the 4D light field and the 3D focal
manifold is a key design factor, and to maximize spectrum content
lenses should concentrate their energy in the focal manifold of the
light field spectrum. We analyze existing computational imaging
designs and show that some do not follow this principle, while others do not achieve a high spectrum over the depth range. Guided by
this analysis we propose the lattice-focal lens, accounting for the
dimensionality gap. This allows us to achieve defocus PSFs with
higher spectra compared to previous designs.
However, the lattice-focal lens does not fully achieve the upper
bound. One open question is whether better designs exist, whether
the upper bound could be tighter, or both. Our intuition is that the
upper bound could be tighter. The proof of Claim 2 is based on
the assumption that an A × A primal support is devoted to every frequency point. However, the fact that the integration surface has to
“cover” a full family of slopes implies that the aperture area has
to be divided between all slopes. Thus the primal support of each
slope is much smaller than A, which implies a wider frequency sup-
port around the focal segment, reducing the height of the spectrum
on the focal segment itself.
We have focused on spectra magnitude, which dominates the deconvolution quality. However, the accuracy of depth estimation is
important as well. Wavefront coding and focus sweep cameras have
an important advantage that they bypass the need to estimate depth.
On the other hand, the lattice-focal lens has the benefit of recovering
a rough depth map in addition to an all-focused image. One future
research question is whether the higher spectrum of the lattice-focal
lens can also be achieved with a depth-invariant design.
Acknowledgments: We thank the Israel Science Foundation,
the Royal Dutch/Shell Group, NGA NEGI-1582-04-0004, MURI
Grant N00014-06-1-0734, NSF CAREER award 0447561. F. Durand acknowledges a Microsoft Research New Faculty Fellowship
and a Sloan Fellowship. S. Hasinoff acknowledges the NSERC
PDF program.
Appendix A: Spectra derivations
Below we complete the budget and spectra derivation of Sec. 3.
Claim 5 For an aperture of size A × A and exposure length 1, the
total energy in each ωx0 ,y0 -slice is bounded by A2 :
ZZ
|k̂ωx0 ,y0 (ωu , ωv )|2 d ωu d ωv ≤ A2 .
(31)
Proof: The proof reviews the budget proof in [Levin et al. 2008c].
Note that k̂ωx0 ,y0 (ωu , ωv ) is the 2D Fourier transform of:
ZZ
k(x, y, u, v)e−2iπ (ωx0 x+ωy0 y) dxdy .
(32)
shift (resulting from the translation of the subsquare center):
k̂ j (ωx , ωy , ωu , ωv ) = ε 2 A2 e−2iπ (ωu u0 +ωv v0 ) sinc(ε Aωu )sinc(ε Aωv ) .
(39)
As in the proof of Claim 3, we note that E[k̂ j ] affects very low
frequencies only, so we use Eq. (21) to approximate
For every clear aperture point |u| ≤ A/2, |v| ≤ A/2:
2 ZZ
2
ZZ
−2iπ (ωx0 x+ωy0 y)
≤
≤1.
dxdy
k(x,
y,
u,
v)dxdy
k(x,
y,
u,
v)e
(33)
Where the first inequality follows from the fact that a phase change
does not increase magnitude, and the second inequality follows
from the unit energy through every clear aperture point (see also
Eqs. (13) and (14)).
Since the aperture size is A2 , the total norm is bounded by A2 :
2
ZZ ZZ
−2iπ (ωx0 x+ωy0 y)
dudv ≤ A2 .
dxdy
(34)
k(x,
y,
u,
v)e
By Parseval’s theorem, the square integral is the same in the dual
and the primal domains, thus:
ZZ
|k̂ωx0 ,y0 (ωu , ωv )|2 d ωu d ωv ≤ A2 .
(35)
E[|k̂|2 ]
1
E[|k̂ j |2 ]
ε2
ε 2 A4
sinc2 (ε Aωu )sinc2 (ε Aωv ) ,
2
≈
=
(41)
where the number of subsquares is 1/ε 2 and the factor 1/2 represents the probability of a blocked subsquare. By selecting an OTFslice, Eq. (38) follows.
Eq. (41) suggests that, ignoring diffraction, the sensor spatial resolution implies a tradeoff in selecting the optimal hole size. If
we use small holes, the power spectrum of the aperture is wider,
and wider spectrum implies that more budget is spread away from
the main focal segment (indeed Eq. (38) shows that the expected
spectrum is multiplied by ε and decreases when ε is small). On
the other hand, the expected power spectrum of k̂ falls off like
sinc2 (ε Aωu )sinc2 (ε Aωv ). That is, since the lens is focused only
at a single depth, to have spectral content at slopes corresponding
to other depths, the spectrum of the aperture must be sufficiently
wide, implying that a small hole size ε is needed.
Focus sweep:
Standard lens:
Claim 6 The power spectrum of a standard lens focused at depth
s0 with aperture A × A is
Claim 8 For |ωx |, |ωy | > (SA)−1 , the power spectrum of the focus
sweep camera asymptotically approaches
|φ̂s (ωx , ωy )|2 = A4 sinc2 (A(s − s0 )ωx )sinc2 (A(s − s0 )ωy ) . (36)
Proof: A lens focused at slope s0 is modeled by a linear integration
surface c(u, v) = (s0 u, s0 v). If the surface were infinite, the Fourier
transform would be an impulse at ωu = −s0 ωx , ωv = −s0 ωy . Given
the finite aperture we need to convolve that with a sinc, and thus
k̂(ωx , ωy , ωu , ωv ) = A2 sinc(A(ωu − s0 ωx ))sinc(A(ωv − s0 ωy )) .
(37)
Eq. (36) follows by selecting an OTF-slice.
The ωx0 ,y0 -slices in Figure 4(b) reveals a sinc around the point
ωu = −s0 ωx , ωv = −s0 ωy . Note that reducing the aperture size
A increases the sinc width and minimizes defocus blur. However,
given a fixed exposure length it also reduces the amount of light collected, which reduces the MTF. Indeed, the sinc height in Eq. (36)
decreases for smaller A values.
A coded aperture is constructed with a standard
lens, w.l.o.g. focused at slope s0 = 0. We construct a coded aperture
by dividing the aperture into squares of size ε A × ε A and randomly
blocking each subsquare with probability 1/2. The expected power
spectrum can then be computed analytically.
|φ̂s |2 ≈
A2 sinc(Aωx (s0 − s))sinc(Aωy (s0 − s)) .
(44)
The spectrum of a focus sweep is obtained by averaging Eq. (44)
over s0 . To compute this integral we make use of the following
identity: for a 2D vector r = [r1 , r2 ],
Z ∞
−∞
sinc(r1t)sinc(r2t)dt =
α (|r|)
.
|r|
(45)
If −S/2 < s < S/2 and S large enough5 , we use Eq. (45) and get:
φ̂s (ωx,y )
=
=
(38)
Proof: We express k̂ = ∑ j k̂ j where k̂ j is the 4D spectrum of an
individual subsquare. For an unblocked hole centered at u0 , v0 we
can express k̂ j analytically as the transform of a box times a phase
(42)
Proof: The spectrum of a standard lens focused at slope s0 is
Claim 7 For a lens focused at s0 = 0, the expected power spectrum
of a random coded aperture with holes size ε A × ε A is
ε 2 A4
sinc2 (ε Asωx )sinc2 (ε Asωy ) .
2
A2 α (ωx,y )2
,
S2 |ωx,y |2
where
√ α (|ωx,y |) is a bounded multiplicative factor in the range
[1, 2]:
|ωx,y |
.
(43)
α (|ωx,y |) =
max(|ωx |, |ωy |)
Coded aperture:
E[|φ̂s (ωx,y )|2 ] ≈
(40)
→
A2
S
Z S/2
−S/2
sinc(Aωx (s0 − s))sinc(Aωy (s0 − s))ds0
A2 S/2+s
sinc(Aωx s0 )sinc(Aωy s0 )ds0
S −S/2+s
Aα (ωx,y )
.
S|ωx,y |
Z
Taking the power of Eq. (46) provides Eq. (42).
5 The
approximation is reasonable for |ωx |,|ωy | > (SA)−1 .
(46)
Figure 4(d) displays ωx0 ,y0 -slices from the power spectrum of a
focus sweep camera. On one hand, this spectrum is concentrated around the main focal segment, with the same narrow width
achieved by the upper bound in Fig. 4(a). However, the magnitude
of the focus sweep is significantly lower. In fact, the total energy at
every ωx0 ,y0 -slice is much lower than the budget of Claim 5, that is:
ZZ
for example, [Levin et al. 2009]):
Iˆest (ωx,y )
=
j
=
|k̂ωx0 ,y0 (ωu , ωv )|2 d ωu d ωv ≪ A2 .
(47)
To understand why, recall that the upper bound in Claim 5 is obtained by noting that when x, y are integrated, the magnitude of the
projection integral is bounded by 1 (Eq. (33)). And indeed, when
the 4D lens kernel is a delta function of u, v, that integral is equal
to 1. By contrast, the effective 4D kernel for a focus sweep camera is the average of standard-lens 4D kernels over all depths, and
therefore is not a delta function. When such a non-delta kernel is
multiplied by a wave of the form e−2iπ (ωx x+ωy y) , interference and
phase cancelations significantly reduce the magnitude of the integral, and Eq. (33) is far below 1.
Wavefront coding:
Claim 9 For a slope s ∈ [−S/2, S/2], the power spectrum of a
wavefront coding lens asymptotically approaches
|φ̂s (ωx , ωy )|2 ≈
A2
.
S2 |ωx ||ωy |
ˆ ωx,y )|2
|I(
1
ˆ ωx,y ) − B̂ j (ωx,y )|2 +
|φ̂ j (ωx,y )I(
η2
σ2
1
φ̂ (ω )∗ B̂ j (ωx,y )
η 2 ∑ j j x,y
1
|φ̂ (ω )|2 + σ12
η 2 ∑ j j x,y
.
(52)
One can also compute the expected reconstruction error (expectaˆ ωx,y ), n j samples from the prior):
tion over all I(
ˆ ωx,y )| ] =
E[|Iˆest (ωx,y ) − I(
2
1
1
|φ̂ j (ωx,y )|2 + 2
η2 ∑
σ
j
!−1
. (53)
Eq. (53) states that the reconstruction error is a function of the sum
of power spectra of the individual OTFs. In the following analysis we evaluate two multiple measurement configurations, the focal
stack and the plenoptic camera, by examining the summed power
spectrum ∑ j |φ̂ j |2 .
Note that this suggests how to merge multiple independent measurements of the same spatial frequency, and should not be confused with summing multiple elements of a single measurement.
All other cameras considered in the main body of this paper have a
single measurement per spatial frequency.
(48)
With additive noise and Wiener filtering, the reconstruction error for a focal stack depends on the summed power spectra of all images in the stack. Suppose we divide exposure time into
N bins and spread the focus setting of all time bins evenly over the
depth range. The OTF of a time bin j focused at slope s j is
Focal stack:
Proof: A wavefront coding lens is a cubic refractive element (as
reported in [Dowski and Cathey 1995]). From Snell’s law, the
integration surface is determined by the lens normal. Therefore
the integration surface is a separable parabola c(u, v) = (au2 , av2 ).
The parabola width a can be set such that the parabola slope covers the slope range of interest [−S/2, S/2], implying a = S/(2A).
The power spectrum of a 1D parabola as computed in [Levin et al.
2008c] is
A
δ
.
(49)
|k̂(ωx , ωu )|2 ≈
S|ωx | |ωu |<S/2|ωx |
The 2D parabola case is a separable extension:
|k̂(ωx , ωy , ωu , ωv )|2 ≈
arg min ∑
A2
. (50)
δ
δ
S2 |ωx ||ωy | |ωu |<S/2|ωx | |ωv |<S/2|ωy |
If s ∈ [−S/2, S/2], we can slice Eq. (50) to get Eq. (48).
φ̂s (ωx,y ) =
A2
sinc Aωx (s j − s) sinc Aωy (s j − s) .
N
(54)
By substituting ε = 1 into Eq. (25), and multiplying by 1/N 2 to
account for the shorter exposure, the integration over all slopes s j
provides
1 A3
E[|φ̂s (ωx,y )|2 ] = 2
.
(55)
N S|ωx,y |
Since we have N such bins, the total sum is
1
A3
∑ |φ̂ j,s (ωx,y )|2 = N S|ωx,y | .
(56)
j
Appendix B: Cameras with multiple measurements
Note that the number of bins N does not cancel out in the integration. By adjusting Claim 4, the minimal number of bins required
to cover the depth range without aliasing is N = ASΩ, resulting in
summed power spectrum of
Many imaging models in the literature, like a focal stack, or a
plenoptic camera involve multiple measurements per pixel. Analyzing these cameras requires additional assumptions about the noise
model. One possible analysis is described below.
∑ |φ̂ j,s (ωx,y )|2 = S2 Ω|ωx,y | .
We consider additive Gaussian noise at the sensor and Wiener reconstruction, with a Gaussian zero-mean prior on the signal. That
is, we assume N different measurements of each spatial image frequency, and denote the measurement j as
ˆ ωx,y ) + n j .
B̂ j (ωx,y ) = φ̂ j (ωx,y )I(
(51)
ˆ ωx,y ) variance
We denote the noise variance by η 2 , and the signal I(
with σω2 x,y . The N-measurement Wiener estimate Iˆest (ωx,y ) is (see,
A2
(57)
j
As we have seen in Eq. (53), the reconstruction error is a function
of the summed power spectrum. From Eq. (57) the summed power
spectrum of a focal stack is lower than the focus sweep spectrum
of A2 /(S2 |ωx,y |2 ), since |ωx,y | < Ω. Thus, under additive noise,
the reconstruction error of a focus sweep camera which measures
different focus depths simultaneously is better than the focal stack
approach which measures each focus depth separately.
The MTF of a plenoptic camera cannot be
compared directly to the other cameras because the imaging model
Plenoptic camera:
is different and involves tradeoffs between spatial and angular resolution. As a result, assumptions about the sensor layout of the
plenoptic camera must also be made. One possible analysis is
described below, which concludes that reconstruction error for a
plenoptic camera is higher than for a lattice-focal lens. Intuitively,
the plenoptic camera captures the entire 4D light field and hence
spends energy away from the focal manifold.
Assume other cameras use N sensor elements. To keep the same
spatial resolution in the final image we assume the plenoptic camera
has N lenslets and 1/ε 2 elements below each lenslet, resulting in
N/ε 2 sensor elements in total. We assume the lens is focused at the
middle of the depth range, at slope s0 = 0.
This lens is then expressed as
R(Au)R(Av)e−iπ
u2 +v2
λf
(63)
where R is a rect function and λ is the wavelength. A general refractive element is expressed relative to this lens as
R(Au)R(Av)e−iπ
u2 +v2
λf
ϕ (u, v)
(64)
where ϕ (u, v) is the aberration, or the deviation from a normal lens
behavior. We also denote by ϕA (u, v) = R(Au)R(Av)ϕ (u, v) the
aberration multiplied by the aperture.
The OTF from the j-th directional view is:
φ̂sj (ωx,y ) = e2π i(γ j,x ωx +γ j,y ωy ) ε 2 A2 sinc(ε Asωx )sinc(ε Asωy ) , (58)
where (γ j,x , γ j,y ) denotes the phase shift of the j-th view.
Recall that the reconstruction error in Eq. (53) is a function of the
sum of power spectra of the individual OTFs. For the case of a
plenoptic camera,
∑ |φ̂ j (ωx,y )|2 = ε 2 A4 sinc2 (ε Asωx )sinc2 (ε Asωy )
(59)
j
In our derivation we study the classical ambiguity function, the
Fourier dual of the Wigner distribution. The ambiguity function
of ϕA is defined as
A (u, v, su , sv ) =
ZZ
ξu,v (t1 ,t2 )ei2π (sut1 +sv t2 ) dt1 dt2 .
(65)
where
ξu,v (t1 ,t2 ) = ϕA (t1 + u/2,t2 + v/2)ϕA∗ (t1 − u/2,t2 − v/2).
(66)
since there are 1/ε 2 terms in the sum.
This suggests that there is a tradeoff in the number of directional
views: more directional views increase the noise contribution and
the reconstruction error. By contrast, if we do not divide the directional views finely enough we have defocus in the depth range.
To avoid defocus over the depth range, the worst-case slope S and
the worst-case frequency Ω should be included in the first sinc lobe,
hence
1
.
(60)
ε≤
ASΩ
This implies that
A2
(61)
∑ |φ̂ j (ωx,y )|2 ≤ S2 Ω2
j
at all spatial frequencies ωx,y , and the plenoptic camera will have
increased noise contribution compared to most single-measurement
cameras we have analyzed.
Appendix C: Fourier optics derivation
Our goal in this appendix is to derive an analogy to the light field
results, while modeling wave optics effects and not only geometric
optics. We use a traditional optics representation- the Wigner distribution [Wigner 1932], and build on the tight relation between light
fields and Wigner distribution derived by Zhang and Levoy [2009].
We will show that an analogy to the budget bounds exist, but while
the geometric optics bound allocates an equal budget at all spatial frequencies, Fourier optics implies that the bound drops as the
diffraction limited OTF. Yet, if the system dimensions are kept well
above the diffraction limit, the approximation provided by the geometric model is reasonable enough.
In our derivation we consider a lens with an A×A aperture and focal
length f , we denote the distance from the lens to the camera sensor
by di and the focus depth by do . By the Gaussian lens formula,
1
1
1
+ = .
do di
f
(62)
We now define a 4D analog of the 4D lens spectrum as
˜ ω , ω , ω , ω ) = A (ρω , ρω , ω , ω )
k̂(
x y u v
x
y u v
(67)
where ρ = λ do .
It is well known in optics that OTFs can be obtained as slices from
the ambiguity function [Brenner et al. 1983]. Below we provide a
revised derivation of this result adjusted to our notation. That is, we
show that the Fourier optics OTF is a slice through k̂˜ exactly in the
same way that the geometric optics OTF is a slice through k̂. We
later show that we can place upper bounds on the norm of k̂˜ in the
same way we have specified bounds the norm of k̂.
Claim 10 Consider an object at depth da and its light field slope
s = (da − do )/da . The Fourier optics OTF φ̂s (ωx,y ) is a slice
˜
through k̂:
˜ ω , ω , −sω , −sω ).
φ̂ (ω ) = k̂(
(68)
s
x,y
x
y
x
y
Proof: If we are imaging an on axis point at distance da , the wavefront reaching the lens is
ei2π
da
λ
eiπ
u2 +v2
λ da
(69)
The wavefront leaving the lens is a product of the lens with the wave
reaching the lens.
A wavefront reaching a point (x′ , y′ ) on the image plane (the image
plane is located at distance di from the lens) is thus the product of
the wave leaving the lens with a spherical wavefront centered at
(x′ , y′ , di ), which reduces to
h(x′ , y′ )
=
ZZ
eiπ
u2 +v2
λ da
ei2π λ e
ϕA (u, v)eiπ (1/da +1/di −1/ f )
u2 +v2
λ
e
da
λ
′ 2
′ 2
)
iπ (u−x ) λ+(v−y
d
di
i
dudv
′2 +y′2
λ di
′
′
v
−i2π x u+y
λd
i
dudv
Claim 11 For every spatial frequency ωx0 ,y0 the norm of a ωx0 ,y0 slice k̂˜ ωx0 ,y0 is bounded by the aperture area times the diffraction
limited OTF:
′2 +y′2
λ di
iπ x
=1 ei2π (di +da )/λ e
ZZ
ei2π
iπ x
= ei2π (di +da )/λ e
ZZ
u2 +v2
λf
ϕA (u, v)e−iπ
−iπ s u
ϕA (u, v)e
2 +v2
ρ
′
′
v
−i2π x u+y
λd
e
i
dudv
Now let
2 2
−iπ s u +v
ρ
ψs (u, v) = ϕA (u, v)e
,
and let Ψs denote its Fourier transform. Then
′
x
y′
′ ′
h(x , y ) = Ψs
eiπ ·const .
,
λ di λ di
(72)
The incoherent PSF is now equal to
φs (x) = |hs (x)| = |Ψs (x/ρ )|
2
k̂˜ ωx0 ,y0 (ωu , ωv ) =
ZZ
ξωx0 ,y0 (t1 ,t2 )ei2π (ωu t1 +ωv t2 ) dt1 dt2 .
(78)
That is, k̂˜ ωx0 ,y0 is the Fourier transform of ξωx0 ,y0 . Therefore we can
use Parseval’s theorem to bound the norm of k̂˜ ωx0 ,y0 :
ZZ
=
≤
|k̂˜ ωx0 ,y0 (ωu , ωv )|2 d ωu d ωv
ZZ
ZZ
|ξωx0 ,y0 (t1 ,t2 )|2 dt1 dt2
R(A(t1 + ρωx0 /2))R(A(t1 − ρωx0 /2))
(79)
· R(A(t2 + ρωy0 /2))R(A(t2 − ρωy0 /2))dt1 dt2
= A Λ A (ωx0 )Λ A (ωy0 )
2
ρ
ρ
= A H (ωx0 , ωy0 ).
2
(74)
φ̂s (ωx , ωy )
=
(77)
2
The OTF φ̂s can thus be obtained from the auto correlation of ψ ,
scaling the coordinates by ρ , which reduces to the definition of k̂˜ in
Eq. (67), since:
ZZ
ψ (t1 + ρωx /2,t2 + ρωy /2)ψ ∗ (t1 − ρωx /2,t2 − ρωy /2)dt1 dt2
ZZ
ξρωx,y (t1 ,t2 )ei2π (ωut1 +ωv t2 ) dt1 dt2
ZZ
|k̂˜ ωx0 ,y0 |2 ≤ A2 H (ωx0 , ωy0 ).
Proof: We have seen in the proof of Claim 10 that
(71)
To map x′ to the spatial coordinates in the rest of this document, we
note that we used coordinates on the xy-plane, and not on the image.
That is, we need to divide by the magnification factor M = di /do ,
and transfer to the coordinate system x = x′ /M,y = y′ /M. That is:
x y
hs (x, y) = hs (x′ /M, y′ /M) = Ψs
,
ei·const
(73)
ρ ρ
=
ZZ
(70)
where 1 follows from the Gaussian lens formula 1/do +1/di = 1/ f ,
implying that the spherical wave coefficient above can be expressed
as 1/λ (1/da + 1/di − 1/ f ) = 1/λ (1/da − 1/do ) = −s/ρ .
=
1969], which can be translated to energy bounds on k̂˜ ωx0 ,y0 [FitzGerrell et al. 1997]. We provide a revised derivation of this result adjusted to our notation. In an analogy to the geometric optics derivation, the norm of a ωx0 ,y0 -slice through k̂˜ is bounded. However,
the more accurate Fourier optics bound is no longer equal across
spatial frequencies, it is falling off like the diffraction limited OTF
H (ωx0 , ωy0 ). This result also suggests that if the system dimensions are kept well above the diffraction limit, the approximation
provided by the geometric model is reasonable enough.
ξρωx,y (t1 ,t2 )ei2π s(ωx t1 +ωy t2 ) dt1 dt2
˜ ω , ω , ω , ω ).
= k̂(
x y u v
(75)
We now let H (ωx , ωy ) denote the diffraction limited OTF of the
system
H (ωx , ωy ) = ΛA/ρ (ωx )ΛA/ρ (ωy )
(76)
where Λ(ω /c) is a width c tent function, and we use the shortcut
notation Λc (ω ) = Λ(ω /c).
The previous claim states that the Fourier optics OTF is a slice
through the ambiguity function. The radars literature has also derived the energy of slices via the ambiguity function [Rihaczek
We have seen that using the Fourier optics model, the budget bound
is decreasing as the diffraction limited OTF. To understand how specific computational imaging designs are affected, we briefly review
the spectrums derivation in the previous section and describe the
required adjustments for the Fourier optics model. In a nutshell, for
most cases Fourier optics implies that the results in Table 2 should
x
be multiplied by the diffraction limit OTF and A → AΛ ρω
A . For
designs with holes (coded aperture and lattice-focal lens),
the hole
x
size reduces the diffraction limit and ε A → ε AΛ ρω
ε A . Therefore,
if the aperture and hole sizes are kept well above the diffraction
limit, the approximation provided by the geometric optics model is
good enough.
The simplest case is that of a standard lens. The diffraction limited OTF of a squared aperture is derived in many Fourier optics
textbooks, e.g., [Goodman 1968]. We briefly review this in the following claim.
Claim 12 The Fourier optics squared MTF of a standard lens focused at s0 is
|φ̂s (ωx , ωy )|2 = A4 Λ2A (ωx )sinc2 AΛ A (ωx )(s − s0 )ωx
ρ
ρ
2
2
·Λ A (ωy )sinc AΛ A (ωy )(s − s0 )ωy . (80)
ρ
ρ
Proof: The derivation above considered a reference lens with focal
length f focused at depth do , and with sensor to lens distance di
such that 1/ f = 1/do + 1/di . We now consider another lens with
focal length fa . We maintain the lens to sensor distance di . This
lens is then focused at depth da such that 1/ fa = 1/da + 1/di . If
s0 is the light field slope at that depth then s0 = do (1/do − 1/da ) =
do (1/ f − 1/ fa ). The aberration ϕ for this lens is
ϕ (u, v) = e−iπ
u2 +v2
λ
(1/ fa −1/ f )
−iπ s0 u
=e
2 +v2
ρ
|φ̂s0 (ωx , ωy )|2 ≈ ε A3 Λ2ε A (ωx )Λ2ε A (ωy )
ρ
where
ρ
˜
Eq. (80) now follows by taking a ωx0 ,y0 -slice through k̂.
ρ
′ )
β (ωx,y
.
′ |
S|ωx,y
′
= (Λ ε A (ωx )ωx , Λ ε A (ωy )ωy ).
ωx,y
(81)
and ξu,v (t1 ,t2 ) = R(ΛA (u) tA1 )R(ΛA (v) tA2 )e−i2π s0 (t1 u+t2 v) . Taking the
Fourier transform of ξ and adjusting the coordinate system provides
˜
k̂:
˜ ω , ω , ω , ω ) = A2 Λ A (ω )sinc AΛ A (ω )(ω − s ω )
k̂(
x y u v
x
x
u
0 x
ρ
ρ
·Λ A (ωy )sinc AΛ A (ωy )(ωv − s0 ωy ) (82)
.
ρ
focal lens with holes size ε A × ε A is
ρ
ρ
(89)
(90)
Proof: The squared MTF for a sublens focused at slope s is
|φ̂s0 (ωx , ωy )|2 = ε 4 A4 Λ2ε A (ωx )sinc2 ε AΛ ε A (ωx )(s − s0 )ωx
ρ
ρ
2
2
·Λ ε A (ωy )sinc ε AΛ ε A (ωy )(s − s0 )ωy .(91)
ρ
ρ
As in the proof of Claim 3, the expected squared MTF of a latticefocal lens is obtained by averaging Eq. (91) over s. Using Eq. (24),
this converges to
′ )
β (ωx,y
.
′
S|ωx,y |
Claim 13 For a lens focused at s = 0, the expected Fourier optics
squared MTF of a random coded aperture with hole size ε A × ε A is
|φ̂s (ωx , ωy )|2 = ε 2 A4 Λ2ε A (ωx )sinc2 ε AΛ ε A (ωx )(s − s0 )ωx
ρ
ρ
2
2
·Λ ε A (ωy )sinc ε AΛ ε A (ωy )(s − s0 )ωy . (83)
Multiplying Eq. (92) by the number of subsquares 1/ε 2 produces
Eq. (89).
Proof: Referring to the notation in the proof of Claim 9, we need
to compute E|K̃ j |2 . Based on Claim 12, the expected MTF of an
ε A × ε A aperture hole is
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ρ
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(85)
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