Focused Image Recovery from Two Defocused Images Recorded

Focused Image Recovery from Two Defocused Images Recorded
Focused Image Recovery from Two Defocused
Images Recorded With Different Camera Settings
Murali Subbarao
Tse-Chung Wei
Gopal Surya
Department of Electrical Engineering
State University of New York
Stony Brook, NY 11794-2350
Phone: (516) 632-8405. email: [email protected]
Abstract
Two new methods are presented for recovering the focused image of an object
from only two blurred images recorded with different camera parameter settings. The
camera parameters include lens position, focal length, and aperture diameter. First
a blur parameter is estimated using one of our two recently proposed depth-fromdefocus methods. Then one of the two blurred images is deconvolved to recover the
focused image. The first method is based on a recently proposed Spatial Domain
Convolution/Deconvolution Transform. This method requires only the knowledge of
of the camera’s point spread function (PSF). It does not require information about
the actual form of the camera’s PSF. The second method, in contrast to the first,
requires full knowledge of the form of the PSF. As part of the second method, we
present a calibration procedure for estimating the camera’s PSF for different values of
the blur parameter . In the second method, the focused image is obtained through
deconvolution in the Fourier Domain using the Wiener filter. For both methods, results
of experiments on actual defocused images recorded by a CCD camera are given. The
first method requires much less computation than the second method. The first method
gives satisfactory results for up to medium levels of blur and the second method gives
good results for up to relatively high levels of blur.
Index Terms
Defocus, image restoration, inverse filtering, spatial domain deconvolution
1
1 Introduction
In machine vision, early processing tasks such as edge-detection, image segmentation,
stereo matching, etc. are easier for focused images than for defocused images of threedimensional (3D) scenes. However, the image of a 3D scene recorded by a camera is in
general defocused due to limited depth-of-field of the camera. Autofocusing can be used
to focus the camera onto a desired target object. But, in the resulting image, only the target
object and those objects at the same distance as the target object will be focused. All other
objects at distances other than that of the target object will be blurred. The objects will
be blurred by different degrees depending on their distance from the camera. The amount
of blur also depends on camera parameters such as lens position with respect to the image
detector, focal length of the lens, and diameter of the camera aperture. In this paper, we
address the problem of recovering the focused image of a scene from its defocused images.
We recently proposed two new methods for estimating the distance of objects in a
scene [14, 15] using image defocus information. In these methods, two defocused images
of the scene are recorded simultaneously with different camera parameter settings. The
defocused images are then processed to obtain the distance of objects in the scene in small
image regions. In this process, first a blur parameter which is a measure of the spread
of the camera’s point spread function (PSF) was estimated as an intermediate step. In
this paper we present two methods for using the same blur parameter for recovering the
focused images of objects in the scene from their blurred images. The main contributions
of this paper are summarized below.
The first method of focused image recovery is based on a new spatial domain convolution/deconvolution transform (S transform) proposed in [13]. This method uses only
the blur parameter which is a measure of the spread of the camera’s PSF. In particular,
the method does not require a knowledge of the the exact form of the camera PSF. The
second method, in contrast to the first, requires complete information about the form of the
camera PSF. For most practical camera systems, the camera PSF cannot be characterized
with adequate accuracy using simple mathematical models such as Gaussian or cylindrical
1
functions. A better model is obtained by measuring experimentally the actual PSF of the
camera for different degrees of image blur and using the measured data. This however
requires camera calibration. An alternative but usually a more difficult solution is to derive
and use a more accurate mathematical model for the PSF based on diffraction, lens aberrations, and characteristics of the various camera components such as the optical system,
image sensor elements, frame grabber, etc. As part of the second method, we present a
camera calibration procedure for measuring the camera PSF for various degrees of image
blur. The calibration procedure is based on recording and processing the images of blurred
step edges. In the second method, the focused image is obtained through a deconvolution
operation in the Fourier domain using the Wiener filter.
For both methods of recovering the focused image, results of experiments on an actual
camera system are presented. The results of the first method are compared with the results
obtained using two commonly used PSF models– cylindrical based on geometric optics,
and a 2D Gaussian. The results of the second method are compared with simulation results.
A subjective evaluation of the results leads to the following conclusions. The first method
performs better and is much faster than the methods based on simple PSF models. The
focused image recovery is good for up to medium levels of image blur (upto an effective
blur circle radius of about 5 pixels). The performance of the second method is comparable
to the simulation results. The simulation results represent the best attainable when all noise,
except quantization noise, is absent. The second method gives good results upto relatively
high levels of blur (upto an effective blur circle radius of about 10 pixels). Overall the
second method gives better results than the first, but it requires estimation of the camera’s
PSF through calibration and is computationally several times (about 4 in practice) more
expensive.
In the next section we summarize the two methods for estimating the blur parameter .
In the subsequent sections we describe methods for recovering the focused image using the
blur parameter and experimental details.
2
2 Estimation of Blur Parameter The blur parameter
is a measure of the spread of the camera PSF. For a circularly
symmetric PSF denoted by h(x; y ) it is defined as
2 =
Z 1Z 1
1 1
(x2 + y 2) h(x; y ) dx dy
(1)
For a PSF model based on paraxial geometric optics, it can be shown that the blur parameter
p
is proportional to the blur circle radius. If R is the blur circle radius, then = R= 2. For
a PSF model based on a 2D Gaussian function, is the standard deviation of the distribution
of the 2D Gaussian function.
We recently proposed two depth-from-defocus methods– DFD1F [14] and STM [15].
In both these methods, the blur parameter is first estimated and then the object distance
is estimated based on . In this paper we will not provide details of these methods, but
summarize below some relevant results.
In addition to object distance, the blur parameter depends on other camera parameters
shown in Figure 1. The parameters include– the distance between the lens and the image
detector denoted by s, the focal length
f
D of the camera
aperture. We denote a particular setting for these camera parameters by e = (s; f; D).
Both DFD1F and STM require at least two images, say g1 (x; y ) and g2 (x; y ), recorded
with different camera parameter settings, say e1 = (s1 ; f1; D1 ) and e2 = (s2 ; f2 ; D2 )
of the lens, and the diameter
respectively, such that at least one, but possibly two or all three, of the camera parameters
are different, i.e.
s1 6= s2 or f1 6= f2, or D1 6= D2.
DFD1F and STM also require a
knowledge of the values of the camera parameters e1 and e2 (or a related camera constant
which can be determined through calibration). Using the two blurred images g1 , g2 , the
camera settings (or related camera constants) e1 and e2 , and some camera calibration data
related to the camera PSF, both DFD1F and STM methods estimate the blur parameter .
A Fourier domain method is used in DFD1F whereas a spatial domain method is used in
STM. The methods are general in that no specific model is used for the camera PSF, such
as a 2D Gaussian or a cylindrical function.
Both DFD1F and STM have been successfully implemented on a prototype camera
system named SPARCS. Experimental results on estimating have yielded a root-meansquare (RMS) error of about 3.7% for DFD1F and about 2.3% for STM. One estimate of 3
can be obtained in each image region of size as small as 48 48 pixels. By estimating in
small overlapping image regions, the scene depth-map can be obtained.
In the following sections we describe two methods for using the blur parameter thus
estimated (using DFD1F or STM) to recover the focused image of the scene.
3 Spatial Domain Approach
In this section we describe the spatial domain method for recovering the focused image
of a 3D scene from a defocused image for which the blur parameter has been estimated
using either DFD1F or STM [15]. The recovery is done through deconvolution of the
defocused image using a new Spatial-Domain Convolution/Deconvolution Transform (S
Transform) [13]. The transform itself is general and applicable to n-dimensional continuous
and discrete signals for the case of arbitrary order polynomials. However, a special case
of the general transform will be used in this section. First we summarize the S-Transform
Convolution and Deconvolution formulas that are applicable here and then discuss their
application for recovering the focused image.
3.1 S Transform
Let f (x; y ) be an image which is a two variable cubic polynomial in a small neighborhood,
defined by
f (x; y) =
3 3;
X
Xm
am;nxmyn
(2)
m=0 n=0
where am;n are the polynomial coefficients [3]. Let h(x; y ) be the PSF of a camera. The
moment hm;n of the PSF is defined by
hm;n =
Z1Z1
;1
m y n h(x; y ) dxdy
x
;1
(3)
g(x; y) be the blurred image obtained by convolving the focused image f (x; y) with
the PSF h(x; y ). Then we have
Z1Z1
g(x; y) =
f (x ; ; y ; )h(; ) dd
(4)
Let
;1 ;1
4
By substituting the Taylor series expansion of f in the above relation and simplifying, the
following relation can be obtained:
g(x; y) =
X (;1)m+n m;n
f (x; y)hm;n
0m+n3 m!n!
(5)
Equation (5) expresses the convolution of a function f (x; y ) with another function h(x; y ) as
a summation involving the derivatives of f (x; y ) and moments of h(x; y ). This corresponds
to the forward S-Transform. If the PSF
h(x; y) is circularly symmetric (which is largely
true for most camera systems) then it can be shown that
h0;1 = h1;0 = h1;1 = h0;3 = h3;0 = h2;1 = h1;2 = 0 and h2;0 = h0;2
(6)
Also, by definition, for the PSF of a camera,
h0;0 = 1
(7)
Using these results Equation (5) can be expressed as
g(x; y) = f (x; y) + h22;0 52 f (x; y)
where
52 is the Laplacian operator.
(8)
Taking the Laplacian on both sides of the above
equation and noting that 4-th and higher order derivatives of
f
are zero as
f
is a cubic
polynomial, we obtain
52 g(x; y) = 52f (x; y)
(9)
Substituting the above equation in Equation (8) and rearranging terms we obtain
f (x; y) = g(x; y) ; h22;0 52 g(x; y)
(10)
Equation (10) is a deconvolution formula. It expresses the original function (focused image)
f (x; y) in terms of the convolved function (blurred image) g(x; y), its (i.e. g’s) derivatives,
and the moments of the point spread function h(x; y ). In the general case this corresponds
to Inverse S-Transform [13].
Using the definitions of the moments of
of h, we have h2;0
written as
h and the definition of the blur parameter = h0;2 = 2=2, and therefore the above deconvolution formula can be
2
f (x; y) = g(x; y) ; 4 52 g(x; y)
5
(11)
The above equation suggests a method for recovering the focused image f (x; y ) from the
blurred image
g(x; y) and the blur parameter .
Note that the above equation has been
derived under the following assumptions (i) the focused image
f (x; y) is modeled by a
cubic polynomial (as in Eq. 2) in a small (3 3 pixels in our implementation) image
neighborhood, and (ii) the PSF h(x; y ) is circularly symmetric. These two assumptions are
good approximations in practical applications and yield useful results.
3.2 Advantages
Equation (11) is similar in form to the previously known result that a sharper image can
be obtained from a blurred image by subtracting a constant times the Laplacian of the
blurred image from the original blurred image [11]. However that result is valid only for
a diffusion model of blurring where the PSF is restricted to be a Gaussian. In comparison,
our deconvolution formula is valid for all PSFs that are circularly symmetric including a
Gaussian. Therefore, the previously known result is a special case of our deconvolution.
Further, the restriction on the circular symmetry of the PSF can be removed if desired in
our method of deconvolution using a more general version of the S-Transform [13]. Such
generalization is not possible for the previously known result. In our deconvolution method,
the focused image can be generalized to be an arbitrarily high order polynomial although
such a generalization does not seem useful in practical applications that we know.
The main advantages of this method are (i) the quality of the focused image obtained (as
we shall see in the discussion on experimental results), (ii) computational complexity, and
(iii) the locality of the computations. Simplicity of the computational algorithm is another
characteristic of this method. Given the blur parameter , at each pixel, estimation of
the focused image involves the following operations (a) estimation of the Laplacian which
can be implemented with a few integer addition operations (8 in our implementation), (b)
floating point multiplication of the estimated Laplacian with 2=4, and (c) one integer
operation corresponding to the subtraction in Eq. (11). For comparison purposes in the
following sections, let us say that these computations are roughly equivalent to 4 floating
N N
4N 2 floating point operations
are required. All operations are local in that only a small image region is involved (3 3
point operations. Therefore, for an
6
image, about
in our implementation). Therefore the method can be easily implemented on a parallel
computation hardware.
Next we describe the camera system on which this method of focused image recovery
was implemented, and then we describe the experiments.
3.3 Camera System
All our experiments were performed on a camera system named StonyBrook Passive
Autofocusing and Ranging Camera System (SPARCS). SPARCS consists of a SONY XC77 CCD camera and an Olympus 35-70 mm motorized lens. Images from the camera are
captured by a frame grabber board (Quickcapture DT2953 of Data Translation ) residing
in an IBM PS/2 (model 70) personal computer. The captured images are processed in the
PS/2.
The lens system consists of multiple lenses and focusing is done by moving the front
lens forward and backward. The lens can be moved under computer control using a stepper
motor. The stepper motor has 97 steps, numbered 0 to 96. Step number 0 corresponds to
focusing an object at distance infinity and step number 96 corresponds to focusing a nearby
object, at a distance of about 55cm from the lens.
There is a one-to-one relation between the lens position specified by the step number
of the stepper motor and the distance of an object that would be in best focus for that
lens position. Based on this relationship, we often find it convenient to specify distances
of objects in terms of lens step number rather than in units of length such as meter. For
example, when the “distance” of an object is specified as step number n, it means that the
object is at such a distance D0 that it would be in best focus when the lens is moved to step
number n.
3.4 Experiments
A set of experiments is described in Section 5 where the blur parameter is first estimated
from two blurred images and then the focused image is recovered. In this section we
describe experiments where is assumed to be given.
7
A poster with printed characters was placed at a distance of step 70 (about 80 cms)
from the camera. The focused image is shown in Figure 3. The camera lens was moved
to different positions (steps 70,60,50,40,30 and 20) to obtain images with different degrees
of blur. The images are shown in figures 4a to 9a. The corresponding blur parameters ( s)
for these images were roughly 2.2, 2.8, 3.5, 4.7, 6.0 and 7.2 pixels. These images were
deblurred using equation (11). The results are shown in Figures 4d-9d. We see that the
results are satisfactory for small to moderate levels of blur corresponding to about = 3:5
pixels. This corresponds to about 20 lens steps or a blur circle radius of about 5 pixels.
In order to evaluate the above results through comparison, two standard techniques were
used to obtain focused images. The first technique was to use a two-dimensional Gaussian
model for the camera PSF. The spread parameter of the Gaussian function was taken to be
equal to the blur parameter , and therefore the PSF was:
x 2 +y 2
1 ; 22
hb (x; y) = 2
(12)
2e
The plots of the PSF for two values of corresponding to about 2.7 pixels and 5.3 pixels
are shown in Figure 2.
The focused image was obtained using the Wiener filter [11] specified in the Fourier
domain by:
where
1
jH (!; )j2
M (!; ) = H (!;
) jH (!; )j2 + ;
(13)
H (!; ) is the Fourier Transform of the PSF and ; is the noise-to-signal power
; was approximated by a constant. The constant was
determined empirically through several trials so as to yield best results. Let g (x; y ) be the
blurred image, and f^(x; y ) be the restored focused image. Let their corresponding Fourier
Transforms be G(!; ) and F^ (!; ) respectively. Then the restored image, according to
density ratio. In our experiments
Wiener filtering is
F^ (!; ) = G(!; )M (!; ):
By taking the inverse Fourier Transform of
f^(x; y).
F^ (!; ),
(14)
we can obtain the restored image
The results are shown in Figures 4c-9c. We see that for small values of
(about 3.5
pixels), the Gaussian model performs well, but not as good as the previous method (Figs.
4d-9d). In addition to the quality of the focused image that is obtained, this method has
8
three important disadvantages. The first is computational complexity. For a given , first
H (!; ), and then the Weiner filter M (!; ). It is
possible to precompute and store M (!; ) for later usage for different values of . But
this would require large storage space. After M (!; ) has been obtained for a given ,
we need to compute G(!; ) from g (x; y ) using FFT algorithm, multiply M (!; ) with
G(!; ) to obtain F^ (!; ), and then compute the inverse Fourier transform of F^ (!; ).
The complexity of the FFT algorithm is O(N 2 logN ) for an N N image. Roughly, at
least (2N 2 + 2N 2 log2 N ) floating point operations are involved. For N = 128 used in our
experiments, the number of computations is at least 16N 2 . In comparison, the number of
computations in the previous case was 4N 2 . Therefore, this method is at least 4 times slower
one needs to compute the the OTF
than the previous method. The second disadvantage of this method is that the computations
are not local because of the computation of the Fourier transform of the entire image. The
third disadvantage is the estimation of the noise parameter ;.
In the second standard technique of focused image recovery, the PSF was modeled by
a cylindrical function based on paraxial geometric optics:
8>
< R1 2
ha(x; y) = >
:0
if x2 + y 2 R2
(15)
otherwise.
where R is the radius of the blur circle. The spread parameter corresponding to the above
PSF can be shown to be related to the radius R by the relation R =
p
2 . The plots of the
PSF for two values of of about 2.7 pixels and 5.3 pixels are shown in Figure 2. With
a knowledge of the blur parameter , it is thus possible to use equation (15) and generate
the entire cylindrical PSF. The focused image was again obtained using the Wiener filter
mentioned earlier, but this time using the cylindrical PSF.
In computing the Wiener filter, computation of the discrete cylindrical PSF at the border
of the corresponding blur circle involves some approximations. The value of a pixel which
lies only partially in the blur circle should be proportional to the area of overlap between
the pixel and the blur circle. Violation of this rule leads to large errors in the restored
image, especially for small blur circles. In our implementation, the areas of partial overlap
were computed by resampling the ideal PSF at a higher rate (about 16 times), calculating
the PSF by ignoring the pixels whose center did not lie within the blur circle, and then
downsampling by adding the pixel values in 16 x 16 non-overlapping regions.
9
The results of this case are shown in Figures 4b-9b for different degrees of blur. The
images exhibit “ripples” around the border between the background and the characters.
Once again we see that the results are not as good as for the S transform method. For
low levels of blur (upto about R
= 5 pixels) Gaussian model gives better results than the
cylindrical PSF, and for higher levels of blur (R greater than about 5 pixels) the cylindrical
PSF gives better results than the Gaussian PSF.
In addition to the quality of the final result, the relative disadvantages of this method in
comparison with the S transform method are same as those for the Gaussian PSF model.
4 Second Method
In the second method, the blur parameter is used to first determine the complete PSF. In
practice, the PSF is determined by using as an index into a prestored table that specifies
the complete PSF for different values of . In theory, however, the PSF may be determined
by substituting into a mathematical expression that models the actual camera PSF. Since
it is difficult to obtain a sufficiently accurate mathematical model for the PSF, we use a
prestored table to determine the complete PSF. After obtaining the complete PSF, Wiener
filter is used to compute the focused image. First we describe a method of obtaining the
prestored table through a calibration procedure.
4.1 Camera calibration for PSF
Theoretically, the PSF of a camera can be obtained from the image of a point light source.
However, in practice, it is difficult to create an ideal point light source that is incoherent
and polychromatic. Therefore the standard practice in camera design is to estimate the PSF
from the image of an edge.
f (x; y) be a step edge along the y-axis on the image plane. Let a be the image
intensity to the left of the y -axis and b be the height of the step. The image can be expressed
Let
as
f (x; y) = a + b u(x)
10
(16)
where u(x) is the standard unit step function. If g (x; y ) is the observed image and h(x; y )
is the camera’s PSF then we have,
g(x; y) = h(x; y) f (x; y)
(17)
where * denotes the convolution operation.
Now consider the derivative of g along the gradient direction. Since differentiation and
convolution commute, we have
where
@g = h(x; y) @f
@x
@x
= h(x; y ) b (x)
(18)
(x) is the dirac delta function along the x axis.
The above expression can be
simplified to obtain
(19)
@g = b (x)
@x
(20)
where (x) is the line spread function of the camera defined by
(x) =
Z1
;1
h(x; y)dy
(21)
For any PSF h(x; y ) of a lossless camera, by definition, we have
Z1Z1
;1 ;1
h(x; y) dx dy = 1
(22)
Z 1 @g(x; y)
dx = b
(23)
;1 @x
Therefore, given the observed image g (x; y ) of a blurred step edge, we can obtain the
line spread function (x) from the expression
Therefore we obtain
@g
(x) = R 1 @[email protected]
;1 @x dx
After obtaining the line spread function
(24)
(x), the next step is to obtain the PSF or
its Fourier Transform, which is known as the Optical Transfer Function (OTF). Here we
outline two methods of obtaining the OTF, one assuming the separability of the OTF and
another using Inverse Abel Transform.
11
4.1.1
Separable OTF
Let the Fourier Transforms of the PSF h(x; y ) and LSF (x) be H (!; ) and (! ) respectively. Then we have [11]
(! ) = H (!; 0)
(25)
If the camera has a circular aperture then the PSF is circularly symmetric. If the PSF is
circularly symmetric (and real), then the OTF is also circularly symmetric (and real), i.e.
H (!; ) is also circularly symmetric. Therefore we get
p
H (!; ) = ( !2 + 2)
(26)
Once we have the Fourier Transform of the LSF, (! ), we can calculate
H (!; ) for any
p
values of ! and . However, in practice where digital images are involved, ! 2 + 2 may
have non integer values, and we may have to interpolate (! ) to obtain H (!; ). Due to the
nature of (! ), linear interpolation did not yield good results in our experiments. Therefore
interpolation was avoided by assuming that the OTF to be separable, i.e. H (!; ) =
H (!; 0)H (0; ) = (!)( ). A more accurate method, however, is to use to the Inverse
Abel Transform.
4.1.2
Inverse Abel Transform
In the case of a circularly symmetric PSF h1(r), the PSF can be obtained from its LSF (x)
directly using the Inverse Abel Transform [5] :
Z 1 0(x)
p 2 2 dx
(27)
r
x ;r
p
where 0(x) is the derivative of LSF (x). Note that h(x; y ) = h1 (r) if r = x2 + y 2 . In our
h(r) = ; 1
implementation the above integral was evaluated using a numerical integration technique.
After obtaining H (!; ), the final step in restoration is to use equations (13) and (14)
and obtain the restored image.
12
4.2 Calibration Experiments
All experiments were performed using the SPARCS camera system. Black and white stripes
of paper were pasted on a cardboard to create a step discontinuity in reflectance along a
straight line. The step edge was placed at such a distance (about 80 cms) from the camera
that it was in best focus when the lens position was step 70. The lens was then moved to 20
different positions corresponding to step numbers 0; 5; 10 90; 95. At each lens position,
the image of the step edge was recorded, thus obtaining a sequence of blurred edges with
different degrees of blur. Twelve of these images are shown in Figure 10. The difference
between the actual lens position and the reference lens position of 70 is a measure of image
blur. Therefore, an image blur of +20 steps corresponds to an image recorded at lens
position of step 50 and an image blur of -20 steps corresponds to an image recorded at lens
position of step 90. The size of each image was 80 200.
In our experiments, the step edge was placed vertically and therefore the image intensity
was almost a constant along columns and the gradient direction was along the rows. To
reduce electronic noise, each image was cut into 16 horizontal strips of size 5 200 and
in each strip, the image intensity was integrated (summed) along columns. Thus each strip
was reduced to just one image row. In each row, the first derivative was computed by simply
taking the difference of gray values of adjacent pixels. Then the approximate location of
the edge was computed in each row by finding the first moment of the derivative, i.e., if i is
the column number where the edge is located, and gx (i) is the image derivative at column
i, then
Pi=200 ig (i)
x
i = Pii=1
=200 g (i)
(28)
i=1 x
The following step was included to reduce the effects of noise further. Each row was
traversed on either side of position i until a pixel was reached where either gx (i) was zero
or its sign changed. All the pixels between this pixel (where for the first time, gx became
zero or its sign changed) and the pixel at the row’s end were set to zero. We found this
noise cleaning step to be very important in our experiments. A small non-zero value of
image derivative caused by noise at pixels far away from the position of the edge affects
the estimation of the blur parameter considerably.
13
From the noise-cleaned gx (i), the line spread function was computed as
gx(i)
(i) = Pi=200
i=1 gx (i)
(29)
Eight LSFs corresponding to different degrees of blur are plotted in Figure 11. It can be
seen that, as the blur increases the LSF function becomes more flat and spread out. The
i was then recomputed using equation (28).
central moment of the LSF, l was computed from
v
u
200
uX
l = t (i ; i)2(i)
location of the edge
i=1
The spread or second
(30)
The computed values of l for adjacent strips were found to differ by only about 2 percent.
The average l was computed over all the strips. It can be shown that l is related to the blur
parameter by p
p
= 2l. The effective blur circle radius R is related to by R = 2 .
The values of R computed using the relation R = 2l for different step edges are shown
in Figure 13. Figure 13 also shows the value of
R predicted by ideal paraxial geometric
optics. The values of R obtained for a horizontal step edge are also plotted in the figure.
The values for the vertical and horizontal edges are in close agreement except for very low
degrees of blur. This minor discrepancy may be due to the asymmetric (rectangular) shape
of the CCD pixels (13 11 microns for our camera).
The PSF’s were obtained from the LSFs using the inverse Abel Transform. Cross
sections of the PSFs thus obtained corresponding to the LSFs in Figure 11 are shown in
Figure 12.
4.3 Experimental Results
Using the calibration procedure described in the previous section, the PSFs and the corresponding OTFs were precomputed for different values of the blur parameter . These
results were prestored in a lookup table indexed by . The OTF data H (!; ) in this table
was used to restore blurred images using the Wiener filter M (!; ). Figures 4e-9e show the
results of restoration using the separability assumption for the OTF and Figures 4f-9f are
the results for the case where the inverse Abel transform was used to compute the PSF from
the LSF. Both these results are better than the other results in Figures 4 (b,c,d) - 9 (b,c,d).
14
The method using the inverse Abel transform is better than all the other methods. We find
that the results in this case are good even for highly blurred images. For example, the
images in Figures 8a and 9a are severely blurred corresponding to 40 and 50 steps of blur
or equal to about 6.0 and 7.2 pixels respectively. It is impossible for humans to recognize
the characters in these images. However, in the restored images shown in Figures 8f and 9f
respectively, many of the characters are easily recognizable.
In order to compare the above results with the best obtainable results, the restoration
method which uses the inverse Abel transform was tested on computer simulated image
data. Two sets of blurred images were obtained by convolving an original image with
a Cylindrical and a Gaussian functions. The only noise in the simulated images was
quantization noise. The blurred images were then restored using the Wiener Filter. The
results are shown in Figures 14 and 15. We see that these results are only somewhat better
but not much better than the results on actual data in Figures 4f-9f. This indicates that our
method of camera calibration for the PSF is reliable.
The main advantage of this method is that the quality of the restored image is the best
in comparison with all other methods. It gives good results for even highly blurred images.
It has two main disadvantages. First, it requires extensive calibration work as described
earlier. Second, the computational complexity is the same as that for the Weiner filter
method discussed earlier. For an
N N
image, it requires at least
2N 2 + 2N 2 log2 N
4N 2 floating point operations for the method
based on spatial domain deconvolution. Therefore, for an image of size 128 128, this
floating point operations as compared with
method is at least 4 times slower than the method based on spatial domain deconvolution.
Another disadvantage is that it requires the estimation of the noise parameter
; for the
Wiener filter.
5 Experiments with unknown and 3D object
In the experiments described earlier, the blur parameter of a blurred image was taken to be
known. We now present a set of experiments where is unknown. It is first estimated using
one of the two depth-from-defocus methods proposed by us recently [15]. Then, of the two
15
blurred images, the one that is less blurred is deconvolved to recover the focused image.
Results are presented for both the first method based on spatial-domain deconvolution and
the second method which uses inverse Abel transform.
The results are shown in Figures 16a-d. The first image in Fig. 16a is the focused
image of an object recorded by the camera. The object was placed at a distance of step
14 (about 2.5 meters) from the camera. Two images of the object were recorded with
two different lens positions–steps 40 and 70 (see Fig. 16a). The blur parameter
was
estimated using the depth-from-defocus method proposed in [15]. It was found to be about
5.5 pixels. Using this, the results of restoring the image recorded at lens step 40 is shown
in Fig. 16a. Similar experiments were done by placing the object at distances steps 36, 56,
and 76 corresponding to 1.31, 0.9 and 0.66 meters from the camera. In each of these cases,
the focused image, the two recorded image at steps 40 and 70, and the restored images are
shown in Figs. b-d. The blur parameters in the three cases were about 1.79, 1.24, and
2.35 pixels respectively. In the last two cases, the images recorded at lens step 70 was less
blurred than the the one recorded at step 40. Therefore the image recorded at lens step 70
was used in the restoration.
In another experiment, a 3D scene was created by placing three planar objects at three
different distances. Two images of the objects were recorded at lens steps 40 and 70. These
images are shown in Figure 17. It can be seen that different image regions are blurred by
different degrees. The image was divided into 9 regions of size 128 x 128 pixels. In each
region the blur parameter was estimated and the image in the region was restored. The
nine different estimated values of are 3.84, 4.76, 4.76, 0.054, 0.15, 0.46 (for image with
lens step 40) and -2.65, -2.55 and -2.55 (for image with lens step 70) respectively. The
different restored regions were combined to yield an image, where the entire image looks
focused. Figure 17 shows the results using both the first and second methods of restoration.
Currently each region can be as small as 48 x 48 pixels, which is a small region in the entire
field of view of 640 x 480 pixels.
In the next experiment, a planar object with posters was placed inclined to the optical
axis. The nearest end of the object was about 50 cms from the camera and the farthest end
was about 120 cms. The blurred images of the object acquired with lens steps 40 and 70
are shown in Figure 18(a) and (b). The images were divided into non-overlapping regions
16
of 64 x 64 pixels and a depth estimate was obtained for each region. The different regions
were then restored separately as before and combined to yield the restored images as shown
in Figure 18(c) and (d). The restored images appear better than either of the blurred images.
However there are some blocking artifacts, which are due to the “wrap around” problem of
the FFT algorithm and the finite filter size in the case of the S -Transform method.
6 Conclusion
The focused image of an object can be recovered using two defocused images recorded with
different camera parameter settings. The same two images can used to estimate the depth
of the object using a depth-from-defocus method proposed by us [14, 15]. For a 3D scene
where the depth variation is small in image regions of size about 64 64, each image region
can be processed separately and the results can be combined to obtain both a focused image
of the entire scene and a rough depth-map of the scene. If, in each image region, at least
one of the two recorded defocused images is blurred only moderately or less (
<= 3:5
pixels), then the focused image can be recovered very fast (computational complexity of
O(N 2) for an N N image) using the new spatial domain deconvolution method described
here. In most practical applications of machine vision, the camera parameter setting can
be arranged so that this condition holds, i.e. in each image region at most only one of the
two recorded defocused images is severely blurred (
> 3:5 pixels). In those cases where
this condition does not hold, the second method which uses the inverse Abel transform
can be used to recover the focused image. This method requires camera calibration for the
PSF and is several times more computationally intensive than the first method above. The
methods in this paper can be used as part of a 3D machine vision system to obtain focused
images from blurred images for further processing such as edge detection, stereo matching,
and image segmentation.
References
17
[1] J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, John Wiley & Sons,
New York, 1978.
[2] P. Grossman, “Depth from focus”, Pattern Recognition Letters 5, pp. 63–69, Jan. 1987.
[3] R. M. Haralick and L. G. Shapiro, Computer and Robot Vision, Addison-Wesley Publishing Company, 1992, Ch. 8.
[4] B. K. P. Horn, “Focusing”, Artificial Intelligence Memo No. 160, MIT, 1968.
[5] B. K. P. Horn, Robot Vision, McGraw-Hill Book Company, 1986, page 143.
[6] J. Ens and P. Lawrence, “A Matrix Based Method for Determining Depth from Focus”,
Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern
Recognition, June 1991.
[7] E. Krotkov, “Focusing”, International Journal of Computer Vision, 1, 223-237, 1987.
[8] S. K. Nayar, “Shape from Focus System” Proceedings of the IEEE Computer Society
Conference on Computer Vision and Pattern Recognition, Champaign, Illinois, pp
302-308 June 1992.
[9] P. Meer and I. Weiss, Smoothed differentiation filters for images, Tech. Report No.
CS-TR-2194, Center for Automation Research, University of Maryland, College Park,
MD 20742-3411.
[10] A. P. Pentland, “A new sense for depth of field”, IEEE Transactions on Pattern
Analysis and Machine Intelligence, Vol. PAMI-9, No. 4, pp. 523–531.
[11] A. Rosenfeld, and A. C. Kak, Digital Picture Processing, Vol. I . Academic Press,
1982.
[12] M. Subbarao, and G. Natarajan, “Depth recovery from blurred edges”, Proceedings of
the IEEE Computer Society Conference on Computer Vision and Pattern Recognition,
Ann Arbor, Michigan, pp. 498-503, June 1988.
[13] M. Subbarao, “Spatial-Domain Convolution/Deconvolution Transform ”, Tech. Report No. 91.07.03, Computer Vision Laboratory, Dept. of Electrical Engineering, State
University of New York, Stony Brook, NY 11794-2350.
18
[14] T. Wei, Three Dimensional Machine Vision Using Image Defocus, Ph.D Thesis, Dept.
of Electrical Engg., State University of New York at Stony Brook, Dec. 1994.
[15] G. Surya, Three Dimensional Scene Recovery from Image Defocus, Ph.D Thesis, Dept.
of Electrical Engg., State University of New York at Stony Brook, Dec. 1994.
19
ID
SCENE
s
u
L
P
D/2
f
Optical Axis
Q
p’
p 2R
v
L: Lens
P: Object
ID: Image Detector
Q: Optical Center
p’: Focused Point
p: Blur Circle
s-v
D: Aperture Diameter
f: Focal Length
R: Blur Circle Radius
Fig. 1 Image Formation in a Convex Lens
19
0.025
0.025
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0.005
64
56
48
40
00
00 8
16 24
32 40
48 56
64
0
8
16
24
32
8 16
24 32
40 48
56 64
0
8
16
24
32
40
48
56
64
(a) Geometric Optics PSF with Radius 3.75 and 7.5 pixels
0.025
0.025
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0.005
64
56
48
40
00
00 8
16 24
32 40
48 56
64
0
8
16
24
32
8 16
24 32
40 48
56 64
(b) Gaussian PSF with Radius 3.75 and 7.5 pixels
Fig. 2 PSF
Fig. 3 Focused Image for Character
20
0
8
16
24
32
40
48
56
64
(a) Blurred Image
(b) Restored by
Geometric PSF Model
(c) Restored by
Gaussian PSF Model
(d) Restored by
S-Transform
(e) Restored by
Separable MTF Model
(f) Restored using Actual
PSF (Abel Transform)
Fig. 4 Restoration with 0 Step of Blur
(a) Blurred Image
(b) Restored by
Geometric PSF Model
(c) Restored by
Gaussian PSF Model
(d) Restored by
S-Transform
(e) Restored by
Separable MTF Model
(f) Restored using Actual
PSF (Abel Transform)
Fig. 5 Restoration with 10 Steps of Blur
21
(a) Blurred Image
(b) Restored by
Geometric PSF Model
(c) Restored by
Gaussian PSF Model
(d) Restored by
S-Transform
(e) Restored by
Separable MTF Model
(f) Restored using Actual
PSF (Abel Transform)
Fig. 6 Restoration with 20 Steps of Blur
(a) Blurred Image
(b) Restored by
Geometric PSF Model
(c) Restored by
Gaussian PSF Model
(d) Restored by
S-Transform
(e) Restored by
Separable MTF Model
(f) Restored using Actual
PSF (Abel Transform)
Fig. 7 Restoration with 30 Steps of Blur
22
(a) Blurred Image
(b) Restored by
Geometric PSF Model
(c) Restored by
Gaussian PSF Model
(d) Restored by
S-Transform
(e) Restored by
Separable MTF Model
(f) Restored using Actual
PSF (Abel Transform)
Fig. 8 Restoration with 40 Steps of Blur
(a) Blurred Image
(b) Restored by
Geometric PSF Model
(c) Restored by
Gaussian PSF Model
(d) Restored by
S-Transform
(e) Restored by
Separable MTF Model
(f) Restored using Actual
PSF (Abel Transform)
Fig. 9 Restoration with 50 Steps of Blur
23
0 step of blur
5 steps of blur
10 steps of blur
15 steps of blur
20 steps of blur
25 steps of blur
30 steps of blur
35 steps of blur
40 steps of blur
45 steps of blur
50 steps of blur
55 steps of blur
Fig. 10 Step Edges for Calibration
0.35
0 step
10 steps
20 steps
30 steps
40 steps
50 steps
60 steps
70 steps
0.30
0.25
LSF
0.20
0.15
0.10
0.05
0
-20
-15
-10
-5
0
5
10
Pixels
Fig. 11 LSF from Step Edges
24
15
20
0.020
0 step
0.018
10 steps
20 steps
30 steps
40 steps
50 steps
60 steps
70 steps
0.016
0.014
PSF
0.012
0.010
0.008
0.006
0.004
0.002
0
-20
-15
-10
-5
0
5
10
15
20
Pixels
Fig. 12 PSF by Inverse Abel Transform
16
14
Psf Radius (Pixels)
12
10
8
6
4
Horizontal Edge
Vertical Edge
Geometric Optics
2
0
-20
-10
0
10
20
30
40
50
Steps of Blur
Fig. 13 PSF Radius from Step Edges
25
60
70
Blurred ( 0 step )
Blurred ( 10 steps )
Blurred ( 20 steps )
Restored ( 0 step )
(a)
Restored ( 10 steps )
(b)
Restored ( 20 steps )
(c)
Blurred ( 30 steps )
Blurred ( 40 steps )
Blurred ( 50 steps )
Restored ( 30 steps )
(d)
Restored ( 40 steps )
(e)
Restored ( 50 steps )
(f)
Fig. 14 Simulation with Geometric Optics PSF
26
Blurred ( 0 step )
Blurred ( 10 steps )
Blurred ( 20 steps )
Restored ( 0 step )
(a)
Restored ( 10 steps )
(b)
Restored ( 20 steps )
(c)
Blurred ( 30 steps )
Blurred ( 40 steps )
Blurred ( 50 steps )
Restored ( 30 steps )
(d)
Restored ( 40 steps )
(e)
Restored ( 50 steps )
(f)
Fig. 15 Simulation with Gaussian PSF
27
Focused Image
(Focus at Step 14)
Blurred Image
(Lens at Step 40)
Blurred Image
(Lens at Step 70)
Restored by
S-Transform
Restored using Actual
PSF (Abel Transform)
Fig. 16(a) Depth Estimation with Restoration for Step 14
Focused Image
(Focus at Step 36)
Blurred Image
(Lens at Step 40)
Blurred Image
(Lens at Step 70)
Restored by
S-Transform
Restored using Actual
PSF (Abel Transform)
Fig. 16(b) Depth Estimation with Restoration for Step 36
28
Focused Image
(Focus at Step 56)
Blurred Image
(Lens at Step 40)
Blurred Image
(Lens at Step 70)
Restored by
S-Transform
Restored using Actual
PSF (Abel Transform)
Fig. 16(c) Depth Estimation with Restoration for Step 56
Focused Image
(Focus at Step 76)
Blurred Image
(Lens at Step 40)
Blurred Image
(Lens at Step 70)
Restored by
S-Transform
Restored using Actual
PSF (Abel Transform)
Fig. 16(d) Depth Estimation with Restoration for Step 76
29
(a) Blurred Image
(Lens Step 40)
(b) Blurred Image
(Lens Step 70)
(c) Restored by
S-Transform
(d) Restored using Actual
PSF (Abel Transform)
Fig. 17 Depth Estimation with Restoration for 3-D Object
30
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