MATCHED FILTERING OF CONTINUOUS IMAGES
617
with and . By the convolution theorem,
(19.2-4)
where is the Fourier transform of . The additive input noise com-
ponent is assumed to be stationary, independent of the signal image, and
described by its noise power-spectral density . From Eq. 1.4-27, the total
noise power at the filter output is
(19.2-5)
Then, forming the signal-to-noise ratio, one obtains
(19.2-6)
This ratio is found to be maximized when the filter transfer function is of the form
(5,8)
(19.2-7)
If the input noise power-spectral density is white with a flat spectrum,
, the matched filter transfer function reduces to
(19.2-8)
and the corresponding filter impulse response becomes
(19.2-9)
In this case, the matched filter impulse response is an amplitude scaled version of
the complex conjugate of the signal image rotated by 180°.
For the case of white noise, the filter output can be written as
(19.2-10a)
x ε= y η=
S εη,()
2
F ω
x
ω
y
,()H ω
x
ω
y
,()i ω
x
εω
y
η+(){}exp ω
x
d ω
y
d
∞
–
∞
∫
∞
–
∞
∫
2
=
F ω
x
ω
y
,() Fxy,()
Nxy,()
W
N
ω
x
ω
y
,()
N W
N
ω
x
ω
y
,()H ω
x
ω
y
,()
2
ω
x
d ω
y
d
∞
–
∞
∫
∞
–
∞
∫
=
S εη,()
2
N
F ω
x
ω
y
,()H ω
x
ω
y
,()i ω
x
εω
y
η+(){}exp ω
x
d ω
y
d
∞
–
∞
∫
∞
–
∞
∫
2
W
N
ω
x
ω
y
,()H ω
x
ω
y
,()
2
ω
x
d ω
y
d
∞
–
∞
∫
∞
–
∞
∫
=
H ω
x
ω
y
,()
F * ω
x
ω
y
,() i ω
x
εω
y
η+()–{}exp
W
N
ω
x
ω
y
,()
=
W
N
ω
x
ω
y
,()n
w
2⁄=
H ω
x
ω
y
,()
2
n
w
F * ω
x
ω
y
,() i ω
x
εω
y
η+()–{}exp=
Hxy,()
2
n
w
F* ε x– η y–,()=
F
O
xy,()
2
n
w
F
U
xy,()
᭺
ء F
∗
ε x– η y–,()=
618
IMAGE DETECTION AND REGISTRATION
or
(19.2-10b)
If the matched filter offset is chosen to be zero, the filter output
(19.2-11)
is then seen to be proportional to the mathematical correlation between the input
image and the complex conjugate of the signal image. Ordinarily, the parameters
of the matched filter transfer function are set to be zero so that the origin of
the output plane becomes the point of no translational offset between and
.
If the unknown image consists of the signal image translated by dis-
tances plus additive noise as defined by
(19.2-12)
the matched filter output for , will be
(19.2-13)
A correlation peak will occur at , in the output plane, thus indicating
the translation of the input image relative to the reference image. Hence the matched
filter is translation invariant. It is, however, not invariant to rotation of the image to
be detected.
It is possible to implement the general matched filter of Eq. 19.2-7 as a two-stage
linear filter with transfer function
(19.2-14)
The first stage, called a whitening filter, has a transfer function chosen such that
noise with a power spectrum at its input results in unit energy
white noise at its output. Thus
(19.2-15)
F
O
xy,()
2
n
w
F
U
αβ,()F
∗
αεx–+ βηy–+,()αd βd
∞
–
∞
∫
∞
–
∞
∫
=
εη,()
F
O
xy,()
2
n
w
F
U
αβ,()F
∗
α x– β y–,()αd βd
∞
–
∞
∫
∞
–
∞
∫
=
εη,()
F
U
xy,()
Fxy,()
F
U
xy,()
∆x ∆y,()
F
U
xy,()Fx ∆x+ y ∆y+,()Nxy,()+=
ε 0= η 0=
F
O
xy,()
2
n
w
F α∆x+ β∆y+,()Nxy,()+[]F
∗
α x– β y–,()αd βd
∞
–
∞
∫
∞
–
∞
∫
=
x ∆x= y ∆y=
H ω
x
ω
y
,()H
A
ω
x
ω
y
,()H
B
ω
x
ω
y
,()=
Nxy,() W
N
ω
x
ω
y
,()
W
N
ω
x
ω
y
,()H
A
ω
x
ω
y
,()
2
1=
MATCHED FILTERING OF CONTINUOUS IMAGES
619
The transfer function of the whitening filter may be determined by a spectral factor-
ization of the input noise power-spectral density into the product (7)
(19.2-16)
such that the following conditions hold:
(19.2-17a)
(19.2-17b)
(19.2-17c)
The simplest type of factorization is the spatially noncausal factorization
(19.2-18)
where represents an arbitrary phase angle. Causal factorization of the
input noise power-spectral density may be difficult if the spectrum does not factor
into separable products. For a given factorization, the whitening filter transfer func-
tion may be set to
(19.2-19)
The resultant input to the second-stage filter is , where
represents unit energy white noise and
(19.2-20)
is a modified image signal with a spectrum
(19.2-21)
From Eq. 19.2-8, for the white noise condition, the optimum transfer function of the
second-stage filter is found to be
W
N
ω
x
ω
y
,()W
N
+
ω
x
ω
y
,()W
N
–
ω
x
ω
y
,()=
W
N
+
ω
x
ω
y
,()W
N
–
ω
x
ω
y
,()[]
∗
=
W
N
–
ω
x
ω
y
,()W
N
+
ω
x
ω
y
,()[]
∗
=
W
N
ω
x
ω
y
,()W
N
+
ω
x
ω
y
,()
2
W
N
–
ω
x
ω
y
,()
2
==
W
N
+
ω
x
ω
y
,()W
N
ω
x
ω
y
,()iθω
x
ω
y
,(){}exp=
θω
x
ω
y
,()
H
A
ω
x
ω
y
,()
1
W
N
+
ω
x
ω
y
,()
=
F
1
xy,()N
W
xy,()+ N
W
xy,()
F
1
xy,()Fxy,()
᭺
ء
H
A
xy,()=
F
1
ω
x
ω
y
,()F ω
x
ω
y
,()H
A
ω
x
ω
y
,()
F ω
x
ω
y
,()
W
N
+
ω
x
ω
y
,()
==
620
IMAGE DETECTION AND REGISTRATION
(19.2-22)
Calculation of the product shows that the optimum filter
expression of Eq. 19.2-7 can be obtained by the whitening filter implementation.
The basic limitation of the normal matched filter, as defined by Eq. 19.2-7, is that
the correlation output between an unknown image and an image signal to be
detected is primarily dependent on the energy of the images rather than their spatial
structure. For example, consider a signal image in the form of a bright hexagonally
shaped object against a black background. If the unknown image field contains a cir-
cular disk of the same brightness and area as the hexagonal object, the correlation
function resulting will be very similar to the correlation function produced by a per-
fect match. In general, the normal matched filter provides relatively poor discrimi-
nation between objects of different shape but of similar size or energy content. This
drawback of the normal matched filter is overcome somewhat with the derivative
matched filter (8), which makes use of the edge structure of an object to be detected.
The transfer function of the pth-order derivative matched filter is given by
(19.2-23)
where p is an integer. If p = 0, the normal matched filter
(19.2-24)
is obtained. With p = 1, the resulting filter
(19.2-25)
is called the Laplacian matched filter. Its impulse response function is
(19.2-26)
The pth-order derivative matched filter transfer function is
(19.2-27)
H
B
ω
x
ω
y
,()
F *
ω
x
ω
y
,()
W
N
–
ω
x
ω
y
,()
i ω
x
εω
y
η+()–{}exp=
H
A
ω
x
ω
y
,()H
B
ω
x
ω
y
,()
H
p
ω
x
ω
y
,()
ω
x
2
ω
y
2
+()
p
F * ω
x
ω
y
,()i
ω
x
εω
y
η+()
–{}exp
W
N
ω
x
ω
y
,()
=
H
0
ω
x
ω
y
,()
F *
ω
x
ω
y
,()
i
ω
x
εω
y
η+()
–{}exp
W
N
ω
x
ω
y
,()
=
H
p
ω
x
ω
y
,()ω
x
2
ω
y
2
+()H
0
ω
x
ω
y
,()=
H
1
xy,()
x
2
∂
∂
y
2
∂
∂
+
᭺
ء H
0
xy,()=
H
p
ω
x
ω
y
,()ω
x
2
ω
y
2
+()
p
H
0
ω
x
ω
y
,()=
MATCHED FILTERING OF CONTINUOUS IMAGES
621
Hence the derivative matched filter may be implemented by cascaded operations
consisting of a generalized derivative operator whose function is to enhance the
edges of an image, followed by a normal matched filter.
19.2.2. Matched Filtering of Stochastic Continuous Images
In the preceding section, the ideal image to be detected in the presence of
additive noise was assumed deterministic. If the state of is not known
exactly, but only statistically, the matched filtering concept can be extended to the
detection of a stochastic image in the presence of noise (13). Even if is
known deterministically, it is often useful to consider it as a random field with a
mean . Such a formulation provides a mechanism for incorpo-
rating a priori knowledge of the spatial correlation of an image in its detection. Con-
ventional matched filtering, as defined by Eq. 19.2-7, completely ignores the spatial
relationships between the pixels of an observed image.
For purposes of analysis, let the observed unknown field
(19.2-28a)
or noise alone
(19.2-28b)
be composed of an ideal image , which is a sample of a two-dimensional sto-
chastic process with known moments, plus noise independent of the image,
or be composed of noise alone. The unknown field is convolved with the matched
filter impulse response to produce an output modeled as
(19.2-29)
The stochastic matched filter is designed so that it maximizes the ratio of the aver-
age squared signal energy without noise to the variance of the filter output. This is
simply a generalization of the conventional signal-to-noise ratio of Eq. 19.2-6. In the
absence of noise, the expected signal energy at some point in the output field
is
(19.2-30)
By the convolution theorem and linearity of the expectation operator,
(19.2-31)
Fxy,()
Fxy,()
Fxy,()
EFxy,(){}Fxy,()=
F
U
xy,()Fxy,()Nxy,()+=
F
U
xy,()Nxy,()=
Fxy,()
Nxy,()
Hxy,()
F
O
xy,()F
U
xy,()
᭺
ء
Hxy,()=
εη,()
S εη,()
2
EFxy,(){}
᭺
ء Hxy,()
2
=
S εη,()
2
E F ω
x
ω
y
,(){}H ω
x
ω
y
,()i ω
x
εω
y
η+(){}exp ω
x
d ω
y
d
∞
–
∞
∫
∞
–
∞
∫
2
=
622
IMAGE DETECTION AND REGISTRATION
The variance of the matched filter output, under the assumption of stationarity and
signal and noise independence, is
(19.2-32)
where and are the image signal and noise power spectral
densities, respectively. The generalized signal-to-noise ratio of the two equations
above, which is of similar form to the specialized case of Eq. 19.2-6, is maximized
when
(19.2-33)
Note that when is deterministic, Eq. 19.2-33 reduces to the matched filter
transfer function of Eq. 19.2-7.
The stochastic matched filter is often modified by replacement of the mean of the
ideal image to be detected by a replica of the image itself. In this case, for
,
(19.2-34)
A special case of common interest occurs when the noise is white,
, and the ideal image is regarded as a first-order nonseparable
Markov process, as defined by Eq. 1.4-17, with power spectrum
(19.2-35)
where is the adjacent pixel correlation. For such processes, the resultant
modified matched filter transfer function becomes
(19.2-36)
At high spatial frequencies and low noise levels, the modified matched filter defined
by Eq. 19.2-36 becomes equivalent to the Laplacian matched filter of Eq. 19.2-25.
N W
F
ω
x
ω
y
,()W
N
ω
x
ω
y
,()+[]H ω
x
ω
y
,()
2
ω
x
d ω
y
d
∞
–
∞
∫
∞
–
∞
∫
=
W
F
ω
x
ω
y
,()W
N
ω
x
ω
y
,()
H ω
x
ω
y
,()
E F *
ω
x
ω
y
,()
{}i
ω
x
εω
y
η+()
–{}exp
W
F
ω
x
ω
y
,()W
N
ω
x
ω
y
,()+
=
Fxy,()
εη0==
H ω
x
ω
y
,()
F *
ω
x
ω
y
,()
W
F
ω
x
ω
y
,()W
N
ω
x
ω
y
,()+
=
W
N
ω
x
ω
y
,()n
W
2⁄=
W
F
ω
x
ω
y
,()
2
α
2
ω
x
2
ω
y
2
++
=
α–{}exp
H ω
x
ω
y
,()
2
α
2
ω
x
2
ω
y
2
++()
F *
ω
x
ω
y
,()
4 n
W
α
2
ω
x
2
ω
y
2
++()+
=
MATCHED FILTERING OF DISCRETE IMAGES
623
19.3. MATCHED FILTERING OF DISCRETE IMAGES
A matched filter for object detection can be defined for discrete as well as continu-
ous images. One approach is to perform discrete linear filtering using a discretized
version of the matched filter transfer function of Eq. 19.2-7 following the techniques
outlined in Section 9.4. Alternatively, the discrete matched filter can be developed
by a vector-space formulation (13,14). The latter approach, presented in this section,
is advantageous because it permits a concise analysis for nonstationary image and
noise arrays. Also, image boundary effects can be dealt with accurately. Consider an
observed image vector
(19.3-1a)
or
(19.3-1b)
composed of a deterministic image vector f plus a noise vector n, or noise alone.
The discrete matched filtering operation is implemented by forming the inner prod-
uct of with a matched filter vector m to produce the scalar output
(19.3-2)
Vector m is chosen to maximize the signal-to-noise ratio. The signal power in the
absence of noise is simply
(19.3-3)
and the noise power is
(19.3-4)
where is the noise covariance matrix. Hence the signal-to-noise ratio is
(19.3-5)
The optimal choice of m can be determined by differentiating the signal-to-noise
ratio of Eq. 19.3-5 with respect to m and setting the result to zero. These operations
lead directly to the relation
f
U
fn+=
f
U
n=
f
U
f
O
m
T
f
U
=
S m
T
f[]
2
=
NEm
T
n[]m
T
n[]
T
{}m
T
K
n
m==
K
n
S
N
m
T
f[]
2
m
T
K
n
m
=
624
IMAGE DETECTION AND REGISTRATION
(19.3-6)
where the term in brackets is a scalar, which may be normalized to unity. The
matched filter output
(19.3-7)
reduces to simple vector correlation for white noise. In the general case, the noise
covariance matrix may be spectrally factored into the matrix product
(19.3-8)
with , where E is a matrix composed of the eigenvectors of and
is a diagonal matrix of the corresponding eigenvalues (14). The resulting matched
filter output
(19.3-9)
can be regarded as vector correlation after the unknown vector has been whit-
ened by premultiplication by .
Extensions of the previous derivation for the detection of stochastic image vec-
tors are straightforward. The signal energy of Eq. 19.3-3 becomes
(19.3-10)
where is the mean vector of f and the variance of the matched filter output is
(19.3-11)
under the assumption of independence of f and n. The resulting signal-to-noise ratio
is maximized when
(19.3-12)
Vector correlation of m and to form the matched filter output can be performed
directly using Eq. 19.3-2 or alternatively, according to Eq. 19.3-9, where
and E and denote the matrices of eigenvectors and eigenvalues of
m
m
T
K
n
m
m
T
f
K
n
1
–
f=
f
O
f
T
K
n
1
–
f
U
=
K
n
KK
T
=
KEΛ
ΛΛ
Λ
n
12⁄
–
= K
n
Λ
ΛΛ
Λ
n
f
O
K
1
–
f
U
[]
T
K
1
–
f
U
[]=
f
U
K
1
–
S m
T
η
ηη
η
f
[]
2
=
η
ηη
η
f
N m
T
K
f
mm
T
K
n
m+=
mK
f
K
n
+[]
1
–
η
ηη
η
f
=
f
U
KEΛ
ΛΛ
Λ
12⁄
–
= Λ
ΛΛ
Λ
IMAGE REGISTRATION
625
, respectively (14). In the special but common case of white noise and a
separable, first-order Markovian covariance matrix, the whitening operations can be
performed using an efficient Fourier domain processing algorithm developed for
Wiener filtering (15).
19.4. IMAGE REGISTRATION
In many image processing applications, it is necessary to form a pixel-by-pixel com-
parison of two images of the same object field obtained from different sensors, or of
two images of an object field taken from the same sensor at different times. To form
this comparison, it is necessary to spatially register the images, and thereby, to cor-
rect for relative translation shifts, rotational differences, scale differences and even
perspective view differences. Often, it is possible to eliminate or minimize many of
these sources of misregistration by proper static calibration of an image sensor.
However, in many cases, a posteriori misregistration detection and subsequent cor-
rection must be performed. Chapter 13 considered the task of spatially warping an
image to compensate for physical spatial distortion mechanisms. This section
considers means of detecting the parameters of misregistration.
Consideration is given first to the common problem of detecting the translational
misregistration of two images. Techniques developed for the solution to this prob-
lem are then extended to other forms of misregistration.
19.4.1. Translational Misregistration Detection
The classical technique for registering a pair of images subject to unknown transla-
tional differences is to (1) form the normalized cross correlation function between
the image pair, (2) determine the translational offset coordinates of the correlation
function peak, and (3) translate one of the images with respect to the other by the
offset coordinates (16,17). This subsection considers the generation of the basic
cross correlation function and several of its derivatives as means of detecting the
translational differences between a pair of images.
Basic Correlation Function. Let and for and ,
represent two discrete images to be registered. is considered to be the
reference image, and
(19.4-1)
is a translated version of where are the offset coordinates of the
translation. The normalized cross correlation between the image pair is defined as
K
f
K
n
+[]
F
1
jk,() F
2
jk,(), 1 jJ≤≤ 1 k
K
≤≤
F
1
jk,()
F
2
jk,()F
1
jj
o
kk
o
–,–()=
F
1
jk,() j
o
k
o
,()
626
IMAGE DETECTION AND REGISTRATION
(19.4-2)
for m = 1, 2, , M and n = 1, 2, , N, where M and N are odd integers. This formu-
lation, which is a generalization of the template matching cross correlation expres-
sion, as defined by Eq. 19.1-5, utilizes an upper left corner–justified definition for
all of the arrays. The dashed-line rectangle of Figure 19.4-1 specifies the bounds of
the correlation function region over which the upper left corner of moves in
space with respect to . The bounds of the summations of Eq. 19.4-2 are
(19.4-3a)
(19.4-3b)
These bounds are indicated by the shaded region in Figure 19.4-1 for the trial offset
(a, b). This region is called the window region of the correlation function computa-
tion. The computation of Eq. 19.4-2 is often restricted to a constant-size window
area less than the overlap of the image pair in order to reduce the number of
FIGURE 19.4-1. Geometrical relationships between arrays for the cross correlation of an
image pair.
Rmn,()
F
1
jk,()F
2
jm– M 1+()2 kn– N 1+()2⁄+,⁄+()
k
∑
j
∑
F
1
jk,()[]
2
k
∑
j
∑
1
2
F
2
jm– M 1+()2 kn– N 1+()2⁄+,⁄+()[]
2
k
∑
j
∑
1
2
=
F
2
jk,()
F
1
jk,()
MAX 1 mM1–()2⁄–,{}j MIN JJ m M 1+()2⁄–+,{}≤≤
MAX 1 nN1–()2⁄–,{}k MIN KK n N 1+()2⁄–+,{}≤≤
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