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2342 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 10, OCTOBER 2009
Fig. 2. (a) Original image Lena; (b) the scaled and rotated image of (a); (c) the
LPT transformed image of (a); (d) the LPT transformed image of (b).
The advantage of using log-polar over the Cartesian coordi-
nate representation is that any rotation and scale in the Carte-
sian coordinates is represented as shifting in the angular and the
log-radius directions in the log-polar coordinates, respectively,
as shown in Fig. 2. Fig. 2(a) is the original image and Fig. 2(b)
is the scaled and rotated version of the original image. Fig. 2(c)
and (d) are the LPT images of Fig. 2(a) and (b), respectively.
The column of the log-polar coordinates represents the angular
direction while the row represents the log-radius. We can see
that rotation and scale in the Cartesian coordinates are repre-
sented as shifting in the log-polar coordinates.
B. Motivation of the Proposed Approach
Although LPT has been widely used in many image pro-
cessing applications, it suffers from nonuniform sampling. As
shown in Fig. 1(a), as the radius of the mapping increases, pixels
in the Cartesian coordinates are sampled with less number of
times. This nonuniform sampling would cause image pixels that
are far away from the center point to be missed. This phenom-
enon yields losses in image information which will eventually
decrease the accuracy of the registration system. To prevent any
image pixel from being missed in the mapping process, large
numbers of samples are required in both the log-radius and the
angular directions. We denotes
and as numbers of sam-
ples in the log-radius and the angular directions, respectively.
We also denote
as the radius size in pixel for ,
which will be sampled equally to cover 360 degrees in the an-
gular direction for
. To prevent undersampling in
the log-radius direction, the condition of
has
to be met. It can be easily found that the following equations
should hold:
(3)
(4)
In the angular direction, the greatest sampling circumference,
, contains approximately image pixels. Hence,
we need to have the number of samples in the angular direction
equal or greater than that amount to prevent image pixels being
missed in the sampling procedure
(5)
For the sake of simplicity,
and can be selected in the
form of multiple power of two. This will not only make the
computation simpler, but also accelerate the matching which is
Fig. 3. Example of inaccurate registration results using the conventional LPT
due to occlusion in the target image: (a) the model image patch cropped from
Lena image is registered to the nonoccluded target image on the upper-right and
occluded target image on the lower-right, where the red rectangular indicate
the registration results, (b) the correction coefficients when registering to the
nonoccluded target image, and (c) the correction coefficients when registering
to the occluded target image.
usually carried out by cross-correlation [7] or by Fourier phase
correlation [8]. Hence, (4) and (5) can be expressed as
(6)
The operation
denotes the nearest integers greater than
or equal to
. With the proposed numbers of samples to be used
in the computation of LPT, image pixels in the Cartesian coor-
dinates will not be missed in the mapping process. The pixels
at the circumference will be mapped to the log-polar coordi-
nates at approximately one-to-one. However, image pixels at the
fovea will be oversampled. Since no information is gained from
oversampling, the computation is wasted. For an image of size
256
256 pixels, the minimum 613 samples in the log-radius
and 805 samples in the angular directions are required (1024
samples in both the log-radius and the angular directions in the
case of power of 2 number). It can be easily seen that the re-
quired number of samples is too large compared to the original
size of the image in the Cartesian coordinates.
Another major problem of using the conventional LPT for
image registration is the bias matching. Since the number of
samples at the fovea of the image is larger than that of the periph-
eral, the matching mechanism will also focus more on the fovea
or central area while the peripheral area is given less considera-
tion. Hence, using the conventional LPT for image registration
yields inaccurate result when there are changes in the image,
especially at the area close to the center point. Fig. 3 shows an
example of registering the Lena image patch to the nonoccluded
target image [the upper image on the right of Fig. 3(a)] and to the
occluded target image [the lower image on the right of Fig. 3(a)],
respectively. In this demonstration, an artificial block of 20
20
pixels is used as occlusion. The sensitivity of the matching is
evaluated through the correlation function between the model
image and the target image (referring to the registration method
using the conventional LPT proposed in [4]). Fig. 3(b) and (c)
shows the correlation coefficients at the point where the model
image and the target image are matched in case of the nonoc-
cluded target image and the occluded target image, respectively.
In this demonstration, it can be seen that the conventional LPT
performs effectively in registering images in the normal con-
dition. Fig. 3(b) shows the strong peak of the correlation co-
efficients and Fig. 3(a) (upper-right image) indicates accurate
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