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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B3, 2012 
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia 
normalized-correlation, (ii) sum of squared brightness 
differences, and (iii) mutual information [8]. 
The image representations commonly used for area-based 
methods are some gradient operators which include: Canny, 
Sobel, Prewitt, Kirsch, Laplacian of Gaussian, and Susan etc.. 
Using these spatial derivative operators, one can emphasize the 
edges, corners, and blobs that represent some illumination 
invariant component of images. However, these gradient 
representations are usually sensitive to noise. Therefore, the 
gradient representations are not able to handle some noisy 
multi-sensor and multi-temporal images efficiently. 
Representation based on local frequency information was 
introduced in [9]. Working in the frequency domain, local 
phase and amplitude information over many different scales and 
orientations were used to construct a dimensionless measure of 
similarity that has high localization. However, due to the un- 
weighted local frequency, the algorithm was unable to clearly 
emphasize common information, such as edges and corners. 
Therefore, the performance of our evaluation experiment is 
unsatisfactory. As phase congruence is condition-independent 
and invariant to illumination changes, it was employed to 
represent images in [10]. However, when calculating phase 
congruence with the denominator representing the sum of the 
amplitude in the Log-Gabor expansion spaces, a division 
operator is inevitably involved. As the value of the denominator 
is usually small in the texture-less regions, the method 
presented in [10] is quite sensitive to noise in the texture-less 
regions, as shown in Figure 2. This paper addresses these 
difficulties by using local frequency information obtained from 
Log-Gabor wavelets over many scales and orientations. A 
compositional similarity measurement and a local best matching 
point detection are also presented to make the presented image 
matching approach more robust and accuracy. 
Figure | The 
results of the 
SIFT feature 
detection and 
matching. From 
top to bottom: the 
reference image 
and searching 
image; the results 
of. the. feature 
detection; the 
results of the 
feature matching 
after outlier 
removal. 
2. LOCAL AVERAGE PHASE AND LOCAL 
WEIGHTED AMPLITUDE 
2.1 Local Average Phase 
In this working, the wavelet transformation is used to obtain the 
frequency information which is local to a point in the signals. 
To preserve phase information, the nonorthogonal wavelets in 
the symmetric/anti-symmetric quadrature pairs are adopted. 
Rather than using Gabor filters, we prefer to use Log-Gabor 
functions, because Log-Gabor filters allow arbitrarily large 
bandwidth filters to be constructed, while maintaining a zero 
DC component in the even-symmetric filter. (A zero DC value 
cannot be maintained in Gabor functions for bandwidths over 
one octave [11].) On the linear frequency scale, the Log-Gabor 
function has a transfer function of the form: 
-(log(0/ ay) 
2(log(x/ 4) 
g(o)-e (1) 
where qj is the filter’s centre frequency. To obtain constant- 
shape ratio filters, the term x /¢ must also be held constant for 
varying qj. Let I denote the signal, M? and M; denote the 
even-symmetric (cosine) and odd-symmetric (sine) wavelets, 
respectively, and e : (39,0, (X) denote the even-symmetric and 
odd-symmetric filter outputs at location x .We can think of the 
responses of each quadrature pair of filters as forming a 
response vector, 
le, (x),0,, ()]=[1(x)x MZ, (3) x M, ] 2) 
The amplitude 4 and phaseó,, at a given wavelet scale is 
given by 
A. (x) = 2 (x) +0, (x) (3) 
9,,(x)=atan2(e,,(x),0,,(x)) 
At each point X in a signal, we will have an array of these 
response vectors, with one vector for each scale and orientation 
of the filter. These response vectors form the basis of our 
localized representation of the signal. An estimate of F(x) can 
be formed by summing the even filter convolutions. Similarly, 
H(x) can be estimated from the odd filter convolutions. 
F(x)= Y S e,, (x) (4) 
H(x)- 2,2... 00 
The average phase is given by 
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where 9 (x)ranging from 0 to 180 can be seen as the phase of 
the sum of the response vectors over many scales and 
orientations. The local average phase which emphasizes the 
phase information of local frequency is used as one of the image 
representations for multi-temporal and multi-sensor images. 
Another representation, Local Weighted Amplitude, is designed 
for extracting the amplitude information of local frequency. 
Apparently, it is independent of LAP. The calculation of the 
LWA is similar to that of phase congruence, except the division