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 
operator is not introduced, which makes our new image 
representation more robust to noise, even in the texture-less 
image region. 
2.2 Local Weighted Amplitude 
In this work, we extend the phase congruence to LWA, which is 
more suitable for multi-temporal and multi-sensor image 
representation. The equation of LWA (A, ) is expressed as the 
summation of orientations r and scales s: 
4,0) 7 €i X. W G2| 4, 60^6,6) - T | (6) 
where| | denotes that the enclosed quantity is not permitted to 
be negative; 4. represents the amplitude at scale S and 
orientation 7^ ; and 7 compensates for the influence of noise and 
is estimated empirically. Ag (x)is a sensitive phase deviation 
of the rthorientation and is defined as: 
^6, (x) = cos (9,, (x)=, (x) - [sin (9, (x) 6, 6) a 
The calculation of this new LWA, 4 (x), can be performed 
using dot and cross products between the filter output response 
vectors to calculate the cosine and sine of (6... (x) -$,(x)) ; 
The unit vector representing the direction of the weighted mean 
phase angle, @ (x), is given by 
ES Uu) (Ze, 0)Xo, 0) (8) 
(6..00.6,,0)) » T ; 
Ze.) + Zo.) 
  
Using dot and cross products one can obtain: 
A, (XA, (x)=4, (x) (cos(a,, 69-6. 69)- [sin(6..69 -660)|) ©) 
-e,,(90,, 09) 0,00, (9 - e, C90, (X) - 0,8, (x) 
  
Clearly, a point of frequency amplitude is only significant if it 
occurs over a wide range of frequencies. Thus, as a measure of 
feature significance, frequency amplitude should be weighted 
by some measure of the spread of the frequencies present. A 
phase significance weighting function can then be constructed 
by applying a sigmoid function to the filter response spread 
value: 
WG)- = (10) 
er s(x) 
where Cis the "cut-off" value of the filter response spread, 
below which the frequency amplitude values become penalized, 
and Y is a gain factor that controls the sharpness of the cut-off. 
Eq.6 — Eq.10 give us the quantities needed to calculate this 
version of the LWA without any division operator. 
The LAP and LWA are both used to extract the common 
components of multi-temporal and multi-sensor images, such as 
edges, contours, and blobs. Note that the two image 
representations do not involve any thresholding and, therefore, 
preserve all the image details. This is in contrast to commonly 
used representations (e.g., edge maps, contours, point features), 
which eliminate most of the detailed variations within the local 
image regions. 
Figure 2. The results of PC and LWA. From left to right: the 
raw image; the PC map; and the LWA map. Note that the LWA 
map is much more robust and stable than the PC map, 
especially in texture-less image regions such as the sky, sea, and 
ground. 
3. THE COMPOSITIONAL SIMILARITY 
MEASUREMENT 
As discussed above, The LAP and LWA are independent of 
each other. In order to combine the information of the LAP and 
LWA, we present a new similarity measurement: CSM, which is 
able to take advantage of more information than those 
commonly used similarity measures [9, 10] and is therefore able 
to improve the robustness and applicability of image matching. 
The LWA is designed to emphasize the common amplitude 
components for multi-temporal and multi-sensor images, and 
has a stronger anti-noise capability than the commonly used 
Phase Congruence. However, from the theoretic analysis and 
experimental results, we know that the LWA is an image 
contrast-dependent variation. To overcome this problem, we 
employ the zero-mean normalized cross-correlation (ZNCC) as 
the similarity measure function, which is a contrast invariant 
variation. If we define f and g as the corresponding LWA 
map pair, 7 and z as the mean value within the template 
window W around pixel (x,y) in f, and (yy) In g, 
respectively, and § as the searching window, where 
(i, ))e W,(u,v) e S , the ZNCC can be expressed as follows: 
  
  
   
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