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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
2.3 Fusion method based on a shift invariant extension of the
DWT
More recently, the wavelet transform has been used for merging
data derived from different resolution sensors (Rockinger,
1996). To overcome the shift dependency of the wavelet fusion
method, the input images must be decomposed into a shift
invariant representation. For convenience, we
summarize this approach for the case of ID input signals.
wavelet
As for the discrete wavelet transform (DWT), each stage of the
shift invariant DWT (SIDWT) splits the input sequence into the
wavelet sequence W;(n) and the scale sequence S;(n) which
serves as input for the next decomposition level (Rockinger,
1996):
W,(n) = Y 2055 (n —k) (9)
k
$5400 & V. Ah! K)s on: K) (10)
k
The zero'th level scale sequence is set equal to the input
sequence. So (n) — f(n). thus defining the complete SIDWT
decomposition scheme. In contrast to the standard DWT
decomposition scheme the subsampling is dropped, resulting in
a highly redundant wavelet representation. The analysis filters
g(2'k) and A(2'k) at level i are obtained by inserting the
appropriate number of zeros between the filter taps of the
prototype filters g(k) and AK).
The reconstruction of the input sequence is performed by the
inverse SIDWT as a convolution of both shift invariant wavelet
sequence and scale sequence with the appropriate
reconstruction filters g(2'.k) and h(2'k) as follows:
s,(n)= y h(2in- K).s; 4 Q0) + > g(2 n =k)w;,,,(n) (11)
k k
2.4. Fusion based on a Laplacian pyramid method
The Laplacian filtered image can be realized as a difference
Gaussian filtered images. Accordingly the Laplacian pyramid is
obtainable Let G* be the
from the Gaussian pyramid.
hg : S a : 1201 i
k (kzL..,N) level of the Gaussian pyramid for an image /.
Then
ES (13)
GA GE], for k=1....N-1
where the kernel w is obtained a discrete Gaussian density,
*' denotes two-dimensional convolution and the notation
Ladys indicates that the image in brackets is down-sampled by 2
(in both, horizontal and vertical directions) which is
accomplished by selecting every other point in the filtered
mage. The Gaussian pyramid is a set of lowpass filtered copies
à the image, each with a cut-off frequency one octave lower
than its predecessor. The Laplacian pyramid is determined by
fu =GY
se (14)
| -G' 49g [o^ I for
k =0,.., N-1
891
where the notation [...]7> Indicates that the image inside the
brackets is up-sampled by 2 (in both the horizontal and vertical
directions). Here, convolution by the Gaussian kernel has the
effect of interpolation by a low-pass filter.
The Laplacian pyramid transform decomposes the image into
multiple levels. Each level in the Laplacian pyramid represents
the result of convolving the original image with a difference of
two Gaussian functions thus each successive level is a band-
passed, sub-sampled and scaled version of the original image.
The Laplacian pyramid has a perfect reconstruction property;
the original image can be reconstructed by reversing the
Laplacian pyramid operations:
(15)
for k=0....,N-1
G is identical to the original image /.
Fusion is performed in the Laplacian pyramid domain by
constructing a fused pyramid. The pyramid coefficient (or
hyperpixel) at each location in the fused pyramid is obtained by
selecting the hyperpixel of the sensor pyramid that has the
largest absolute value. Let. L4 and Lpgbe the Laplacian
pyramids of two images A and B. With L; the fused pyramid is
denoted which is determined by
[thai i ool»
; iij |
D (i, j) Otherwise
where k is the level of the pyramid and (i,j) denotes a hyperpixel
at that level (Sharma, 1999).
2.5. Fusion method based on Contrast pyramids
Toet (1990) introduced an image fusion technique which
preserves local luminance contrast in the sensor images. The
technique is based on selection of image features with
maximum contrast rather than maximum magnitude (Sharma.
1999). It is motivated by the fact that the human visual system
is based on contrast and hence the resulting fused image will
provide better details to a human observer. The pyramid
decomposition used for this technique is related to luminance
processing in the early stages of the human visual system which
are sensitive to local luminance contrast (Toet, 1990). Fusion is
performed using the multiresolution contrast pyramid. The j^
level R; of the contrast pyramid is obtained by:
for'’k=1,..N-1
(17)
The hyperpixels of the contrast pyramid R are related to the
local luminance contrast. Luminance contrast C is defined as:
L-L L
Ca DL
Ly Ly
=] (18)
where L is the luminance at a certain location in the image and
Ly is the luminance of the local background. The denominator