The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008
Number of band: 49
Figure 3: Spatial simplification: Comparing the filtering result of
ADF, ML (channel by channel process) and the proposed vecto
rial AML. Two line plots with the cross-sections along the y-axis
of the different filters are shown for bands #49 (top) and #64
(down). The proposed AML did simplify the initial image by en
larging and creating new flat zones and at the same time followed
more constantly and closely original image’s intensity values and
variation. AML did retain all its elegant 2D properties.
smoother hypercubes in 7Z :i . The constructed levelings are re
spectively
9i=Z, 92 = A(gi,hi), g 3 = A(52,h 2 ),
(4)
54 = A(53 j ^3) j ••• j 9n — A(c7 n _i, h n —i)
A series g n of simpler and simpler hypercubes, with fewer and
fewer smooth zones are produced forming a 4D scale space with
g : Cl C TZ 4 and g(x,y,z,n) = g n (x,y,z). Similar to the
2D case the introduced, here, vectorial morphological levelings
AMLs can be associated to an arbitrary or an alternating family
of marker functions. Examples with openings, closings, alter
nate sequential filters and isotropic and anisotropic markers can
be found in the literature for scalar images [Meyer, 1998, Meyer
and Maragos, 2000, Meyer, 2004, Karantzalos et al., 2007]. For
specific tasks one may take advantage of the possible prior knowl
edge for scene’s content and design accordingly the family of
markers.
3.2 Anisotropic Diffused Markers
For the construction of the simplified hypercubes anisotropic dif
fused markers were chosen, since they have proven to be effec
tive for scalar images [Karantzalos et al., 2007]. In addition,
since levelings are highly constrained by the type of the marker
used [Meyer and Maragos, 2000], only those markers who are
fully suitable for hyperspectral imagery were appropriate for our
case. The recent formulations of [Martin-Herrero, 2007] provide
a suitable diffusion framework which respects the special char
acteristics of hyperspectral data by separating the elegant vector-
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10 20 30 4Q SO 60 /0
Band Number
Spectral Cross Section
Figure 4: Spectral simplification: Comparing the filtering result
of ADF, ML and AML. Two line plots with the cross-sections
along the spectral axis of the different filtered hypercubes are
shown. The proposed AML did surpassed broad spectral varia
tions (spike-like features) among adjacent spectral signatures and
at the same time followed more constantly the initial intensity.
valued diffusion approach of [Tschumperle and Deriche, 2005]
in the spatial and spectral space. For a hypercube 2 : Cl <Z TZ 3
the anisotropic diffusion process is expressed by the following
equation:
■dl
— = trace(TH) (5)
with H and T the 3x3 Hessian and diffusion tensor matrices, re
spectively. The tensor separates the diffusion in the spatial and
spectral directions while suitable edge-stoping functions Ti con
trol the diffusion:
T = r x 0+0+ + r yO-d'E. + r z zz T (6)
with 9 the eigenvectors of a 2x2 metric tensor D which depends
on the spatial derivatives:
N
D = Ga * ^2 VliVI? (7)
i
where G a is a gaussian smoothing for regularizing the spatial 2D
derivatives of V7 at every channel N. In [Martin-Herrero, 2007]
the edge stoping functions 7~i, which act differently in the spa
tial and spectral directions, have been defined in such a way so
as to allow all possible adjustments regarding their regularization
effect. One should tune all the coefficients according to image
characteristics and the filtering purpose. For a scale space repre
sentation, however, where
J : Cl C TZ 4 , 2(x,y, z,n) = l n {x,y,z) (8)
(n is the scale of diffusion) one may avoid tuning the vector
edge strength n(v) -with v = -^/trace(D)- and rely on the
