The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008
Figure 2: Simplifying hyperspectral imagery with the proposed scale space vectorial leveling (AML). First row: The initial spectral
band #100 (left) and three of its increasingly simplified versions (scales n=2, 3 and 4). Second row: Zoom on a crop of the images
above.
the following pseudo-code: in cases where {h < /}, replace the
values of h with / A 5h and in cases where {h > /}, replace the
values of h with / V eh. The algorithm can be repeated until the
above equation has been satisfied everywhere. Its convergence
is certain, since the replacements on the values of h are point-
wise monotonic.This makes function g be flat on {<7 < /} and
{g > /} and the procedure continues until convergence.
Under this framework MLs form a general class of morphologi
cal operators which can elegantly simplify images and possess a
number of desirable nonlinear scale space characteristics. Lev
elings do satisfy the following properties [Meyer and Maragos,
2000, Karantzalos et al., 2007]: i) the invariance by spatial trans
lation, ii) the isotropy, invariance by rotation, iii) the invariance
to a change of illumination, iv) the causality principle, v) the
maximum principle, excluding the extreme case where g is com
pletely flat. In addition levelings: vi) do not produce new ex
trema at larger scales, vii) enlarge smooth zones, viii) they, also,
create new smooth zones ix) are particularly robust (strong mor
phological filters) and x) do not displace edges. The aforemen
tioned properties have made them a very useful simplification
tool for a number of computer vision and remote sensing appli
cations [Meyer and Maragos, 2000, Meyer, 2004, Paragios et al.,
2005, Karantzalos and Argialas, 2006].
3 MULTISCALE VECTORIAL LEVELINGS FOR THE
HYPERCUBE
Lets denote with I : Q C TZ d —> TZ N a hyperspectral image
with a normalized hyperspectrum of N spectral channels. The
pseudo-scalar and autarkical vector levelings, that have been al
ready proposed [Gomila and Meyer, 1999], are not suitable for
hyperspectral imagery since they do not account for the special
spatial/spectral specificities of hyperspectral data. In addition,
the first ones do not efficiently enlarge flat zones and the second
ones produce annoying visual artifacts due to their formulation
on color propagation [Gomila and Meyer, 1999].
Excluding atmospheric effects which are tackled during a specific
atmospheric correction stage, the dark or photon shot noise and
the readout noise, which appears as uncorrelated high-frequency
variations in the spatial and spectral space without forming a co
herent structure, is what a filtering procedure should be able to
address [Martin-Herrero, 2007]. However, unconstrained spatial
smoothing is not desirable and in addition, spectral resolution and
band adjacency are, usually, high enough to assume that the spec
tral vector is a good approximation to the spectral signature of
the pixel, i.e the mixture of the spectral signatures of the objects
within the pixel plus atmospheric, scatter and radiometric effects.
Last but not least, in the spatial directions all the aforementioned
in the previous section properties of the 2D levelings must be re
tained. To sum up a sophisticated vectorial leveling formulation
should retain all its 2D properties for the spatial directions and
at the same time respect gross variations among adjacent spectral
signatures and only suppress the broad spectral variations (spike
like features).
Towards this end, the levelings construction mechanism was kept
the same in order to carry out the same effect on the spatial di
rections and reformulated in a way to include in the inequalities a
comparison with the adjacent spectral signatures. Thus, the equa
tion for the vectorial leveling takes, now, the following form:
/ A (ôg s V 6'g c ) <g<fy (eg s A e g c ) (2)
where 5g s denotes an extensive marker in the spatial axis and 5'g c
an extensive marker in the spectral one (the anti-extensive opera
tors eg are equally defined). The spatial g s marker acts as in the
2D case ensuring an elegant simplification in the spatial neigh
borhood of a pixel and the spectral g c accounts for the spike-like
features by enforcing its relevant operators (S' and e) to have a
much broader effect. Under this framework and employing al
ways a marker function h for levelings’ construction the process
is decomposed and the spectral and spatial spaces are treated dif
ferently according to the posed constrains. Rephrasing Equation
(2) and in a unique parallel step we have that:
g = A (/,/1) = (/ A (8h s V S'he)') V (eh s A e'h c ) (3)
Hence, the proposed vectorial levelings can be considered as trans
formations A(/, h) where a marker h is transformed to a func
tion g , which is a leveling of the reference signal /. Where
{(8h s V 8'h c ) < /}, h is increased as little as possible until a
flat zone is created or function g reaches the reference function /
and where {(eh s A e'h c ) > /}, h is decreased as little as possi
ble until a flat zone is created or function g reaches the reference
function / . This process simplifies the hypercube by enlarging
and by creating new flat zones and this procedure continues until
convergence.
3.1 Scale Space Hypercubes
Hyperspectral data can be viewed like any video data, where the
wavelength corresponds to time or like MRI volumes in medical
imaging, where wavelength corresponds to another spatial axis.
Instead of defining the stack of a hyperspectral image as I : Q C
TZ d —> TZ N , where N is the number of spectral channels and
I = (h(x,y),..., lN(x,y)) G 1Z N , a hypercube can be defined,
also, as a 3D function X : Q C TZ 3 —> TZ, where X(x,y,z) =
h{x,y)).
Following this notation, multiscale levelings can be constructed
when the initial (reference) hypercube X is associated with a se
ries of marker functions {hi, /12, •••, h n } -all h are increasingly