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