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Figure 1: Smoothing and simplifying hyperspectral imagery (©Norsk Elektro Optikk). First column: A 3D view of the initial hyper 
cube (top) and a zoom on two spectral bands i.e band number #33 (middle) and and band number #87 (bottom). Second and third 
columns: the resulting hypercubes and the corresponding spectral bands after anisotropic diffusion filtering ADF (second column) and 
after the proposed vectorial leveling AML (third). Contrary to ADF, which smoothed but created spurious extrema and intensity shifts, 
AML simplified and stayed constantly closer to the initial hypercube’s intensity and structure. 
ground (peaks) and background (valleys) in an asymmetrical man 
ner, causing spectral shifts [Meyer and Maragos, 2000, Karantza- 
los et al., 2007]. Thus, they pass on these drawbacks to the suc 
ceeding classification and object detection procedures, harming 
their outcome significantly. A recent solution for scalar images 
(77 2 ), came from the development of a more general and pow 
erful class of self-dual morphological filters, the Morphological 
Levelings (MLs) [Meyer, 1998] which have been further stud 
ied and applied for image simplification and image segmentation 
by [Meyer and Maragos, 2000, Meyer, 2004]. 
In this paper, we aim to overcome anisotropic diffusion draw 
backs and exploit all the properties that make MLs powerful. 
Hence, we introduce a novel 4D (the 3D hypercube plus one non 
linear diffusion scale) morphological scale space representation 
for denoising and simplifying hyperspectral imagery. The devel 
oped nonlinear scale space is based on the extension of the 2D 
morphological levelings’ formulation to a multidimensional vec 
tor valued one. The novelty of our approach lies also, in the fact 
that our formulation takes into account the following consider 
ations which are customized to hyperspectral data specificities, 
both during levelings and markers construction. The proposed 
vectorial scale space filtering does: 
i) tackle the kind of noise that never forms a coherent structure 
both in spatial and spectral directions, 
ii) take into account the fact that signal continuity in spectrum 
is, usually, more plausible than continuity in space, i.e the 
assumption that the spectral vector is a good approximation 
to the spectral signature of a particular pixel usually holds 
iii) take into account the fact that object boundaries in the spa 
tial directions should be enhanced, smoothed and elegantly 
simplified while their contours/edges must remain perfectly 
spatially localized: no edge displacements, intensity shifts 
or spurious extrema should occur. 
Integrating spatial and spectral information while respecting the 
aforementioned criteria, the developed scale space morphologi 
cal filtering was applied to a number of hyperspectral images and 
its evaluation was carried out by both a qualitative and a quan 
titative assessment. The remainder of this paper is organized as 
follows: Starting with a brief review on conventional 2D mor 
phological levelings in Section 2, a detailed description of the 
introduced vectorial extension for hyperspectral imagery is given 
in Section 3, along with a reference on the construction of the 
anisotropic markers. In Section 4, experimental results together 
with a discussion on the qualitative and quantitative evaluation 
are presented. Finally, conclusions and perspectives for future 
work are on Section 5. (Supplemental material can be found in 
http://www.mas.ecp.fr/vision/Personnel/karank/Demos/4D). figure 
2 MORPHOLOGICAL 2D LEVELINGS 
Given an image / at domain (bounded) fl G TZ 2 —► 7Z and 
following the definitions from [Meyer, 2004, Karantzalos et al., 
2007], one can consider as f x and f y the values of a 2D func 
tion / at pixels x and y and then define the relations: f y < f x 
(f y is lower than f x ), f y > f x (f y is greater or equal than f x ) 
and f y = f x (the similarity between f x and f y , which are at 
level). Based on these relations, the zones in an image without 
inside contours (isophotes, contour lines with constant brightness 
values) are called smooth/ flat zones. Being able to compare the 
values of neighboring pixels, a general and powerful class of mor 
phological filters the levelling can be defined [Meyer, 1998]. MLs 
are a particular class of images with fewer contours than a given 
image /. A function g is a leveling of a function / if and only if 
f A 8g < g < f V eg 
(1) 
where S is an extensive operator (Sg > g) and e an anti-extensive 
one (eg < g). 
For the construction of MLs a class Inter(g, f) of marker func 
tions h is defined, which separates function g and the reference 
function /. For the function h we have that h G Inter(g, /) and 
so: gAf<h<g\/f. Algorithmically and with the use of h, 
one can ’interpreter’ above equation and construct levelings with 
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