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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008
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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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