Function-valued image segmentation
using functional kernel density estimation
Laurent Delsol1 , Cécile Louchet1,
?
1. MAPMO, UMR CNRS 7349, Université d’Orléans, Fédération Denis Poisson FR CNRS 2964
? Contact author: cecile.louchet@univ-orleans.fr
Keywords: Hyperspectral imaging, functional kernel density estimation, minimal partition.
Introduction. More and more image acquisition systems record high-dimensional vectors for
each pixel, as it is the case for hyperspectral imaging or dynamic PET imaging, to cite only them.
In hyperspectral imaging, each pixel is associated with a light-spectrum containing up to several hundreds of radiance values, each corresponding to narrow spectral bands. In dynamic PET
images, each pixel is associated with a time activity curve giving a radioactivity measurement
throughout the (discretized) time after radiotracer injection. In both cases, the data accessible for
each pixel is a vector coming from the discretization of a function with physical meaning. It is thus
interesting to extend classical image processing techniques on function-valued images.
Image segmentation is the task of partitioning the pixel grid of an image such that the obtained
regions correspond to distinct physical objects; it can be described as automatic search of regions
of interest. It is based on the assumption that each region has different effects on the image values.
Many research works have been about segmenting single- or color-valued images, but little has
been done on function-valued images.
In symbolic data analysis terminology, each pixel is an object whose symbolic description is a
function, and image segmentation can be seen as a symbolic clustering method.
In our work, we recall the most famous single-valued image segmentation method, namely the
minimal partition model (Mumford, Shah, 1989), and adapt it to our case using functional estimation tools, namely functional kernel density estimation, depending on a distance between functions.
We discuss the choice of this distance, and test it on a hyperspectral image and give an application
to gray-level texture image segmentation.
Function-valued image segmentation. Let y be an image defined on a finite grid Ω, with values
in a function set F. We aim at segmenting y into L regions, that is at finding x : Ω → {1, . . . , L}
such that each region {s ∈ Ω : xs = l} (1 ≤ l ≤ L) corresponds to one object. A Bayesian
approach is to consider the segmentation x̂ that maximizes P (X = x|Y = y), or equivalently,
that maximizes fY |X=x (y)PX (x) among x : Ω → {1, . . . , L}. The widely used minimal partition
model estimates the data term fY |X=x using an isotropic Gaussian model. For the regularity term,
it uses a parameter β > 0 and chooses the Potts (1952) model
PX (x) ∝ e−β
P
s∼t
1xs 6=xt
,
(1)
where ∼ refers
P to a neighborhood system in Ω, typically the 4 or the 8 nearest neighbor system. The
potential s∼t 1xs 6=xt corresponds to a discrete boundary length between regions: short boundary
segmentations are preferred.
In our context, the isotropic Gaussian assumption on the data term is not admissible because it is
not able to account for any function regularity. The only assumption we make is that the functions
Q
(ys )s∈Ω are independent conditionally to the segmentation, yielding fY |X=x (y) = s∈Ω fYs |Xs =xs (ys ),
each term of which can be estimated using functional kernel density estimation (Dabo-Niang,
2004)
X
d(y
,
y
)
t
s
fˆYs |Xs =xs (ys ) ∝
K
,
(2)
h
t:x =x
t
s
where K is a positive kernel, h > 0 is the K’s bandwidth, and d is a distance on the space of
functions F. The normalizing factor, not written here, is linked to the small ball probability, but
luckily enough does not depend on the segmentation.
Results. The maximization algorithm details can be found in Delsol, Louchet (2014), so we prefer focusing here on the choice of the distance d. In many hyperspectral images, the L2 distance
between the derivatives of the functions (Tsai, Philpot, 1998) is able to accurately measure a discrepancy between functions because it is more robust to vertical shifts of the functions, even if it
is more sensitive to noise than usual L2 distance. This distance was used in Figure 1 (a-c) where a
fake lemon slice was well distinguished from a real one.
Another application to gray-level image texture segmentation is proposed. A texture will be described by the density of gray levels estimated at each pixel’s neighborhood: a simple gray-level
image becomes a function-valued image, where each function is a density. In this particular case,
natural choices of distance are transport distances: they allow for small horizontal shifts, and are
straighforward to compute in dimension 1. The L2 transport distance was used in a mouse brain
MRI to extract the cerebellum region which is more textured than the others (Figure 1 (d-e)).
(a)
(b)
(c)
(d)
(e)
Figure 1: (a) One slice of hyperspectral image (fake and real lemon) (b) Functions attached to a
sample of pixels (c) Segmentation using the distance between derivatives (d) MRI of mouse brain
(e) Cerebellum segmentation using L2 transport distance.
References
S. Dabo-Niang (2004). Kernel density estimator in an infinite-dimensional space with a rate of
convergence in the case of diffusion process. Applied Mathematics Letters, 17(4), pp. 381–386.
L. Delsol, C. Louchet (2014). Segmentation of hyperspectral images from functional kernel density
estimation. Contributions in infinite-dimensional statistics and related topics, 101–105.
D. Mumford, J. Shah (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Comm. on Pure and Appl. Math., XLII (5), 577–685.
R. B. Potts (1952). Some generalized order-disorder transformations. Mathematical Proceedings,
48(1), 106–109.
F. Tsai, W. Philpot (1998). Derivative analysis of hyperspectral data. Remote Sensing of Environment, 66(1), 41–51.