IEEE Geoscience and Remote Sensing Magazine - September 2015 - 64

multicomponent remote sensing data is used in the reconstruction process (with maybe just a few components being used), the related data/components will reconstitute the
new 3D data with a given rule (discussed in Section VI-B).
In our framework, the diverse algorithms for reconstructing the missing information of remote sensing data
are divided into four main classes, based on the supplementary information sources: 1) spatial-based methods; 2)
spectral-based methods; 3) temporal-based methods; and
4) hybrid methods. Note that we usually say that data with
missing information are corrupted or incomplete.
III. SPATIAL-BASED METHODS
The spatial-based methods are the most traditional and
most basic approach of the four kinds of methods. This
category of methods fill in the missing data regions using the remaining parts of the data. This approach is
based on the assumption that the missing data and the
remaining data share the same statistical or geometrical
structures [6]. In other words, the spatial-based methods
utilize the correlations of the local or nonlocal information in the corrupted data itself. Most of these methods
have been intensively studied in the field of digital image processing [7-11], and can also be applied to remote
sensing image processing. The representative methods
include interpolation methods [12], [13], propagated
diffusion methods [7-9],
THE SPATIAL-BASED
[14], variation-based methMETHODS OnLy rELy On THE
ods [10], [11], [15-18], and
exemplar-based methods
SPATIAL cOrrELATIOn Of
[19-21]. It should be noted
THE cOrruPTED DATA ITSELf.
that because of the lack
of other reference data to
judge the recovery fidelity,
the goal of spatial-based reconstruction is that the recovered region shows a smooth transition at the junctions,
and is as physically plausible as possible, without unconnected edges, blurring, or inconsistent texture [6].
A. InterpolAtIon Methods
In the spatial-based reconstruction methods, interpolation
algorithms are the fundamental ones. Most spatial interpolation methods can be represented as weighted averages of
sampled values. They all share the same general estimation
formula, as follows:
N

tI ^ x h = / w I ^ x h
i
0
i

(1)

i =1

where tI ^ x 0 h is the estimated value of an attribute at the
point of interest p x 0, I ^ x ih is the observed value at the sampled point p x i, w i is the weight assigned to the sampled
point p x i, and N represents the number of sampled points
used for the interpolation [22]. The most commonly used
interpolation algorithms include nearest neighbor interpolation, bilinear interpolation, and cubic convolution
64

interpolation. Geostatistical interpolation methods [13],
[23] are also very useful in remote sensing data processing. Geostatistics includes several methods that use kriging
algorithms for estimating intermediate values. Kriging is a
kind of best linear unbiased prediction. The kriging weights
are estimated by minimizing the variance v 2E:
2
v 2E = E e $7I^ x 0 h - tI^ x 0 hA .
2
= E e 8_ tI^ x 0 hi +^I^ x 0 hh2 - 2tI^ x 0 hI^ x 0 hB

= C e ^ x 0, x 0 h- 2 / w j C e ^ x 0, x j h+/
N

N

j =1

/ w i w j C e ^ x i, x jh (2)
N

i =1 j =1

where C e ^ $ h is the covariance between observations,
E e ^ $ h represents the expectation, w j is the interpolation weight, and x i (x j) is the position of point p x i (p x j).
A typical example of the use of an interpolation method
for missing information reconstruction was described by
Zhang et al. [12], who filled ETM+ image gaps using an
ordinary kriging technique. This case study showed that
the geostatistical methods can be a useful tool for interpolating missing pixels.
In general, the interpolation methods are efficient and
easy to operate. However, the spatial information is not
fully utilized in most interpolation methods. Therefore,
these methods only work well in cases with relatively simple ground features.
B. propAgAted dIffusIon Methods
The propagated diffusion methods aim to recover the missing areas in such a way as to propagate the local information from the exterior to the interior of the missing areas,
which is analogous to the physical phenomenon of heat
propagation in physical structures. This effect can be formalized with partial differential equations (PDEs); therefore, the diffusion inpainting is performed using PDEs [24].
The simplest linear diffusion equation at time t 0 is:
2I
2t 0 = D I

(3)

where I represents the image data, and DI denotes the image Laplacian. The diffusion processing at point p x at time
t 0 + 1 has the discrete form:
I ^x h = I^x h + D I^x h

t 0 +1

t0

t0

(4)

This PDE evolution is isotropic diffusion, which propagates the image variations in all directions, as a low-pass
linear filtering. For this reason, this diffusion method introduces a lot of blurring to the recovered region.
Many anisotropic PDE models, which limit the diffusion around specific directions such as edges, have been
proposed for image inpainting. The diffusion inpainting
method was first proposed by Bertalmio et al. [7] in 2000.
This approach uses an anisotropic PDE model that propagates the image Laplacians from the surrounding neighborhood into the interior of the missing area. The directions of
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