IEEE Geoscience and Remote Sensing Magazine - September 2015 - 65

the propagation are the directions of the isophotes, which
are lines of constant intensity within an image, and they are
estimated by the perpendicular direction to the image gradient at each point. The diffusion at point p x at time t 0 + 1
is governed by the equation:
I ^ x h = I ^ x h + d ` D I ^ x hj d = I ^ x h

t 0 +1

t0

t0

t0

(5)

where d is the gradient operator, and d = represents the
perpendicular direction of d (i.e., the direction of the
isophote). In this method, the image information is
propagated inside the missing zone in a way that aims to
preserve the isophote directions, as shown in Fig. 4(a).
The propagation directions are generally the key to the
process of diffusion. In order to precisely determine the
propagation directions, the method proposed in [14] employs an advanced tool called a "bandelet", which accurately calculates the geometrical flow of the image structures,
as shown in Fig. 4(b), and then the values are propagated
into the missing areas following the directions of the geometrical flow. The propagation at point p x i +1 is implemented by the equation:
I ^ x i +1 h = I ^ x ih + E d ^ x i +1, x ih Il ^ x ih

(7)

where E d ^ x i +1, x ih is the Euclidean distance between pixels
p x i +1 and p x i. I^ x i +1 h corresponds to the value of the nearest pixel that belongs to the missing areas, in the direction of the geometrical flow determined by the bandelet.
Il ^ x h is the directional derivative of I with respect to the
geometrical direction i that is calculated by the bandelet
transformation.
For the diffusion methods, many variants exist using
different PDE models, including linear, nonlinear, isotropic [25], and anisotropic models [26], to favor the propagation in particular directions. These methods tend to prolong structures to the interior of the missing areas. Thus,
they are suitable for recovering strong structures or filling
small regions. However, they are not suitable for reconstructing large areas and textures, in which they
lead to blurring.
C. Variation-Based Methods
Reconstructing the missing information of remote sensing data can also be seen as solving an
ill-posed inverse problem. It is standard to use
a regularization technique to make an inverse
problem well posed. Image regularization can be
formulated as a variational problem, where the
image amounts to a function of the bounded variation (BV), and the solution is the minimization
of a global energy function:
september 2015

(8)

I

where I is the clear target image data, J is the observed corrupted data, and M is a mask signifying the missing data.
The term MI - J qq is the data fidelity term, which expresses the fidelity between the observed corrupted data and the
target data, and the term U^ I h is the regularization term,
which gives a prior model of the target data. m is the regularization parameter, which controls the tradeoff between
the data fidelity and the regularization term. It is worth
noting that the regularization term ensures the uniqueness
of the recovery solution; therefore, how to select an appropriate regularization is very significant. Some frequently
used regularizations are introduced in the following.
1) , 2 Norm Regularization: , 2 norm regularizations, such
as Laplacian regularization [27], Tikhonov regularization
[28], and Gauss-Markov regularization [29], are the most
common regularization models for image processing, because they are simple and can be solved quickly and easily.
One of the representative , 2 norm regularizations is Laplacian regularization, which is defined as follows:
U^ I h = / x DI (x)

(6)

with
Il (x) = xl (x) cos i + yl (x) sin i

TI = argmin # MI - J qq + mU^ I h-

2
2

(9)

The goal of Laplacian regularization is to minimize the
energy of the high-frequency component of the images;
therefore, this model is able to suppress the high-frequency
information and impose smoothness constraints on the
images. For this reason, the Laplacian regularization inpainting model is suitable for the recovery of flat regions
or for use in low-resolution images. However, it performs
poorly in edge and detail information reconstruction, for
which it results in blurring.
2) variation (TV) regularization, which has been widely applied for image inpainting, such as in [1]. On the basis of (8), the TV regularization can be expressed as the
following equation:
U^ I h = / x dI^ x h

(10)

TV regularization is an effective inpainting technique
that is capable of recovering sharp edges. However, it

(a)

(b)

Figure 4. Propagation direction and geometrical flow. (a) Propagation along

isophote directions (from [7]). (b) Geometrical flow of the image structures.

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Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - September 2015
