IEEE Geoscience and Remote Sensing Magazine - September 2013 - 26

tf l = arg min f l / n

df l (n)

+ m / n W f l (n) - H [W gl (n)]

(49)

where H [$] denotes the hard-thresholding operator.
Since the above estimator is prone to bias, the authors
of [122] propose to compute the despeckled image as
tf = exp (tf ') (1 + z 1 (L)/2), where z 1 (L) is the first-order
polygamma function and represents the variance of L-look
log-transformed speckle [33].
In general, the solution of the aforementioned optimization problems requires a suitable minimization scheme.
According to the properties of the functional to be minimized, several schemes can be used, including gradient projection [118], iterative splitting methods [122], [123], inverse
scale space flow [120]. The details of such minimization
schemes are beyond the scope of this tutorial and the related
literature is really vast. The interested reader is referred to
the above cited papers and the references therein.
An example of the application of the filter proposed in
[122] to the COSMO-SkyMed image in Fig. 10-(a) is given
in Fig. 12.
VIII. ASSESSMENT OF DESPECKLING FILTERS
One of the most challenging tasks is the validation and
quality assessment of data processed for speckle reduction.
The most evident problem is that the noise-free reflectivity that we wish to estimate is unknown, so that no
comparison can be carried out between the output of the
despeckling process and the actual ground truth. Another
important issue is the relationship between quality and
fidelity of despeckled SAR data. Like many other denoising
frameworks, the quality of a processed SAR image is usually evaluated in terms of blurring of homogeneous areas,
i.e., suppression of speckle noise, and detail preservation
in heterogeneous areas. Nonetheless, in incoherent SAR
imagery, a fundamental part of the information is represented by the relative values of the reflectivity of the targets, which allow measurements and inferences on the target scene. Consequently, the radiometric preservation of
the signal is an important requirement: a good despeckling
filter should not introduce bias on the reflectivity.
An immediate and subjective approach for quality
assessment is represented by visual inspection of filtered
images. Visual inspection permits detection of the main
human-visible features that characterize the behavior of a
despeckling filter. Such features include edge preservation
capability, degree of blur, point target preservation, as well
as structural artifacts which are hardly detected by objective and direct measurements. On the other hand, visual
assessment does not allow either quantitative comparisons
between the performances of different despeckling filters
to be made or the bias introduced by the filter to be effectively estimated.
In order to overcome the limitations of visual comparison, several objective performance indexes have been
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proposed in the literature for the quality assessment of
despeckling filters. They can be mainly divided into two
classes: with-reference and without-reference indexes.
With-reference indexes are commonly used in the
image denoising field. Their use implies that the noise-
free, or reference, image is known. A typical approach consists in choosing a reference image, either optical or synthetic, representing the actual reflectivity or ground-truth,
and creating a synthetically degraded version according to
a given signal model. These indexes permit a quantitative
and objective comparison between the performances of
different filters, which are expected to perform similarly
on real SAR images. Moreover, insights on filters behavior on specific image features, like edge preservation and
homogeneous areas smoothing, can be easily highlighted
by choosing appropriate reference images and even synthetic-generated patterns. Unfortunately, experimental
results carried out on simulated SAR images often are not
sufficient to infer the performances of despeckling filters
on real SAR images, since the synthetically speckled image
may not be consistent with the actual SAR image formation and acquisition processes. Furthermore, the statistical properties of the chosen reference image and of a real
ground-truth reflectivity can substantially differ.
On the contrary, without-reference indexes do not trust
on the knowledge of the ground-truth. They are uniquely
based on specific statistical hypotheses on the signal model.
Since the signal model is strongly dependent on the degree
of scene heterogeneity, a supervised selection of the most
appropriate areas for the computation of a specific index,
e.g., homogeneous areas, may be required.
In the following, the most used indexes belonging to
both the above mentioned classes are presented. Note that
the statistical operator of expectation E [$] and the moments
of the involved variables, such as the variance and covariance, here denoted as Var[$] and Cov [$] for the sake of simplicity, should be replaced by their empirical versions based
on spatial averages when evaluating the indexes.
A. With-reference indexes
The mean square error (MSE), or Euclidean distance, between
the ground-truth f and the despeckled image tf, and other
measures derived from the MSE, like the signal-to-noise
ratio (SNR), the peak signal-to-noise ratio (PSNR) and the
energy signal-to-noise ratio (ESNR), have been widely used
for the quality assessment of both denoising and despeckling [33], [57], [76]. Unlike the case of additive signal-independent noise, in the presence of signal-dependent noise
the MSE is strongly influenced by the average signal level of
the ground truth. Consequently, a quantitative evaluation
of despeckling filters using this kind of indexes is strongly
dependent on the content of the ground-truth image, even
though performance hierarchy is usually preserved across
different images.
MSE-based measurements are useful to obtain a global
performance assessment on the whole image, but usually
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