Displacement Estimation by Maximum-Likelihood Texture Tracking
ABSTRACT This paper presents a novel method to estimate displacement by maximum-likelihood (ML) texture tracking. The observed polarimetric synthetic aperture radar (PolSAR) data-set is composed by two terms: the scalar texture parameter and the speckle component. Based on the Spherically Invariant Random Vectors (SIRV) theory, the ML estimator of the texture is computed. A generalization of the ML texture tracking based on the Fisher probability density function (pdf) modeling is introduced. For random variables with Fisher distributions, the ratio distribution is established. The proposed method is tested with both simulated PolSAR data and spaceborne PolSAR images provided by the TerraSAR-X (TSX) and the RADARSAT-2 (RS-2) sensors.
- SourceAvailable from: David Baratoux[show abstract] [hide abstract]
ABSTRACT: A complete and detailed map of the ice-velocity field on mountain glaciers is obtained by cross-correlating SPOT5 optical images. This approach offers an alternative to SAR interferometry, because no present or planned RADAR satellite mission provides data with a temporal separation short enough to derive the displacements of glaciers. The methodology presented in this study does not require ground control points (GCPs). The key step is a precise relative orientation of the two images obtained by adjusting the stereo model of one “slave”' image assuming that the other “master” image is well georeferenced. It is performed with numerous precisely-located homologous points extracted automatically. The strong ablation occurring during summer time on the glaciers requires a correction to obtain unbiased displacements. The accuracy of our measurement is assessed based on a comparison with nearly simultaneous differential GPS surveys performed on two glaciers of the Mont Blanc area (Alps). If the images have similar incidence angles and correlate well, the accuracy is on the order of 0.5 m, or 1/5 of the pixel size. Similar results are also obtained without GCPs. An acceleration event, observed in early August for the Mer de Glace glacier, is interpreted in term of an increase in basal sliding. Our methodology, applied to SPOT5 images, can potentially be used to derive the displacements of the Earth's surface caused by landslides, earthquakes, and volcanoes.Remote Sensing of Environment. 03/2005;
- [show abstract] [hide abstract]
ABSTRACT: The contribution of radar interferometry to the field of digital terrain modeling is important because this technique offers specific features which optical instruments cannot attain. However, the complexity of the height restitution and the accuracy of the result strongly depend on the orbital geometry at the time of the data takes. The present study aims at assessing the potential of a given image pair with regard to interferometry and at automatically reducing the phase ambiguity intrinsic to such processing. Particular applications of differential interferometry are also discussed in order to estimate their requirements and prepare future experimentsIEEE Transactions on Geoscience and Remote Sensing 04/1993; · 3.47 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: Geophysical applications of radar inter-ferometry to measure changes in the Earth's surface have exploded in the early 1990s. This new geodetic technique calculates the interference pattern caused by the difference in phase between two images acquired by a spaceborne synthetic aperture radar at two distinct times. The resulting interferogram is a contour map of the change in distance between the ground and the radar instrument. These maps provide an unsurpassed spatial sampling density (100 pixels km 2), a competitive pre-cision (1 cm), and a useful observation cadence (1 pass month 1). They record movements in the crust, pertur-bations in the atmosphere, dielectric modifications in the soil, and relief in the topography. They are also sensitive to technical effects, such as relative variations in the radar's trajectory or variations in its frequency standard. We describe how all these phenomena contribute to an interferogram. Then a practical summary explains the techniques for calculating and manipulating interfero-grams from various radar instruments, including the four satellites currently in orbit: ERS-1, ERS-2, JERS-1, and RADARSAT. The next chapter suggests some guide-lines for interpreting an interferogram as a geophysical measurement: respecting the limits of the technique, assessing its uncertainty, recognizing artifacts, and dis-criminating different types of signal. We then review the geophysical applications published to date, most of which study deformation related to earthquakes, volca-noes, and glaciers using ERS-1 data. We also show examples of monitoring natural hazards and environ-mental alterations related to landslides, subsidence, and agriculture. In addition, we consider subtler geophysical signals such as postseismic relaxation, tidal loading of coastal areas, and interseismic strain accumulation. We conclude with our perspectives on the future of radar interferometry. The objective of the review is for the reader to develop the physical understanding necessary to calculate an interferogram and the geophysical intu-ition necessary to interpret it.
398 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 5, NO. 3, JUNE 2011
Displacement Estimation by Maximum-Likelihood
Olivier Harant, Lionel Bombrun, Member, IEEE, Gabriel Vasile, Member, IEEE,
Laurent Ferro-Famil, Member, IEEE, and Michel Gay, Member, IEEE
Abstract—This paper presents a novel method to estimate
displacement by maximum-likelihood (ML) texture tracking. The
observed polarimetric synthetic aperture radar (PolSAR) data-set
is composed by two terms: the scalar texture parameter and the
speckle component. Based on the Spherically Invariant Random
Vectors (SIRV) theory, the ML estimator of the texture is com-
puted. A generalization of the ML texture tracking based on the
Fisher probability density function (pdf) modeling is introduced.
For random variables with Fisher distributions, the ratio distri-
bution is established. The proposed method is tested with both
simulated PolSAR data and spaceborne PolSAR images provided
by the TerraSAR-X (TSX) and the RADARSAT-2 (RS-2) sensors.
Index Terms—Maximum-likelihood (ML), offset tracking, po-
larimetric synthetic aperture radar (SAR), spherically invariant
random vectors, texture.
climate changes makes these investigations more and more
required as the glaciers are good indicators for local climate
Different approaches using both optical and SAR sensors
have been proposed to derive displacement fields. In the optical
domain, optical flow methods have been successfully validated
, but those methods are strongly dependent by weather
phenomenon (snow fall, etc.) which will change the scene
illumination. In addition, optical flow methods required cloud-
less images. Because of its all weather and all-day monitoring
capabilities, SAR imagery offers a number of advantages for
Earth-surface and feature observation. Different approaches
is investigated for years. The progressive awareness to
Manuscript received April 20, 2010; revised November 17, 2010; accepted
November 28, 2010. Date of publication December 17, 2010; date of current
(ANR) through the EFIDIR Project (ANR-2007-MCDC0-04, The associate ed-
itor coordinating the review of this manuscript and approving it for publication
was Prof. Jocelyn Chanussot.
O. Harantiswith the GIPSA-Lab, CNRSINPG-961, 46-38402Saint-Martin-
d’Hères, France,and alsowith the IETR Laboratory, SAPHIR Team,University
ofRennes 1,35042Rennes, France(e-mail:email@example.com-
L. Bombrun, G. Vasile, and M. Gay are with the GIPSA-Lab, CNRS INPG-
961, 46-38402 Saint-Martin-d’Hères, France (e-mail: lionel.bombrun@gipsa-
L. Ferro-Famil is with the IETR Laboratory, SAPHIR Team, University of
Rennes 1,35042 Rennes, France(e-mail:firstname.lastname@example.org).
Color versions of one or more of the figures in this paper are available online
Digital Object Identifier 10.1109/JSTSP.2010.2100365
have been proposed to derive displacement fields with SAR
imagery: Differential Interferometric SAR (D-InSAR) and
offset tracking techniques.
Although the potential of D-InSAR methods  have been
successfully validated on different geophysical objects such as
volcanoes, landslides , glaciers –, its applicationis lim-
ited to coherence preservation. For the generation of new high
resolution sensors (RADARSAT-2, TSX), the repeat time ob-
servation intervals becomes larger. It varies from 11 days for
TSX to 24 days for RADARSAT-2 compared to 1 day during
the ERS-1/2 tandem mission. For TSX data, coherence is not
preserved at 11 days in winter on Alpine glaciers . Another
limitation of D-InSAR techniques concerns the fact that the dis-
placement estimated is only a projection in the line-of-sight
(LOS) direction. Different hypothesis have been proposed to re-
trieve the three components. A common assumption is to con-
sider a flow parallel to the glacier surface and in the direction of
maximum averaged downhill slope . Another approach is to
combine both ascending and descending passes .
Nevertheless, the new generation of recently launched SAR
sensors are now able to produce high quality images of the
Earth’s surface with meter resolution. The decrease of the reso-
features. Offset tracking methods in the SAR domain are now
more and more studied. It exists different techniques:
• The speckle tracking technique correlates small blocks to
determine the relative displacement in the range and az-
imuth (along-track) directions  . This technique
does not depend on image feature tracking but rather on
the fact that there is coherence between the blocks. As co-
herence is not preserved with TSX for temperate Alpine
glaciers, the speckle tracking method is not well suited.
• Recently, a noveltrackingmethod based on Isolated Point
Scatterer (IPS) has been proposed . It corresponds to
the matched filter between the signal backscattered by an
IPS and the ideal response: a double cardinal sinus. This
method is valid only for particular objects such as corner
• The classical intensity tracking technique based on the
normalized cross-correlation (NCC) criterion  .
• The Maximum-Likelihood (ML) texture tracking al-
gorithm which takes into account the statistics of the
backscattered signal .
ture tracking algorithm introduced by Erten et al. . It takes
into accounts the statistics of the texture parameter extracted
from the PolSAR data. Fig. 1 shows the global scheme of this
1932-4553/$26.00 © 2010 IEEE
HARANT et al.: DISPLACEMENT ESTIMATION BY ML TEXTURE TRACKING 399
Fig. 1. Global scheme of the generalized ML texture tracking method.
This paper is organized as follows. In Section II, the Spher-
ically Invariant Random Vectors (SIRV) model is introduced
to extract the texture component from PolSAR data. Next, we
focus on the texture modeling. Then, in Section III, the ML tex-
texture model with both uncorrelated and correlated texture be-
tween images. Section IV presents results on simulated data, on
tière glacier. Finally, some conclusion and perspectives of this
work are discussed.
II. SIRV MODEL
With the new generation of airborne and spaceborne SAR
sensors, the number of scatterers present in each resolution
cell decreases considerably, homogeneous hypothesis of the
PolSAR clutter can be reconsidered. Heterogeneous clutter
models have therefore recently been studied with POLSAR
data with the SIRV processes .
From a PolSAR point of view, the target vector
defined as the product of a square root of a positive random
(representing the texture) with an independent com-
plex Gaussian vector
with zero mean and covariance matrix
(representing the speckle):
where the superscript
denotes the complex conjugate trans-
the mathematical expectation.
B. SIRV Estimation Scheme
For a given covariance matrix
, the ML estimator of the
is given byfor the pixel
for the reciprocal case).
The ML estimator of the normalized covariance matrix under
the deterministic texture case is the solution of the following
is the dimensionof thetarget scattering vector(
ized covariance matrix depends on the texture pdf
given by 
is the density generator function defined by 
is limited to the “approximate” ML estimator (3). Pascal et al.
have established the existence and the uniqueness, up to a scalar
matrix, as well as the convergence of the recursive algorithm
whatever the initialization  . In this paper, the trace of
the covariance matrix is normalized to
scattering vector. In practice, the normalized covariance matrix
is first computed because it does not depend on the texture com-
ponent. Then, the ML estimator of the texture parameter is esti-
mated according to (2).
It is important to notice that in the SIRV definition, the prob-
ability density function (pdf) of the texture random variable is
not explicitly specified. As a consequence, SIRVs describe a
whole class of stochastic processes. This class includes the con-
ventional clutter models having Gaussian,
pdfs which correspond respectively to Dirac, Gamma, Inverse
Gamma, and Fisher distributed texture –.
the dimension of target
C. Texture Modeling
1) Fisher pdf: The Fisher pdf is the Pearson type VI distri-
bution, it is defined by three parameters as –
As Fisher pdfs can be viewed as the Mellin convolution of a
Gamma pdf by an Inverse Gamma pdf , they can fit distri-
butions with either heavy heads or heavy tails.
2) Benefit of Fisher PDF: A glacier area (80
from theX-bandTSXdata overtheChamonixMont-Blanctest-
400 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 5, NO. 3, JUNE 2011
Fig. 2. ? ?? plan for a glacier area over the Chamonix Mont-Blanc test-site
site has been extracted. Then, the covariance matrix
the texture parameter
are estimated according to (2) and (3).
To see the benefit of Fisher pdfs to model the texture of PolSAR
plan has been plotted in Fig. 2. It shows the
evolution of the second log-cumulant
. In this plan, Gamma and Inverse Gamma pdf are
log-cumulants estimates are
versus the third log-
case of Fig. 2, a 7
to compute the log-cumulants. Fisher pdfs cover all the space
between the blue and red line .
In this example, 15.53% of the pixels are Beta distributed
the red line) and 84.47% are Fisher distributed. It shows that
Fisher pdfs are well adapted to model PolSAR clutter . In
the following, the texture parameter will be considered to be
After having shown the benefit of Fisher pdfs to model the
texture of high-resolution PolSAR data, we propose to imple-
ment this distribution in the ML texture tracking algorithm. As
Fisher pdfs are a generalization of Gamma pdfs, the proposed
algorithm can be seenasan extensionof thealgorithm proposed
by Erten et al. .
is the number of pixels in the sliding window. In the
7 pixels sliding window has been used
III. TEXTURE TRACKING
Classical algorithms estimate the shift between images by
maximizing the normalized cross-correlation coefficient. This
criterion is the ML solution for optical data corrupted by
additive noise  or for complex SAR data having circular
Gaussian statistics . With increasing the resolution of
PolSAR data, the number of scatterers in each resolution de-
creases. The central limit theorem may not be respected and the
Gaussian hypothesis may be reconsidered. Consequently, the
NCC criterion may not be optimal for high-resolution PolSAR
data. In this section, the texture tracking algorithm is improved
based on the SIRV model and the Fisher distribution for texture
blocks of the PolSAR data-set containing
resent respectively the slave and master images. According to
(2) and (3), the texture blocks
master one with a displacement
limited to a translation and has only two components
in range and azimuth. The ML texture tracking algorithm esti-
mates the shift vector
by maximizing for each slave block
the conditional density function (cdf) . It yields
pixels. They rep-
. In this study, the vectoris
of the texture ratio
the shift vector. This study has to be done when the texture be-
tween images is uncorrelated and correlated.
must be established to estimate
B. Texture Model With Uncorrelated Texture Between Images
andare two independent and identically distributed
(i.i.d.) random variables, the pdf of the texture ratio
by [30, Eq. 6.56]
For Fisher distributed texture, the pdf of the ratio of two un-
correlated texture has been established (see Appendix A), its
expression is given by
pergeometric function and the Euler Beta function
andare respectively the Gauss hy-
the two shape parameters of the Fisher pdf. The scale parameter
simplifies because the texture ratio variable is studied.
According to , (8) is equivalent to
HARANT et al.: DISPLACEMENT ESTIMATION BY ML TEXTURE TRACKING 401
By taking the natural logarithm of (12), one can prove that
the criterion to maximize to estimate the shift vector for uncor-
related texture between images is
C. Texture Model With Correlated Texture Between Images
In the case of correlated texture between two images, the bi-
variate Fisher distribution with marginal Fisher pdf should be
used. Its pdf is defined by six parameters as 
For Fisher distributed texture, the pdf of the ratio of two cor-
related texture has been established (see Appendix B), its ex-
pression is given by
procedureByfollowing thesameas described in
Section III-B, the criterion to maximize to estimate the
shift vector for correlated texture between images is given by
It plays a role similar as the cross-correlation coefficient which
takes into account the spatial arrangement of pixels.
A. On Simulated Data
Simulations have been performed to test the reliability of
the ML shift estimators. One Master/Slave image pair is sam-
pled from the same Fisher pdf for each region. The master and
Fig. 3. NCC, uncorrelated and correlated Gamma ML, uncorrelated and cor-
related Fisher ML criteria computed on simulations. (a) Simulated texture data.
(b) Detection surfaces for the multiplicative noised dataset: NCC, uncorrelated
and correlated Gamma ML, uncorrelated and correlated Fisher ML.
The border is sampled from
and corresponds to the size of the estimation
neighborhood. The simulated dataset has been corrupted with
independent multiplicative noise sampled from a Gamma pdf
. Fig. 3(a) shows an example of one simulation.
Five shift estimators are then computed: NCC, uncorrelated
and correlated Gamma ML, uncorrelated and correlated Fisher
ML. Since there is no motion between the two texture images,
the detection surface should be flat except in the center, where
a peak is expected.
Fig. 3(b) illustrates the results obtained where 1000 Monte
Carlo simulations have been performed for the shift estimators
(NCC and MLs). The NCC, both uncorrelated and correlated
Gamma ML and the uncorrelated Fisher ML estimators fail to
detect the no-motion: the criteria are very noisy without any
peak. However, the correlated Fisher ML gives a smoother
detection surface with a more pronounced detection peak. This
recommends the correlated Fisher ML texture tracking with
. The center is sampled
402IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 5, NO. 3, JUNE 2011
Fig. 4. D-InSAR results from 2009-01-06/2009-01-17 couple. (a) Amplitude.
(b) Coherence. (c) Phase images.
B. On Real Data
1) Argentière Glacier Test-Site: This work introduces some
preliminary TSX and RADARSAT-2 observations of the Ar-
gentière glacier in order to estimate its displacement. It is lo-
cated in the Mont-Blanc massif, its head (catchment area) starts
near 3000 m. Its slope is quite regular and not steep except at
the bottom where the seracs fall breaks the slope and discon-
nects the terminal part from the rest of the glacier. 14 TSX and
4 RADARSAT-2 images have been acquired on the Argentière
glacier during the winter and spring 2008–2009.
2) D-InSARPotential: Asthesurfaceofthetemperateglacier
erally at X-band, the interferometry on the surface of temperate
glaciers is difficult and its potential is quite limited . Fig. 4
shows amplitude, coherence and phase of an 11-day interfero-
gram acquired in winter (2009-01-06/2009-01-17). For this in-
terferometric couple, coherence is not preserved on the Argen-
tière glacier. The interferometric phase cannot be used to derive
3) Texture Tracking: As the new sensors provide higher res-
olution and according to the limitation of the interferometric
methods on the temperate glaciers, incoherent methods seems
promising. From October 2007 to June 2009, 14 TSX complex
dual-pol images in stripmap mode and 4 RADARSAT-2 com-
plex fine quad-pol images have been acquired over the Cha-
monix Mont-Blanc test-site. Table I summaries the main de-
tails of the two images pairs on which we have worked. Only
a coarse coregistration has been processed in each images pair
to avoid any distortion which could affect the polarimetric and
statistical properties. The subpixel coregistration values should
be subtracted to the texture tracking results.
a) On TSX data: Fig. 5 shows the displacement field de-
rived over one crevasses area of the Argentière glacier. The tex-
ture image pair is extracted using the SIRV estimation scheme
. A 64
256 pixels sliding window has been used to de-
DETAILS OF TSX AND RADARSAT-2 PRODUCTS
Fig. 5. Displacement estimation over a crevasse field of the Argentière glacier,
dual-pol TSX data, 2009-01-06/2009-02-08. (a) Master texture estimated using
SIRV model. (b) Displacement field in LOS. (c) Orientation map.
border of the glacier is closed to zero. Note that the mean dis-
placement over the crevasse field is about two times higher than
over the homogeneous area of the glacier. This corresponds to
the annual displacement estimation provided by glaciologists.1
Further studies need to quantitatively assess the derived shift
the conventional temperate glacier flow model: from the upper
left to the bottom right of the image.
Contrary to the NCC criterion, the confidence interval for the
ML similarity measure is hard to qualify. Further investigations
should be necessary to derive the false alarm probability for
Fisher distributed texture. In , Erten et al. have introduced
index defined by
is a measure of confidence. The higher is
accurate is the displacement estimation.
Table II shows the mean and variance of two samples ex-
tracted respectively from a crevasses area and an homogeneous
area of the glacier. The mean of
area and its variance is lower. It highlights the sensitivity of the
results are obtained on the crevasses areas.
b) Regularisation: For the study of geophysical objects, a
, the more
is higher on crevasses
HARANT et al.: DISPLACEMENT ESTIMATION BY ML TEXTURE TRACKING403
Fig. 6. von Mises pdf is the circular analogue of the normal distribution.
(a) von Mises pdf. ? ? ?. (b) Normal and von Mises pdfs. ? ? ???. ? ?? ?:
circularity of the von Mises pdf on ?????.
MEAN AND VARIANCE OVER TWO AREAS ON THE ARGENTIÈRE
GLACIER COMPUTED FROM TSX DATA
the displacement field. According to Bayes’ rule, the problem
andbe, respectively, the two components of the
along the distance and azimuth direc-
tion. In (18), the prior term
can be rewritten as
For the Argentière glacier, the assumptions of a flow parallel
hill slope have been successfully validated with in situmeasure-
parameter follows the von Mises distribution (also known as the
. They are linked with the distance and azimuth com-
are the polar coordinates of the displacement
Fig. 7. Displacement estimation over the Argentière glacier with reg-
ularisation according to Bayes’ rule, quad-pol RADARSART-2 data,
2009-01-29/2009-02-22. (a) Displacement field in LOS. (b) Orientation map.
circular normal distribution) which is the circular analog of the
normal distribution  (Fig. 6)
the maximum downhill slope issued from a digital elevation
is fixed here to .
tion of order . Concerning the absolute value of the displace-
, no constraint is imposed.
uniformly distributed in the search neighborhood.
c) On RADARSAT-2 data: Fig. 7 illustrates a displace-
ment estimation using quad-pol RADARSAT-2 data. On this
example, a regularization process has been applied according
to a Bayes’ rule formulation. The orientation map has been ex-
tracted from a DEM of the Mont-Blanc massif with a resolution
mean and standard deviation.is the direction in
is the modified Bessel func-
is therefore assumed to be
404 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 5, NO. 3, JUNE 2011
tation map [Fig. 7(b)] is consistent with the glacier flow model.
In this paper, the study has been focused on the generaliza-
the ability to fit a wide range of texture scenes. For glacier tex-
ture tracking, we recommend to use low-frequency bands as the
C-band or less to penetrate under the dry snow. This will permit
to observe structures (dust, eratic blocks, relief, etc.) present on
the ice surface which is more heterogeneous and more stable in
time than the snow.
In this paper, the ML texture tracking algorithm has been ap-
plied on Alpine glaciers. They present the advantage to have
largedisplacement. Nevertheless,theirsurface isquitehomoge-
neous and does not highlight the benefit of Fisher pdfs for tex-
ture modeling.This kind of method can be relevantfor the mon-
itoring of other geophysical objects such as volcanoes, earth-
quakes, etc., which have rough surfaces. The more heteroge-
neous the ground is and/or the higher the resolution is, the more
relevant the texture information is for displacement estimation,
segmentation and denoising.
form and second kind statistics define a well-suited formalism
a generalization of the well-known Gamma pdf. Here, the 3-pa-
rameter Fisher pdf has been used but other statistics such as the
KWBU pdfs system may be considered . Those 4-param-
eter distributions should permit a better texture modeling. Fur-
of models with more freedom degrees.
With the new generation of launched PolSAR sensors, the
Earth’s surface is imaged with meter resolution. Small spatial
features can then be observed from the space. Recently, more
and more studies are dedicated to texture extraction and mod-
eling. Based on this consideration, this paper has presented a
new texture tracking method to derive displacement fields from
PolSAR data. According to the SIRV estimation scheme, the
component. The proposed algorithm estimates a shift vector
through maximizing the cdf of two matched texture blocks.
Due to their capability to fit distributions with either heavy
heads or heavy tails, Fisher pdfs are well adapted to model
the texture variable. This observation has been illustrated on
real PolSAR data. Next, based on the assumption of Fisher dis-
tributed texture, the pdfs of the ratio of two texture variables
have been established for both uncorrelated and correlated tex-
ture between images. Then, a ML criterion has been established
and one for the slave texture images. This similarity measure is
which leads to the largest log-likelihood value yields to the es-
Then, the ML texture tracking algorithm has been applied on
simulated and real PolSAR data. The proposed algorithm has
been compared to the NCC criterion and the ML tracking algo-
rithm based on Gamma assumption for the texture component.
Contrary to the NCC criterion, the confidence interval for the
ML similarity measure is hard to qualify. Further investigations
should be necessary to derive the false alarm probability for
Fisher distributed texture.
As discussed before, the ML texture tracking confidence in-
factor which provides some information on the behavior of the
ML distribution. Nevertheless, some works are necessary to de-
fine the PFA for the ML criteria. This will permit to threshold
the similarity image and conclude or not on the relevance of the
Further works will deal with the addition of the covariance
matrix information to estimate displacement. Indeed, only the
texture variable is used in the ML tracking algorithm. The po-
the scattering mechanisms. This type of information has been
widely used in classification of PolSAR data. It should prob-
ably improve tracking performances.
This appendix gives the mathematical details of the pdf of
for uncorrelated Fisher distributed texture. In such case, the pdf
is obtained by replacing the expression of the Fisher pdf
(6) in (9), it yields
Next, by using the substitution
in (21), it
It has been shown the following relation which links an inte-
gral to the Gauss hypergeometric function 
. By identification between (22) and
HARANT et al.: DISPLACEMENT ESTIMATION BY ML TEXTURE TRACKING 405
(23), one can express
with the Gauss hypergeometric function
By combining(21)and (24),onecanobtaintheanalyticalpdf
One can rewrite (25) with the Euler Beta function it yields to
andare two correlated random variables, the pdf of
the texture ratio
is given by [30, Eq. 6.60]:
For correlated Fisher distributed texture, the bivariate Fisher
pdf (14) should be used to derive the pdf of
expression in (26), it leads
. By replacing its
By replacing in (27) the Gauss hypergeometric function by
its expression defined with the Pochhammer symbols
and if we swap the sum and the integral, it yields
fined by parameters
looks like an integral of a Fisher pdf de-
and , it yields
By combining (29), (30), and (31), one can prove that
the pdf of the ratio of two correlated Fisher distributed texture
shown in (15).
and. It yields to
406 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 5, NO. 3, JUNE 2011
 E. Berthier,H.Vadon,D.Baratoux,Y.Arnaud,C.Vincent,K.L.Feigl,
F. Rémy, and B. Legrésy, “Mountain glacier surface motion derived
from satellite optical imagery,” Remote Sens. Environ., vol. 95, no. 1,
pp. 14–28, 2005.
 D. MassonnetandT. Rabaute,“Radarinterferometry, limitsandpoten-
tial,” IEEE Trans. Geosci. Remote Sens., vol. 31, no. 2, pp. 455–464,
 D. Massonnet and K. Feigl, “RADAR interferometry and its applica-
tion to changes in the Earth’s surface,” Rev. Geophys., vol. 36, no. 4,
pp. 441–500, 1998.
 K. E. Mattar, P. W. Vachon, D. Geudtner, A. L. Gray, I. G. Cumming,
and M. Brugman, “Validation of alpine glacier velocity measurements
using ERS tandem-mission SAR data,” IEEE Trans. Geosci. Remote
Sens., vol. 36, no. 3, pp. 974–984, May 1998.
 B. T. Rabus and D. R. Fatland, “Comparison of SAR-interferometric
and surveyed velocities on a mountain glacier: Black rapids glacier,” J.
Glaciol., vol. 152, no. 46, pp. 119–128, 2000.
 N. Reeh, J. J. Mohr, S. N. Madsen, H. Oerter, and N. S. Gunder-
strup, “Three-Dimensional surface velocities of Storstømmen Glacier,
Greenland, derived from radar interferometry and ice-sounding radar
measurements,” J. Glaciol., vol. 49, no. 165, pp. 201–209, 2009.
 E. Trouvé, G. Vasile, M. Gay, L. Bombrun, P. Grussenmeyer, T.
Landes, J. M. Nicolas, Ph. Bolon, I. Petillot, A. Julea, L. Valet, J.
Chanussot, and M. Koehl, “Combining airborne photographs and
spaceborne SAR data to monitor temperate glaciers. potentials and
limits,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 4, pp. 905–924,
 R. Fallourd, O. Harant, E. Trouvé, J.-M. Nicolas, F. Tupin, M. Gay, G.
Vasile, L. Bombrun, A. Walpersdorf, J. Serafini, N. Cotte, L. Moreau,
and P. Bolon, “Monitoring temperate glacier: Combined use of multi-
MultiTemp’09, Groton, CT, 2009.
 I. R. Joughin, R. Kwok, and M. A. Fahnestock, “Interferometric esti-
mation of three-dimensional ice-flow using ascending and descending
passes,” IEEE Trans. Geosci. Remote Sens., vol. 36, no. 1, pp. 25–37,
 A. L. Gray, N. Short, K. E. Matter, and K. C. Jezek, “Velocities and
ice flux of the filchner ice shelf and its tributaries determined from
speckle tracking interferometry,” Can. J. Remote Sens., vol. 27, no. 3,
pp. 193–206, 2001.
 N. H. Short and A. L. Gray, “Potential for RADARSAT-2 interfer-
merometry: Glacier monitoring using speckle tracking,” Can. J. Re-
mote Sens., vol. 30, no. 3, pp. 504–509, 2004.
 F. Serafino, “SAR image coregistration based on isolated point scat-
terers,” IEEE Geosci. Remote Sens. Lett., vol. 3, no. 3, pp. 354–358,
 R. Michel, J. P. Avouac, and J. Taboury, “Measuring ground displace-
ments from SAR amplitude images: Application to the landers earth-
quake,” Geophys. Res. Lett., vol. 26, no. 7, pp. 875–878, 1999.
 T. Strozzi, A. Luckman, T. Murray, U. Wegmuller, and C. L. Werner,
“Glacier motion estimation using SAR offset-tracking procedures,”
IEEE Trans. Geosci. Remote Sens., vol. 40, no. 11, pp. 2384–2391,
 E. Erten, A. Reigber, O. Hellwich, and P. Prats, “Glacier velocity mon-
itoring by maximum likelihood texture tracking,” IEEE Trans. Geosci.
Remote Sens., vol. 47, no. 2, pp. 394–405, Feb. 2009.
 G. Vasile, J.-P. Ovarlez, F. Pascal, and C. Tison, “Coherency matrix
estimation of heterogeneous clutter in high resolution polarimetric
SAR images,” IEEE Trans. Geosci. Remote Sens., vol. 48, no. 4, pp.
1809–1826, Apr. 2010.
 L. Bombrun, G. Vasile, M. Gay, and F. Totir, “Hierarchical segmenta-
tion of polarimetric SAR images using heterogeneous clutter models,”
IEEE Trans. Geosci. Remote Sens., vol. 49, no. 2, pp. 726–737, Feb.
Distributions.London, U.K.: Chapman & Hall, 1990.
 S. Zozor and C. Vignat, “Some results on the denoising problem in the
elliptically distributed context,” IEEE Trans. Signal Process., vol. 58,
no. 1, pp. 134–150, Jan. 2010.
 F. Pascal, Y. Chitour, J. P. Ovarlez, P. Forster, and P. Larzabal,
“Covariance structure maximum-likelihood estimates in compound
gaussian noise: Existence and algorithm analysis,” IEEE Trans. Signal
Process., vol. 56, no. 1, pp. 34–48, Jan. 2008.
 F.Pascal,P. Forster,J.P. Ovarlez,andP.Larzabal, “Performanceanal-
ysis of covariance matrix estimates in impulsive noise,” IEEE Trans.
Signal Process., vol. 56, no. 6, pp. 2206–2216, Jun. 2008.
 J. S. Lee, D. L. Schuler, R. H. Lang, and K. J. Ranson, “K-distribution
for multi-look processed polarimetric SAR imagery,” in Proc. Geosci.
Remote Sens., IGARSS’94, Pasadena, CA, 1994, pp. 2179–2181.
 C. C. Freitas, A. C. Frery, and A. H. Correia, “The polarimetric G dis-
tribution for SAR data analysis,” Environmetrics, vol. 16, pp. 13–31,
 L. Bombrun and J.-M. Beaulieu, “Fisher distribution for texture mod-
eling of polarimetric SAR data,” IEEE Geosci. Remote Sens. Lett., vol.
5, no. 3, pp. 512–516, Jul. 2008.
 C. Tison, J.-M. Nicolas, F. Tupin, and H. Maître, “A new statistical
model for markovian classification of urban areas in high-resolution
SAR images,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 10, pp.
2046–2057, Oct. 2004.
plications des Logs-moments et des Logs-cumulants á l’Analyse des
 J.-M.Nicolas,Applicationde laTransforméedeMellin:étudedesLois
 M. G. Strintzis and I. Kokkinidis, “Maximum likelihood motion es-
timation in ultrasound image sequences,” IEEE Signal Process. Lett.,
vol. 4, no. 6, pp. 156–157, Jun. 1997.
 R. Bamler, “Interferometric stereo radargrammetry: Absolute height
determination from ERS-ENVISAT interferograms,” in Proc. Geosci.
Remote Sens., IGARSS’00, 2000, vol. 2, pp. 742–745.
 A. Papoulis, Probability, Random Variables, and Stochastic Processes,
4th ed.New York: McGraw-Hill, 2002.
 A. H. El-Bassiouny and M. Jones, “A bivariate F distribution with
, pp. 465–481, 2008.
 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Func-
tions With Formulas, Graphs, and Mathematical Tables.
These de Doctorat, Univ. de Rennes I, Paris, France, 1993.
Olivier Harant was born in 1982. He received the
Engineer degree in electrical engineering from the
Ecole Suprieure de Chimie Physique et Electron-
ique, Lyon, France, in 2008. He is currently working
toward the Ph.D. degree in the SAR Polarimetry,
Holography, Interferometry, and Radargrammetry
(SAPHIR) Team, Institut of Electronic and Telecom-
munication, Rennes, in collaboration with the
Grenoble Image sPeech and Automatic laboratory
His research interests include information mod-
eling in polarimetric SAR images and glaciers monitoring.
France, in 1982. He received the M.S. and Ph.D. de-
grees in signal, image, speech, and telecommunica-
tions from the Grenoble National Polytechnic Insti-
tute (INPG), Grenoble, France, in 2005 and 2008,
In 2008, he was a Teaching Assistant at Phelma,
Grenoble. From 2009 to 2010, he was a Postdoc-
toral Fellow from the French National Council for
Scientific Research (CNRS) between the Grenoble
Image Speech Signal Automatics Laboratory and
SONDRA, Gif-sur-Yvette, France. Since October 2010, he has been a Postdoc-
toral Fellow at the IMS Laboratory. His research interests include signal and
image processing, texture analysis, synthetic aperture radar remote sensing,
polarimetry, and interferometry.
HARANT et al.: DISPLACEMENT ESTIMATION BY ML TEXTURE TRACKING 407
Gabriel Vasile (S’06–M’07) received the M.Eng.
degree in electrical engineering and computer
science and the M.S. degree in image, shapes, and
artificial intelligence from the POLITEHNICA
University, Bucharest, Romania, in 2003 and 2004,
respectively, and the Ph.D. degree in signal and
image processing from Savoie University, Annecy,
France, in 2007.
From 2007 to 2008, he was a Postdoctoral Fellow
with the French Space Agency (CNES) and was
with the French Aerospace Laboratory (ONERA),
Palaiseau, France. In 2008, he joined the French National Council for Scientific
Research (CNRS), where he is currently a Research Scientist and a member of
the Grenoble Image Speech Signal Automatics Laboratory, Grenoble, France.
His research interests include signal and image processing, synthetic aperture
radar remote sensing, polarimetry, and interferometry.
Laurent Ferro-Famil (M’00) received the laurea
degree in electronics systems and computer engi-
neering, the M.S. degree in electronics, and the Ph.D.
degree, all from the University of Nantes, Nantes,
France, in 1996, 1996, and 2000, respectively.
Since 2001, he has been an Assistant Professor at
the University of Rennes I, Rennes, France. He is a
member of the Radar Polarimetry Remote Sensing
Group, Institute of Electronics and Telecommunica-
tions of Rennes (IETR). His current activities in edu-
cation concern analog electronics, digital communi-
cations microwave theory, and polarimetric radar imaging. He is especially in-
terested in SAR signal processing, radar polarimetry theory, and natural media
remote sensing from polarimetric interferometric SAR data, with application to
segmentation, classification, electromagnetic scattering modeling, physical pa-
rameter inversion, and time–frequency analysis.
Michel Gay (M’10) received the Engineer degree in
electrical engineering from the Institut des Sciences
de l’Ingéieur de Montpellier, Montpellier, France, in
1987 and the Ph.D. degree in physics from the Uni-
versity Joseph Fourier, Grenoble, France, in 1999.
From 1979 to 1985, he worked with the Botanic
Institute and with the local education authority
of Montpellier. From 1988 to 2003, he was with
Cemagref Grenoble, where he worked on electrical
engineering for environmental applications. Since
2004, he has been a Research Engineer with the
Grenoble Image Speech Signal Automatic Laboratory, Institut National
Polytechnique de Grenoble, Centre National de la Recherche Scientifique,
Saint-Martin-d’Hères, France. His current research interests include remote
sensing, synthetic aperture radar (SAR) image processing, and survey of Alpine
glaciers. He has been Co-Manager of four National Scientific projects and of
three international projects and Leader Project of one international project.
Dr. Gay is a member of the IEEE Geoscience Remote Sensing Society.