Polarimetric SAR Image Analysis with IPython and Docker

Mort Canty (mort.canty@gmail.com)

May 2016

Introduction

The Docker container mort/sardocker provides easy access to Python scripts for analysis of polarimetric synthetic aperture radar (polSAR) imagery. Some of these scripts are described in Canty(2014), Image analysis, Classification and Change Detection in Remote Sensing, 3rd Revised Ed. The change detection scripts are based upon the work of Conradsen et al (2016). In addition to scripts for polSAR speckle filtering, ENL estimation and change detection, the container encapsulates the command line interface of the ASF MapReady software for terrain correction and geocoding of SAR images. The user interacts with the software in an IPython notebook served from within the Docker container.

Software Installation

1. Install Docker
2. In an X-terminal, run the command
   sudo docker run -d -p 433:8888 --name=sar -v your_image_directory:/home/imagery mort/sardocker
   where your_image_directory is the path to your SAR data. (For Windows or Mac OS. see the instructions for sharing volumes in the Docker documentation.)
3. Point your browser to
   localhost:433
4. Click on this tutorial, or open a new notebook with New/Python 2.

Here is a listing of the main directory /home in the container. It contains the various Python and bash scripts required for preprocessing and change detection:

The /home/imagery directory contains the polarimetric SAR data and is shared with the host. In the present example there are 12 Radarsat-2 quadpol images in SLC (single-look complex) format along with a dem (digital elevation model). Acquistion times range from May 25, 2009 (20090525) to October 11, 2010 (20101011):

The images are level one SLC (single look complex format). For example, below are the contents of the image directory corresponding to acquistion date 20100426 (April 26, 2010). The four polarization combinations HH, HV,VH and VV are are stored as complex numbers in GeoTiff format:

The subdirectory polsarpro/T3 contains the mult-looked polarimetric coherency matrix elements generated from the polarization combinations. This was done with the Sentinel-1 Toolbox provided as freeware by the European Space Agency (ESA). (See the discussion below in the Section on the processing chain.) The image files are in ENVI format:

We will return to these images shortly, but first a little theory:

Statistical Properties of polSAR Images

Vector and matrix representations

A fully polarimetric SAR measures a $2\times 2$ scattering matrix $S$ at each resolution cell on the ground. The scattering matrix relates the incident and the backscattered electric fields $E^i$ and $E^b$ according to

$$ \pmatrix{E_h^b \cr E_v^b} =\pmatrix{S_{hh} & S_{hv}\cr S_{vh} & S_{vv}}\pmatrix{E_h^i \cr E_v^i}. $$

Here $E_h^{i(b)}$ and $E_v^{i(b)}$ denote the horizontal and vertical components of the incident (backscattered) oscillating electric fields directly at the target. These can be deduced from the transmitted and received radar signals via the so-called far field approximations. If both horizontally and vertically polarized radar pulses are emitted and discriminated then they determine, from the above Equation, the four complex scattering matrix elements. The per-pixel polarimetric information in the scattering matrix $S$, under the assumption of reciprocity ($S_{hv} = S_{vh}$), can then be expressed as a three-component complex vector

$$ s = \pmatrix{S_{hh}\cr \sqrt{2}S_{hv}\cr S_{vv}}, $$

where the $\sqrt{2}$ ensures that the total intensity (received signal power) is consistent. It is essentially these vectors which are provided in the SLC level one data discussed above. The total intensity is referred to as the span and is the complex inner product of the vector $s$,

$$ {\rm span} = s^\top s = |S_{hh}|^2 + 2|S_{hv}|^2 + |S_{vv}|^2. $$

This is a real number and the corresponding gray-scale image is called the span image. The observation vector $s$ can be shown to be a realization of a complex multivariate normal random variable. An equivalent and often preferred representation is in terms of the coherency vector

$$ k = {1\over\sqrt{2}}\pmatrix{S_{hh} + S_{vv}\cr S_{hh} - S_{vv} \cr 2S_{hv}}. $$

The polarimetric signal is can also be represented by taking the complex outer product of $s$ with itself:

$$ C = s s^\top = \pmatrix{ |S_{hh}|^2 & \sqrt{2}S_{hh}S_{hv}^* & S_{hh}S_{vv}^* \cr \sqrt{2}S_{hv}S_{hh}^* & 2|S_{hv}|^2 & \sqrt{2}S_{hv}S_{vv}^* \cr S_{vv}S_{hh}^* & \sqrt{2}S_{vv}S_{hv}^* & |S_{vv}|^2 }. $$

The diagonal elements of $C$ are real numbers, with span $= {\rm tr}(C)$, and the off-diagonal elements are complex. This matrix representation contains all of the information in the polarized signal.

Multi-looking

The matrix $C$ can be averaged over the number of looks (number of adjacent cells used to average out the effect of speckle) to give an estimate of the covariance matrix of each multi-look pixel:

$$ \bar{C} ={1\over m}\sum_{\nu=1}^m s(\nu) s(\nu)^\top = \langle s s^\top \rangle = \pmatrix{ \langle |S_{hh}|^2\rangle & \langle\sqrt{2}S_{hh}S_{hv}^*\rangle & \langle S_{hh}S_{vv}^*\rangle \cr \langle\sqrt{2} S_{hv}S_{hh}^*\rangle & \langle 2|S_{hv}|^2\rangle & \langle\sqrt{2}S_{hv}S_{vv}^*\rangle \cr \langle S_{vv}S_{hh}^*\rangle & \langle\sqrt{2}S_{vv}S_{hv}^*\rangle & \langle |S_{vv}|^2\rangle }, $$

where $m$ is the number of looks. This matrix (or alternatively the equivalent coherency matrix $\langle k k^\top \rangle$) is generated with the Sentinel-1 Toolbox and stored in the subdirectory polsarpto/T3 which we listed in a cell above. Rewriting the first of the above equalities,

$$ m\bar{C} = \sum_{\nu=1}^m s(\nu) s(\nu)^\top =: x. $$

This quantity $x$ is a realization of a complex random matrix which is known to have a complex Wishart distribution with $N\times N$ covariance matrix $\Sigma$ (here $N=3$) and $m$ degrees of freedom:

$$ p_{W_c}( x) ={|x|^{(m-N)}\exp(-{\rm tr}(\Sigma^{-1} x)) \over \pi^{N(N-1)/2}|\Sigma|^{m}\prod_{i=1}^N\Gamma(m+1-i)},\quad m \ge N, $$

provided that the vectors $s(\nu)$ are independent and drawn from a complex multivariate normal distribution.

Dual and single polarimetric imagery

The scattering vector given above corresponds to so-called full, or quad polarimetric SAR. Satellite-based SAR sensors often operate in reduced, power-saving polarization modes, emitting only one polarization and receiving two (dual polarization) or one (single polarization). The look-averaged covariance matrices are reduced in dimension correspondingly. For instance for dual polarization with horizontal transmission and horizontal and vertical reception,

$$ \bar{C} = \pmatrix{ \langle |S_{hh}|^2\rangle & \langle S_{hh}S_{hv}^*\rangle \cr \langle S_{hv}S_{hh}^*\rangle & \langle |S_{hv}|^2\rangle }, $$

and, for single polarization and horizontal transmission/reception, we get simply the intensity image

$$ \bar{I} = \langle |S_{hh}|^2\rangle \quad {\rm or} \quad \bar{I} = \langle |S_{vv}|^2\rangle. $$

Equivalent number of looks

When multi-look averaging takes place, the observation vectors $s(\nu)$ are not completely independent; they will generally be correlated somewhat. In order to account for this, the complex Wishart distribution is often parameterized with ENL (rather than $m$) degrees of freedom, where ENL is the so-called equivalent number of looks. This quantity can be estimated from the image itself.

Returning to the Radarsat-2 image acquired April 26, 2010, we will geocode it with MapReady and then try to determine the equivalent number of looks.. The bash script /home/mapready.sh takes two arguments, the acquisition date in the format yyymmdd and the sensor (presently rs2quad or tsxdual):

We see that the multi-look images were created with the Sentinel-1 Toolbox with $4\times 3 = 12$ looks. This corresponds to a square pixel size of close to $20.5\times 20.5$ meters. The georeferenced coherency matrix image at this resolution is stored in a single 9-band image geotiff format.

Before we can display the image, we have to enable Matplotlib functionality within the notebook with the so-called magic command

We will use the Python script /home/dispms.py for displaying. Here is the help:

The command below will generate a $1000\times 1000$ pixel RGB color composite of the three diagonal elements:

The scene above was acquired over the city of Bonn, Germany (upper right hand corner with the Rhine river). The blueish, featurless areas are mixed forest (in this coherency decomposition, volume scattering), the green-yellow areas are built-up (double-bounce) and the redish fields are single-bounce.

To check the number of looks, We will run the /home/enlml.py script on a spatial subset which includes a lot forested land cover. This script is mased on an multivariate estimation method due to Anfinsen et al. (2009).

The spatial subset is entered with the -d flag as in the /home/dispms script. Here we choose -d [800,400,500,500]:

There are two modes (maxima) at about 6 and 12 looks. Here is the ENL image

The speckle statistics are most well-developed in the homogeneous forested areas (bright pixels) corresponding to the mode at ENL=12, so this value will be adopted in the sequel.

Speckle Filtering (The gamma-MAP filter)

Let us represent the $m$ look-averaged SAR intensity image by the random variable $G$ with mean $\langle G\rangle=x$, where $x$ is the underlying signal. Then ${\rm var}(G) = x^2/m$ and $G$ is gamma-distributed with density function

$$ p(g\mid x) = {1\over (x/m)^m\Gamma(m)}g^{m-1}e^{-gm/x}. $$

Let $G=xV$. Then it follows that $V$ has the density

$$ p(v) = {m^m\over\Gamma(m)}v^{m-1}e^{-vm}. $$

Therefore, in terms of the observed pixel intensities $g$ (realizations of $G$), we can write

$$ g = x v, $$

where v is distributed as above, and has mean 1 and variance $1/m$.

Because of this special multiplicative noise nature of speckle, conventional smoothing filters are not particularly suitable as an aid to SAR image interpretation.

The gamma maximum a posteriori (gamma MAP) de-speckling filter may be derived from Bayes' Theorem. The a posteriori conditional probability for $x$, given intensity measurement $g$ is

$$ {\rm pr}(x\mid g) = { p(g\mid x){\rm pr}(x)\over p(g) }, $$

where $p(g\mid x)$ is given above, ${\rm pr}(x)$ is the prior probability for $x$ and $p(g)$ is the total probability density for $g$. This formulation allows us to include prior knowledge of the signal statistics (or texture) if available. An empirical statistical model for $x$ is suggested by measurements of backscatter from ocean waves, namely

$$ {\rm pr}(x) \sim \left({\alpha\over\mu}\right)^\alpha {x^{\alpha-1}\over\Gamma(\alpha)}e^{-\alpha x/\mu}. $$

This is just the gamma probability density with $\beta=\mu/\alpha$, and hence with mean $\alpha\beta= \mu$ and variance

$$ {\rm var}(x) = \alpha\beta^2 = \mu^2/\alpha. $$

The parameters $\mu$ and $\alpha$ can be estimated as follows. By passing an $n\times n$ window over the image we can obtain $\bar g = \langle g\rangle$ and ${\rm var}(g)$. Then the estimates are

$$ \hat\mu = \bar g, $$

and $$ \hat\alpha = {\hat\mu^2\over {\rm var}(x)} = {\bar g^2\over {\rm var}(x)} ={1 + 1/m \over {{\rm var}(g)/\bar g^2 - 1/ m}}. $$

Hence the posterior probability for $x$ given measurement $g$ is

$$ {\rm pr}(x\mid g) \sim {1\over (x/m)^m\Gamma(m)}g^{m-1}e^{-gm/x}\left({\alpha\over\mu}\right)^\alpha {x^{\alpha-1}\over\Gamma(\alpha)}e^{-\alpha x/\mu} =: L $$

Taking the logarithm,

$$ \log L =\ m\log m -m\log x +(m-1)\log g - \log\Gamma(m)- mg/x +\alpha\log\alpha - \alpha\log\mu + (\alpha-1)\log x - \log\Gamma(\alpha) -\alpha x/\mu. $$

We now get the maximum a posteriori (MAP) value for $x$ given the observed pixel intensity $g$ by maximizing $\log L$ with respect to $x$:

$$ {d\over dx}\log L = -m/x + mg/ x^2 + (\alpha-1)/x - \alpha/\mu = 0. $$

This leads to a quadratic equation for the most probable signal intensity $x$,

$$ {\alpha\over\mu}x^2 + (m+1-\alpha)x - mg = 0, $$

where the parameters $\mu$ and $\alpha$ are estimated locally. Note that in homogeneous regions where $m\approx \bar g^2/{\rm var}(g)$, $\hat\alpha\to\infty$. In that case $x\approx \hat\mu = \bar g$.

The gamma MAP filter is not appropriate to the complex off-diagonal matrix elements as their {\it a priori} statistics are not well understood.

We will illustrate the Gamma-MAP speckle filter on the April 26 image. The Python script gamma_filter.py takes as input a polSAR image in covariance or coherency matrix form and filters the diagonal elements only:

Because of the use of $7\times 7$ directional filters to preserve detail, the the script is rather slow, and will attempt parallel execution on IPython engines if these are available. On a multi-core processor, IPython engines can be initiated with a Jupyter terminal command, e.g.,

ipcluster start -n 4

Note the good preservation of detailed structure.

It should be noted that this kind of adaptive filter is not an acceptable pre-processing step for change detection, since the equivalent number of looks after filtering is no-longer a characteristic of the entire scene.

Change Detection

Bitemporal data

The following is discussion is based on Conradsen et al (2003).

As we have seen, we can represent a pixel vector in an $m$ look-averaged polSAR image in covariance matrix format by $\bar C$, where

$$ m\bar C = x = \sum_{\nu=1}^m s(\nu) s(\nu)^\top $$

is a realization of a random matrix $X$ with a complex Wishart distribution.

In order to derive a change detection procedure for two polarimetric SAR images, we formulate a statistical test. For each pixel in the two image, define the null (or no-change) simple hypothesis as

$$ H_0:\quad \Sigma_1 = \Sigma_2 = \Sigma, $$

and the alternative composite hypothesis as

$$ H_1:\quad \Sigma_1 \ne \Sigma_2. $$

Under $H_0$ the maximum likelihood for $\Sigma$ can be shown to be given by

$$ L(\hat\Sigma) = { |x_1|^{m-3}|x_2|^{m-3}\exp(-2m\cdot{\rm tr}(I)) \over \left({1\over 2m}\right)^{3\cdot 2m}| x_1+ x_2|^{2m}\Gamma_3(m)^2 }, $$

where $I$ is the $3\times 3$ identity matrix and ${\rm tr}(I)=3$. Under $H_1$ the maximum likelihood for $\Sigma_1$ and $\Sigma_2$ is

$$ L(\hat\Sigma_1,\hat\Sigma_2) = { |x_1|^{m-3}|x_2|^{m-3}\exp(-2m\cdot{\rm tr}(I)) \over \left({1\over m}\right)^{3m}\left({1\over m}\right)^{3m} |x_1|^m |x_2|^m\Gamma_3(m)^2 } $$

Then the likelihood ratio test has the critical region for rejection of the no-change hypothesis

$$ Q = {L(\hat\Sigma) \over L(\hat\Sigma_1,\hat\Sigma_2) } = 2^{6m}{ |x_1|^m |x_2|^m \over |x_1 + x_2|^{2m} } \le k. $$

Finally, one can derive (Conradsen et al (2003)) the following approximation for the statistical distribution of the test statistic $Q$:

$$ {\rm prob}(-2\rho\log Q\le z) \simeq P_{\chi^2;N^2}(z) + \omega_2\left[ P_{\chi^2;N^2+4}(z) - P_{\chi^2;N^2}(z) \right], $$

where $P_{\chi^2;m}(z)$ is the chi square distribution wth m degrees of freedom, $$ \rho = 1 - {2N^2-1\over 6N}\cdot{3\over 2 m}, $$

and

$$ \omega_2 = - {N^2\over 4}\cdot\left(1-{1\over \rho}\right)^2 + {N^2(N^2-1)\over 24 \rho^2}\cdot{7\over 4m^2}. $$

In practice we choose a significance level $\alpha$, e.g., $\alpha = 0.01$, and decision threshold $z$ such that

$$ {\rm prob}(-2\rho\log Q\le z) = 1-\alpha $$

and interpret all pixels with larger values of $-2\rho\log Q$ as change.

Multitemporal data

The preceding discussion generalizes in a straightforward way to a time series of $k$ images (Conradsen et al (2016)). In the equation for the test statistic $Q$ above, the numerator consists of a product $k$ determinants $|x_1|\cdot|x_2|\cdot\dots|x_k|$ and the denominator similarly the determinant of the sum of the $k$ observations $|x_1+x_2+\dots x_k|$. The multitemporal test is referred to as the omnibus test will in general be more powerful (have a higher detection probability) for the same significance level than a bitemporal test just involving the first and last images. Furthermore, the test statistic can be factored into a product of test statistics that can be used to ascertain the time or times at which significant changes occur in the series.

A Processing Chain

The change detection method implies the following processing sequence in order to generate a change map from a time series of polarimetric SAR images provided at the single look complex (SLC) processing level:

1. First of all the multi-look polarimetric SAR images in covariance or coherency matrix format are generated from from the Sentinel 1 Toolbox available from the European Space agency. Presently this must be done outside of the Docker container (and IPython). Geocoding and terrain correction can optionally be applied within the Sentinel toolbox, or deferred to the next step. The coherenecy matrix has the same eigenvalues and hence the same determinant as the covariance matrix, so that the hypothesis test described above can be applied unchanged to either format. The rest of the processing takes place in the IPython notebook.

2. The matrix images can then be processed by MapReady for georeferencing and terrain correction if this has not yet been carried out (see above). The bash script /home/mapready.sh automates the procedure. MapReady will output the geocoded covariance/coherency matrix image in the form of co-registered GeoTiff files, one for each diagonal matrix element and two (real and imaginary parts) for each off-diagonal component. A python script /home/ingest.py is called automatically to combine these files to a single, multi-band image in floating point format.

3. The ENL (equivalent number of looks) can (optionally) be estimated with the script enlml.py. A multivariate estimator is used to accomplish this as described by Anfinsen et al. (2009).

4. The non-sequential multitemporal change detection algorithm for the Radarst-2 images is invoked with the bash script /home/omnibus_rs2.sh. This script calls the Python programs /home/register.py to co-register the sequence of $k$ images to the first in the series and then /home/omnibus.py to perform the pixel-wise hypothesis tests. The test statistics $-2\rho\log Q$ and change probabilities ${\rm prob}(-2\rho\log Q\le z)$ are written to a two-band GeoTiff file. Additionally a change map showing changes at the chosen significance level (default 0.01) in red overlayed onto the span image is writen to a three-band (RGB) GeoTiff file. The sequential algorithm is run by the bash script /home/sar_seq_rs2.sh which calls the routines /home/register.py and /home/sar_seq.py. At present it generates four change image files: three -single-band images showing, per pixel, the the period in which the first change occured (smap), the last change occurrence (cmap), and the change frequency (fmap) and a $k-1$-band file showing where changes occured in each of the $k-1$ intervals.

Change detection with Radarsat-2 Quadpol Images

We can illustrate with the sequential algorithm applied to all 12 Radarsat-2 images:

Here is a subset of the change map showing the time of the first change:

Here is the frequency of change map overlayed onto the 20090525 image, showing a "hotspot" at the moving dredges in the flooded sand quarry:

There are several "hotspots" in the urban area of Bonn. The one along the Rhine (upper middle, near the central Kennedy bridge) corresponds to the docking of excusion boats.