ABSTRACT
The InSAR coherence magnitude encapsulates important information on the underlying scattering mechanism. Hence the importance of achieving accurate and unbiased estimators of the true coherence.
From the theoretical probability density function (PDF) of the sample coherence magnitude, as a function of the true coherence and the number of samples in the averaging estimator, two problems can be pointed out: large variance for low true coherence values; and a significant bias with a low number of samples in the average.
The problem might be solved by averaging more samples in the neighborhood of each data element (multilooking), but the cost would be to reduce the resolution. We propose here a method that overcomes this difficulty and achieves a better estimator expected value and variance by adding a time dimension, i.e. by averaging samples of realizations of the same random process (eventually with different moments) measured at different dates.
The filter works in a combined space–scale–time domain. In the scale–space domain, a wavelet soft thresholding technique provides, for each co-registered coherence amplitude data set, local estimates of the coherence modulus. Wavelet thresholding is equivalent to local signal averaging with a kernel that adapts to the signal regularity in the neighborhood of each sample, and therefore can cope with both stationary areas and discontinuities. Best wavelet thresholds are defined by estimating the wavelet coefficients noise variance as a function of the coherence mean value on a grid of points in the datasets.
This step computes local (in space) estimates of the coherence amplitude mean value within each dataset. The local mean values are used as weights in the average in the time dimension in the stack of datasets.
The filter is first validated using testing data-holding pseudorandom coherence samples generated by a Monte Carlo technique. Quantitative measures of the filter performance are provided using the simulated images. In particular, the reduction of the estimator’s variance is proven.
Several test cases are presented using InSAR coherence data acquired by -X in a project aimed at characterizing forest heterogeneity in a disturbance gradient of the Sungai Wain Protection Forest (SWPF) (Indonesia). The testing experiment included:
A K-means clustering exercise of homogeneous areas in a forest/non-forest mapping context. Cluster statistic of the filtered and original data, and Jeffries–Matusita distances prove the filter effectiveness in improving cluster separability.
Transects across the boundaries (discontinuities) between homogeneous targets (such as forest/non-forest) or across non-homogeneous targets, such as buildings. The gain achieved by the filtering method is measured, in an edge detection context, by the improvement of signal-to-noise ratio, by checking the data level of randomness using the entropy concept, and by spatial statistics provided by power spectra.
Finally, it is anticipated that the coherence filter can also pave the way for improved time-series analysis. In fact, clean trajectories of each resolution element (temporal features) can be exploited to track changes of the imaged ecosystems in time. An example is provided.