This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 1 Bivariate Gamma Distribution for Wavelength- Resolution SAR Change Detection Viet Thuy Vu , Member, IEEE, Natanael Rodrigues Gomes , Mats I. Pettersson, Member, IEEE, Patrik Dammert, Senior Member, IEEE , and Hans Hellsten, Senior Member, IEEE Abstract—A gamma probability density function (pdf) is shown to be an alternative to model the distribution of the magnitudes of high-resolution, i.e., wavelength-resolution, syn- thetic aperture radar (SAR) images. As investigated in this paper, it is more appropriate and more realistic statistical in comparison with, e.g., Rayleigh. A bivariate gamma pdf is con- sidered for developing a statistical hypothesis test for wavelength- resolution incoherent SAR change detection. The practical issues in implementation of statistical hypothesis test, such as assump- tions on target magnitudes, estimations for scale and shape parameters, and implementation of modified Bessel function, are addressed. This paper also proposes a simple processing scheme for incoherent change detection to validate the proposed statistical hypothesis test. The proposal was experimented with 24 CARABAS data sets. With an average detection probability of 96%, the false alarm rate is only 0.47 per square kilometer. Index Terms— Bivariate gamma, CARABAS, change detection, synthetic aperture radar (SAR). I. I NTRODUCTION S YNTHETIC aperture radar (SAR) change detection is an application of SAR imagery for detecting the changes in ground scene between measurements at different times. The causes of changes in ground scene can be deforestation and appearance or disappearance of ground targets. Change detection methods based on only SAR image magnitudes are categorized by incoherent change detection that enables detect- ing change in the order of resolution. For coherent change detection, both amplitude and phase are considered allowing detection in the order of wavelength. For SAR systems with the resolution around the wavelength that will be focused in this paper, there is no big difference in performance of coherent and incoherent change detection. Wavelength-resolution SAR change detection has been researched for decades. The research was initialized with Manuscript received February 26, 2018; revised May 9, 2018; accepted June 24, 2018. This work was supported by the KK Foundation. (Corresponding author: Viet Thuy Vu.) V. T. Vu and M. I. Pettersson are with the Blekinge Institute of Technology, 37179 Karlskrona, Sweden (e-mail: viet.thuy.vu@bth.se; mats.pettersson@bth.se). N. R. Gomes is with the Department of Electronics and Computer, Fed- eral University of Santa Maria, Santa Maria 97105-900, Brazil (e-mail: natanelrodriguesgomes@ufsm.br). P. Dammert and H. Hellsten are with Saab Surveillance, 41289 Gothenburg, Sweden (e-mail: patrik.dammert@saabgroup.com; hans.hellsten@saabgroup.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2018.2856926 CARABAS change detection experiments and followed by SAR change detection method development. A detail of the experiments can be found in [1] and [2], and the data are available for downloading online. One of the early publications on wavelength-resolution change detection method develop- ment is presented in [1], proposing a coherent method and an incoherent method. Both methods are based on statistical hypothesis tests that are simplified into the form of space–time adaptive processing. Basically, space–time adaptive processing exploits differences between target, clutter, and noise in space– time characteristics. By this, clutter and noise are adaptively suppressed. A similar statistical hypothesis test derived for small stacks of magnitude SAR images (three images) is pre- sented in [3]. Following the suggestion in [1], a new statistical hypothesis test based on the bivariate Rayleigh probability density function (pdf) has been investigated recently in [4]. However, the significant mismatch between Rayleigh and data histogram, especially in the tail of Rayleigh pdf, may degrade the performance of wavelength-resolution SAR change detec- tion. More appropriate and more realistic statistical models than Rayleigh can provide better change detection results. Beside Rayleigh distribution, gamma distribution is an alternative for statistically modeling the magnitudes of SAR images. A gamma likelihood ratio test statistic (the simplified form of a test statistic of two complex Wishart dis- tributions) has also been considered for SAR change detection, edge detection, line detection, and segmentation [5]. Gamma distribution belongs to a two-parameter family of probability distributions (Rayleigh requires only a single-scale parameter). This increases the adaptability and flexibility of the gamma pdf in modeling the magnitudes of SAR images. More importantly, the pdf for bivariate gamma distribution is available. This is a necessary condition for developing a likelihood ratio test. Several bivariate gamma pdfs are available in the statistics literature. A review of bivariate gamma distributions can be found in [7] and [8]. A bivariate gamma pdf is derived in [9] for the case that two random variables share a common scale parameter. In the case that two random variables share a common shape parameter, the bivariate gamma pdf introduced in [10] can be used. One of the general bivariate gamma pdfs without limits on shape and scale parameters is suggested in [11] (Izawa’s bivariate gamma). Since the expression is not in closed form (integral with Bessel and exponential functions), the computational cost can be very high if it is considered for a statistical hypothesis test. Fortunately, 0196-2892 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.