Hitting Corners The Lipschitz Regularity, a Measure of Discontinuities in Radar Images Connected with Forest Spatial Distribution

• Published in: Spatial Analysis for Radar Remote Sensing of Tropical Forests Vol. 1; pp. 173 - 235
• Main Authors: De Grandi, Gianfranco (Frank), De Grandi, Elsa Carla
• Format: Book Chapter
• Language: English
• Published: CRC Press 2021
• Edition: 1
• Subjects: HH Backscatter; Wavelet Frame; Wavelet Coefficients; HV Backscatter; Low Coherence Area; Lipschitz Regularity; Shallow Incidence Angle; Co-polar Channels; Mueller Matrices; Wavelet Modulus; Real SAR Data; Wavelet Transform; SAR Backscatter; Maxima Trajectory; InSAR Coherence; Wavelet Maxima; Smoothed Step Function; Polarization Orientation Angle; Backscattering Coefficients; Incidence Angle; Speckle Noise; Forest Segment; Lipschitz Condition
• ISBN: 0367259400; 9780367742669; 0367742667; 9780367259402
• DOI: 10.1201/9780429290657-9

Measures of discontinuities (edges, point targets) in radar images can provide valuable information for the detection of thematic features, such as the margin between forest and grassland, or forest degradation. For the purpose, a technique is proposed here for the characterization of image discontinuities in a combined space-scale- polarization domain. This analysis is based on a wavelet frame that acts as a differential operator. The underpinning mathematical theory for the local characterization of the image singular structures hinges on the concept of Lipschitz regularity, which affords a refinement to the approximation of a function by Taylor polynomials. While the Taylor polynomial provides an integer upper bound to the approximation error by nth order differentiability, the Lipschitz condition provides a non-integer exponent α of the bound K (hereafter dubbed Lip parameters). The mathematics of the Lipschitz regularity concept is reviewed, and a link is established with the scaling properties of the wavelet frame. Moreover, an extension to the theory of distributions is carried out to derive the case of singularity with α = − 1, these corresponding to a Dirac δ distribution. Estimates of the Lipschitz regularity are performed by following the trajectory in scale of the wavelet transform modulus maxima. Examples of singularities, such as step function α = 0, cusp α = 1, and Dirac δ α = − 1, with estimates of the Lip exponent in the continuous wavelet domain are provided. The concept of smoothed singularity is introduced to account for the fact that, in a real imaging system, only finite approximations of pure functions are available. This step introduces a third parameter for the characterization of the singularity, i.e. the variance σ 2 of a smoothing Gaussian function. The effect of speckle on the Lip estimators and the case of non-isolated singularities are also treated. A Monte Carlo simulator of polarimetric backscatter discontinuities is proposed. The purpose is to study in controlled situations the dependence of the estimated Lip parameters (α, K, σ 2) on the backscattering parameters, these being the polarization ellipse φ, χ angle and the incidence angle θ. The formalism of Lip signatures is introduced, these being functional representations (graphs) of each Lip parameter in the polarization state, incidence angle domain. Experiments are conducted using simulated signals including discontinuities, such as edges portraying the margin between forest and grassland, and the effect of a sloping terrain in the radar acquisition cross-range direction. Several test cases are next explored using real SAR data. TanDEM-X backscatter and InSAR coherence data are used. The experiments consist of the supervised definition of profiles of SAR data including discontinuities (e.g. at the margin between forest and grassland), and the estimation of the Lip parameters from the non-linear function defined by the wavelet modulus maxima trajectory with scale. Ways of representing approximations of the Lip parameters in the image space using a 2D discrete wavelet frame transform are proposed. This formalism produces a 3-band map, including for each image pixel the estimated Lip α, Lip K parameters, and the angle of the gradient provided by the two-dimensional wavelet differential operator. Examples of these representations are given using TanDEM-X imagery over tropical forests. Beyond thematic visual interpretation, the Lip map may be used in support of image processing tasks, such as edge detectors. A case in point will be developed in Chapter 8 of the book. Measures of discontinuities (edges, point targets) in radar images can provide valuable information for the detection of thematic features, such as the margin between forest and grassland, or forest degradation. For the purpose, a technique is proposed in this chapter for the characterization of image discontinuities in a combined space-scale- polarization domain. The chapter focuses on the upper part of the hierarchy of functions, on the class of functions that satisfy a Lipschitz condition. This observation has a practical impact on the use of Lipschitz conditions in synthetic aperture radars filtering algorithms, since impulses are associated at short scale to speckle noise. To gain an insight into the influence of speckle on the estimation of the Lip parameters, the probability of detecting the wavelet maximum of interest as a function of speckle strength and the underlying discontinuity shape is calculated.