An EM Algorithm for Spatial Regression with Double Censoring Using a Curvelet Expansion

ABSTRACT: We present a nonparametric Expectation-Maximization (EM) algorithm for spatial regression in image processing applications, specifically addressing the challenge of double censoring. In scenarios like environmental monitoring, medical imaging, and remote sensing, image pixel intensities, representing true spatial responses Y(s) at locations s, are often subject to right-censoring (sensor saturation for bright values) and left-censoring (detection limits for dark values). We model the regression function using a finite curvelet dictionary, which is particularly effective at capturing anisotropic and curvilinear features (edges, ridges) prevalent in 2D images. Assuming an additive Gaussian noise model, our EM algorithm's E-step computes closed-form conditional moments for censored responses, while the M-step updates curvelet coefficients via weighted least squares. We establish theoretical guarantees, including monotonicity of the observed-data likelihood, stationarity of limit points, and consistency of the estimator in a fixeddictionary setting. A core contribution of this work is its empirical validation through extensive experiments on synthetic images and real-world camera test images. Our method was evaluated against several baselines (Naive LS, Winsorized LS, Tobit, Wavelet-EM, Total Variation) across various censoring severities and noise levels. Results demonstrate that our Curvelet-EM approach consistently outperforms existing methods in latent field recovery, achieving lower RMSE in censored regions, higher PSNR, and superior SSIM scores. Keywords: EM algorithm, spatial regression, double censoring, curvelets, truncated normal, consistency 2020 Mathematics Subject Classification. Primary 46L55; Secondary 44B20