Estimation of Piecewise Continuous Regression Function in Finite Dimension using Oblique Regression Tree with Applications in Image Denoising

Decision trees are one of the most widely used nonparametric methods for regression and classification. In existing literature, decision tree-based methods have been used for estimating continuous functions or piecewise-constant functions. However, they are not flexible enough to estimate the complex shapes of jump location curves (JLCs) in two-dimensional regression functions. In this article, we explore the Oblique-axis Regression Tree (ORT) and propose a method to efficiently estimate piece-wise continuous functions in a general finite dimension with fixed design points. The central idea involves clustering the local pixel intensities by recursive tree partitioning and using the local leaf-only averaging for estimation of the regression function at a given pixel. The proposed method can preserve complex shapes of the JLCs well in a finite-dimensional regression function. Due to a different set of assumptions on the underlying regression function, the overall framework of the proofs is different from what is available in the literature on regression trees. Theoretical analysis and numerical results, particularly on image denoising, indicate that the proposed method effectively preserves complicated edge structures while efficiently removing noise from piecewise continuous regression surfaces.