Statistical analysis based on complex representation of real-valued two-dimensional signal derived from modified Hilbert transform

Complex representation of one-dimensional (1-D) real-valued signals, or named as the analytic signal is defined by combination of original signal and its Hilbert transform. With associated instantaneous frequency, amplitude and phase, it plays a great role in 1-D statistical analysis. As an extension to two-dimensional (2-D) condition, complex representation of 2-D real-valued signal derived from modified 2-D Hilbert transform is introduced as retrievable expression without redundant information. Based on this definition, we explore and present 2-D statistical properties involving correlations functions of 2-D real signal and its corresponding complex expression in this paper. Obvious similarity of their forms to well researched 1-D condition reveals the close relation and provides new approaches to explore 2-D stochastic analysis.