Phases under Gaussian additive noise

The phase of complex transforms like Fourier, Complex wavelet and curvelet of an image is more immune to noise than its magnitude. This article analyses its immunity to additive white Gaussian noise (AWGN) both mathematically and quantitatively. We have derived noise sensitivity (i.e. the rate of change of noisy image phase or magnitude with respect to AWGN magnitude) and used Structural Similarity Index Measure (SSIM) and Peak Signal to Noise Ratio (PSNR) to quantify its effects. The results indicate that the magnitude of these transforms deteriorates faster than that of phase with increasing noise strength. The noise sensitivity of phases for different transforms is compared. It is observed that the wavelet phase retains more structural similarity while the curvelet phase is more immune to noise.