On The Pascal Transform and Threshold Selection

The Pascal transform has been introduced recently and it has applications in signal processing and image processing. Applications of the Pascal transform to edge or bump detection often require processing of noisy signals. Since noise can seriously distort the values of the signal transform coefficients, it is important to understand the effects of noise on the Pascal transform of a signal. We focus on evaluating the mean square of the transform coefficients of Gaussian noise and the magnitude of the transform coefficients of the noise. An analytical expression is found for mean square of the transform coefficients of zero mean Gaussian noise. Simulation results are presented for the magnitude of the Pascal transform coefficients of the noise from which models are developed for predicting the magnitude of the transform coefficients. We also present an approach for selecting a threshold to determine a bump using the Pascal transform.