In this thesis, we develop statistical methods for extracting significant information from biomedical signals. Biomedical signals are not only generated from a complex system but also affected by various random factors during their measurement. The biomedical signals may then be studied in two aspects: observational noise that biomedical signals experience and intrinsic nature that noise-free signals possess. We study Magnetic Resonance (MR) images and speech signals as applications in the one- and two-dimensional signal representation. In MR imaging, we study how observational noise can be effectively modeled and then removed. Magnitude MR images suffer from Rician-distributed signal-dependent noise. Observing that the squared-magnitude MR image follows a scaled non-central Chi-square distribution on two degrees of freedom, we optimize the parameters involved in the proposed Rician-adapted Non-local Mean (RNLM) estimator by minimizing the Chi-square unbiased risk estimate in the minimum mean square error sense. A linear expansion of RNLM's is considered in order to achieve the global optimality of the parameters without data-dependency. Parallel computations and convolution operations are considered as acceleration techniques. Experiments show the proposed method favorably compares with benchmark denoising algorithms. Parametric modelings of noise-free signals are studied for robust speech applications. The voiced speech signals are often modeled as the harmonic model with the fundamental frequency, commonly assumed to be a smooth function of time. As an important feature in various speech applications, pitch, the perceived tone, is obtained by way of estimating the fundamental frequency. In this thesis, two model-based pitch estimation schemes are introduced. In the first, an iterative Auto Regressive Moving Average technique estimates harmonically tied sinusoidal components in noisy speech signals. Dynamic programming implements the smoothness of the fundamental frequency. The second introduces the Continuous-time Voiced Speech (CVS) model, which models the smooth fundamental frequency as a linear combination of blockwise continuous polynomial bases. The model parameters are obtained via a convex optimization with constraints, providing an estimate of the instantaneous fundamental frequency. Experiments validate robustness and accuracy of the proposed methods compared with some current state-of-the-art pitch estimation algorithms.

During their acquisition, magnetic resonance (MR) images are affected by random noise, causing the observed magnitude image samples to be Rician distributed. In order to reduce the noise level while preserving the relevant image features, we develop an optimized Non-Local Means (NLM) denoising algorithm. The most sensitive parameters of the proposed NLM estimator are optimized on the squared-magnitude image, which follows a non-central chi-square distribution on two degrees of freedom. This minimum MSE optimization is performed via the minimization of the so-called chi-square unbiased risk estimate (CURE). Taking advantage of some acceleration techniques involving convolutions and parallel computation, we show that the proposed CURE-optimized NLM outperforms some state-of-the-art NLM algorithms with no increase in computation time.

In this article, we propose an innovative way of estimating pitch from speech waveform data, using an iterative ARMA technique that efficiently estimates multiple frequency components of a time series. Additionally, the harmonic structure of voiced speech and the smoothness of its pitch period are incorporated into the iterative ARMA technique, and this novel integration results in an efficient, robust technique for pitch estimation. The KED-TIMIT database was used to evaluate the performance of our proposed algorithm against that of other state-of-the-art pitch estimators in terms of both root mean square error and gross error rate.

In this paper we propose a model-based approach to instantaneous pitch estimation in noisy speech, by way of incorporating pitch smoothness assumptions into the well-known harmonic model. In this approach, the latent pitch contour is modeled using a basis of smooth polynomials, and is fit to waveform data by way of a harmonic model whose partials have time-varying amplitudes. The resultant nonlinear least squares estimation task is accomplished through the Gauss-Newton method with a novel initialization step that serves to greatly increase algorithm efficiency. We demonstrate the accuracy and robustness of our method through comparisons to state-of-the art pitch estimation algorithms using both simulated and real waveform data.