Parameter Estimation Based on Scale-Dependent Algebraic Expressions and Scale-Space Fitting

We present our results of applying wavelet theory to the classic problem of estimating the unknown parameters of a model function subject to noise. The model function studied in this context is a generalization of the second-order Gaussian derivative of which the Gaussian function is a special case. For all five model parameters (amplitude, width, location, baseline, undershoot-size), scale-dependent algebraic expressions are derived. Based on this analytical framework, our first method estimates all parameters by substituting into a given expression numerically obtained values, such as the zero-crossings of the multiscale decompositions of the noisy input signal, using Gaussian derivative wavelets. Our second method takes these estimates as starting values for iterative least-squares optimization to fit our algebraic zero-crossing model to observed numeric zero-crossings in scale-space. For evaluation, we apply our method together with three reference methods to the three-parameter Gaussian model function. The results show that our method is on average 3.7 times more accurate than the respective best reference method for signal-to-noise ratios (SNR) from $-{\text{10}}$ to 70 dB, using a synthetic test scenario proposed by a competitor. For our full five-parameter model, we investigate overall estimation error as well as per-parameter error and per-parameter uncertainty as a function of SNR and various noise models, including correlated noise. To demonstrate practical effectiveness and relevance, we apply our method to the well-studied problem of QRS complex delineation in electrocardiography signals. Out-of-the-box results show a performance comparable to the best algorithms known to date, without relying on problem-specific heuristic decision rules.