Wavelet Spectra for Multivariate Point Processes

Wavelets provide the flexibility to analyse stochastic processes at different scales. Here, we apply them to multivariate point processes as a means of detecting and analysing unknown non-stationarity, both within and across data streams. To provide statistical tractability, a temporally smoothed wavelet periodogram is developed and shown to be equivalent to a multi-wavelet periodogram. Under a stationary assumption, the distribution of the temporally smoothed wavelet periodogram is demonstrated to be asymptotically Wishart, with the centrality matrix and degrees of freedom readily computable from the multi-wavelet formulation. Distributional results extend to wavelet coherence; a time-scale measure of inter-process correlation. This statistical framework is used to construct a test for stationarity in multivariate point-processes. The methodology is applied to neural spike train data, where it is shown to detect and characterize time-varying dependency patterns.

Analysis of Point Based Image Registration Errors With Applications in Single Molecule Microscopy

We present an asymptotic treatment of errors involved in point-based image registration where control point (CP) localization is subject to heteroscedastic noise; a suitable model for image registration in fluorescence microscopy. Assuming an affine transform, CPs are used to solve a multivariate regression problem. With measurement errors existing for both sets of CPs this is an errors-in-variable problem and linear least squares is inappropriate; the correct method being generalized least squares. To allow for point dependent errors the equivalence of a generalized maximum likelihood and heteroscedastic generalized least squares model is achieved allowing previously published asymptotic results to be extended to image registration. For a particularly useful model of heteroscedastic noise where covariance matrices are scalar multiples of a known matrix (including the case where covariance matrices are multiples of the identity) we provide closed form solutions to estimators and derive their distribution. We consider the target registration error (TRE) and define a new measure called the localization registration error (LRE) believed to be useful, especially in microscopy registration experiments. Assuming Gaussianity of the CP localization errors, it is shown that the asymptotic distribution for the TRE and LRE are themselves Gaussian and the parameterized distributions are derived. Results are successfully applied to registration in single molecule microscopy to derive the key dependence of the TRE and LRE variance on the number of CPs and their associated photon counts. Simulations show asymptotic results are robust for low CP numbers and non-Gaussianity. The method presented here is shown to outperform GLS on real imaging data.

Measurement Errors in Fluorescence Microscopy Image Registration

Image registration is an important processing step in fluorescence microscopy, for example in tracking or super-resolution methods. Precision localization of single fluorescent molecules from a quantum limited photon detection process, subject to Gaussian readout noise, is key to the use of single molecule microscopy. It is therefore important to know the effect that registration has on the accuracy of localizing a single molecule. Here we demonstrate a suitable approach to image registration that accounts for point-wise errors in localizing the control points typically used in fluorescence microscopy. This allows expressions for the localization errors caused by the registration process to be derived, showing dependence on the number of control points and their associated photon counts.

IMAGE REGISTRATION ERROR ANALYSIS WITH APPLICATIONS IN SINGLE MOLECULE MICROSCOPY

This paper is concerned with assessing localization errors emanating from the image registration of two monochromatic fluorescence microscopy images. Assuming an affine transform exists between images, registration in this setting typically involves using control points to solve a multivariate linear regression problem; however with measurement errors existing in both sets of variables the use of linear least squares is inappropriate. It is shown that image registration is an errors-in-variable problem and as such the correct method is to use generalized least squares. Traditionally this requires the measurement errors to be independent and identically distributed (iid); an assumption that is rarely satisfied in practical situations. An extension of the multivariate generalized least squares estimator that allows non-iid noise is applied. The distributional properties of the estimators are used to derive localization errors emanating from the image registration process in terms of photon counts and experimental parameters.