On Bounds of Shift Variance in Two-Channel Multirate Filter Banks

Critically sampled multirate FIR filter banks exhibit periodically shift variant behavior caused by nonideal antialiasing filtering in the decimation stage. We assess their shift variance quantitatively by analysing changes in the output signal when the filter bank operator and shift operator are interchanged. We express these changes by a so-called commutator. We then derive a sharp upper bound for shift variance via the operator norm of the commutator, which is independent of the input signal. Its core is an eigensystem analysis carried out within a frequency domain formulation of the commutator, leading to a matrix norm which depends on frequency. This bound can be regarded as a worst case instance holding for all input signals. For two channel FIR filter banks with perfect reconstruction (PR), we show that the bound is predominantly determined by the structure of the filter bank rather than by the type of filters used. Moreover, the framework allows to identify the signals for which the upper bound is almost reached as so-called near maximizers of the frequency-dependent matrix norm. For unitary PR filter banks, these near maximizers are shown to be narrow-band signals. To complement this worst-case bound, we derive an additional bound on shift variance for input signals with given amplitude spectra, where we use wide-band model spectra instead of narrow-band signals. Like the operator norm, this additional bound is based on the above frequency-dependent matrix norm. We provide results for various critically sampled two-channel filter banks, such as quadrature mirror filters, PR conjugated quadrature filters, wavelets, and biorthogonal filters banks.     

Analysis of Multiple Orientations

Estimation of local orientations in multivariate signals is an important problem in image processing and computer vision. This general problem formulation also covers optical flow estimation, which can be regarded as orientation estimation in space-time-volumes. Modelling a signal using only a single orientation, however, is often too restrictive, since occlusions and transparencies occur frequently, thus necessitating the modelling and analysis of multiple orientations. We, therefore, develop a unifying mathematical model for multiple orientations: Beyond describing an arbitrary number of orientations in scalar- and vector-valued image data such as color image sequences, it allows the unified treatment of additively and occludingly superimposed oriented structures as well as of combinations of these. Based on this model, we describe estimation schemes for an arbitrary number of additively or occludingly superimposed orientations in images. We confirm the performance of our framework on both synthetic and real image data.     

Automated segmentation and tracking for large-scale analysis of focal adhesion dynamics

Cell adhesion, a process mediated by the formation of discrete structures known as focal adhesions (FAs), is pivotal to many biological events including cell motility. Much is known about the molecular composition of FAs, although our knowledge of the spatio-temporal recruitment and the relative occupancy of the individual components present in the FAs is still incomplete. To fill this gap, an essential prerequisite is a highly reliable procedure for the recognition, segmentation and tracking of FAs. Although manual segmentation and tracking may provide some advantages when done by an expert, its performance is usually hampered by subjective judgement and the long time required in analysing large data sets. Here, we developed a model-based segmentation and tracking algorithm that overcomes these problems. In addition, we developed a dedicated computational approach to correct segmentation errors that may arise from the analysis of poorly defined FAs. Thus, by achieving accurate and consistent FA segmentation and tracking, our work establishes the basis for a comprehensive analysis of FA dynamics under various experimental regimes and the future development of mathematical models that simulate FA behaviour.     

Simultaneous reconstruction of activity and attenuation for PET/MR

Medical investigations targeting a quantitative analysis of the position emission tomography (PET) images require the incorporation of additional knowledge about the photon attenuation distribution in the patient. Today, energy range adapted attenuation maps derived from computer tomography (CT) scans are used to effectively compensate for image quality degrading effects, such as attenuation and scatter. Replacing CT by magnetic resonance (MR) is considered as the next evolutionary step in the field of hybrid imaging systems. However, unlike CT, MR does not measure the photon attenuation and thus does not provide an easy access to this valuable information. Hence, many research groups currently investigate different technologies for MR-based attenuation correction (MR-AC). Typically, these approaches are based on techniques such as special acquisition sequences (alone or in combination with subsequent image processing), anatomical atlas registration, or pattern recognition techniques using a data base of MR and corresponding CT images. We propose a generic iterative reconstruction approach to simultaneously estimate the local tracer concentration and the attenuation distribution using the segmented MR image as anatomical reference. Instead of applying predefined attenuation values to specific anatomical regions or tissue types, the gamma attenuation at 511 keV is determined from the PET emission data. In particular, our approach uses a maximum-likelihood estimation for the activity and a gradient-ascent based algorithm for the attenuation distribution. The adverse effects of scattered and accidental gamma coincidences on the quantitative accuracy of PET, as well as artifacts caused by the inherent crosstalk between activity and attenuation estimation are efficiently reduced using enhanced decay event localization provided by time-of-flight PET, accurate correction for accidental coincidences, and a reduced number of unknown attenuation coefficients. First results achieved with measured whole body PET data and reference segmentation from CT showed an absolute mean difference of 0.005 cm?1 (< 20%) in the lungs, 0.0009 cm?1 (< 2%) in case of fat, and 0.0015 cm?1 (< 2%) for muscles and blood. The proposed method indicates a robust and reliable alternative to other MR-AC approaches targeting patient specific quantitative analysis in time-of-flight PET/MR.     

Analysis of nerve fibers and their distribution in histologic sections of the human brain

The field of quantitative analysis and subsequent mapping of the cerebral cortex has developed rapidly. New powerful tools have been applied to investigate large regions of complex folded gyrencephalic cortices in order to detect structural transition regions that might partition different cortical fields of disjunct neuronal functions. We have developed a new mapping approach based on axoarchitectonics, a method of cortical visualization that previously has been used only indirectly with regard to myeloarchitectonics. Myeloarchitectonic visualization has the disadvantage of producing strong agglomerative effects of closely neighbored nerve fibers. Therefore, single and neurofunctional-relevant parameters such as axonal branchings, axon areas, and axon numbers have not been determinable with satisfying precision. As a result, different staining techniques had to be explored in order to achieve a suitable histologic staining for axon visualization. The best results were obtained after modifying the Naoumenko-Feigin staining for axons. From these contrast-rich stained histologic sections, videomicroscopic digital image tiles were generated and analyzed using a new fiber analysis framework. Finally, the analysis of histologic images provided topologic ordered parameters of axons that were transferred into parameter maps. The axon parameter maps were analyzed further via a recently developed traverse generating algorithm that calculated test lines oriented perpendicular to the cortical surface and white matter border. The gray value coded parameters of the parameter maps were then transferred into profile arrays. These profile arrays were statistically analyzed by a reliable excess mass approach we recently developed. We found that specific axonal parameters are preferentially distributed throughout granular and agranular types of cortex. Furthermore, our new procedure detected transition regions originally defined by changes of cytoarchitectonic layering. Statistically significant inhomogeneities of the distribution of certain axon quantities were shown to indicate a subparcellation of areas 4 and 6. The quantification techniques established here for the analysis of spatial axon distributions within larger regions of the cerebral cortex are suitable to detect inhomogeneities of laminar axon patterns. Hence, these techniques can be recommended for systematic and observer-supported cortical area mapping and parcellation studies.     

Comparative Analysis of Shift Variance and Cyclostationarity in Multirate Filter Banks

Multirate filter banks introduce periodic time-varying phenomena into their subband signals. The nature of these effects depends on whether the signals are regarded as deterministic or as random signals. We analyze the behavior of deterministic and wide-sense stationary (WSS) random signals in multirate filter banks in a comparative manner. While aliasing in the decimation stage causes subband energy spectra of deterministic signals to become shift-variant, imaging in the interpolation stage causes WSS random signals to become WS cyclostationary (WSCS). We provide criteria to quantify both shift variance and cyclic nonstationarity. For shift variance, these criteria separately assess the shift dependence of energy and of energy spectra. Similarly for nonstationarity, they separately assess the nonstationary behavior of signal power and of power spectra. We show that, under aliasing cancellation and perfect reconstruction constraints of paraunitary and biorthogonal filter banks, these criteria evaluate the behavior of deterministic and WSS random signals in a consistent, dual way. We apply our criteria to paraunitary and biorthogonal filter banks as well as to orthogonal block transforms, and show that, for critical signals such as lines or edges in image data, the biorthogonal 9/7 filters perform best among these     

Analysis of superimposed oriented patterns.

Estimation of local orientation in images may be posed as the problem of finding the minimum gray-level variance axis in a local neighborhood. In bivariate images, the solution is given by the eigenvector corresponding to the smaller eigenvalue of a 2 x 2 tensor. For an ideal single orientation, the tensor is rank-deficient, i.e., the smaller eigenvalue vanishes. A large minimal eigenvalue signals the presence of more than one local orientation, what may be caused by non-opaque additive or opaque occluding objects, crossings, bifurcations, or corners. We describe a framework for estimating such superimposed orientations. Our analysis is based on the eigensystem analysis of suitably extended tensors for both additive and occluding superpositions. Unlike in the single-orientation case, the eigensystem analysis does not directly yield the orientations, rather, it provides so-called mixed-orientation parameters (MOPs). We, therefore, show how to decompose the MOPs into the individual orientations. We also show how to use tensor invariants to increase efficiency, and derive a new feature for describing local neighborhoods which is invariant to rigid transformations. Applications are, e.g., in texture analysis, directional filtering and interpolation, feature extraction for corners and crossings, tracking, and signal separation.