An upper bound on the Bayesian error bars for generalized linear regression

In the Bayesian framework, predictions for a regression problem are expressed in terms of a distribution of output values. The mode of this distribution corresponds to the most probable output, while the uncertainty associated with the predictions can conveniently be expressed in terms of error bars. In this paper we consider the evaluation of error bars in the context of the class of generalized linear regression models. We provide insights into the dependence of the error bars on the location of the data points and we derive an upper bound on the true error bars in terms of the contributions from individual data points which are themselves easily evaluated.

On the relationship between Bayesian error bars and the input data density

We investigate the dependence of Bayesian error bars on the distribution of data in input space. For generalized linear regression models we derive an upper bound on the error bars which shows that, in the neighbourhood of the data points, the error bars are substantially reduced from their prior values. For regions of high data density we also show that the contribution to the output variance due to the uncertainty in the weights can exhibit an approximate inverse proportionality to the probability density. Empirical results support these conclusions.

Edges and bars: where do people see features in 1-D images?

There have been two main approaches to feature detection in human and computer vision - based either on the luminance distribution and its spatial derivatives, or on the spatial distribution of local contrast energy. Thus, bars and edges might arise from peaks of luminance and luminance gradient respectively, or bars and edges might be found at peaks of local energy, where local phases are aligned across spatial frequency. This basic issue of definition is important because it guides more detailed models and interpretations of early vision. Which approach better describes the perceived positions of features in images? We used the class of 1-D images defined by Morrone and Burr in which the amplitude spectrum is that of a (partially blurred) square-wave and all Fourier components have a common phase. Observers used a cursor to mark where bars and edges were seen for different test phases (Experiment 1) or judged the spatial alignment of contours that had different phases (e.g. 0 degrees and 45 degrees ; Experiment 2). The feature positions defined by both tasks shifted systematically to the left or right according to the sign of the phase offset, increasing with the degree of blur. These shifts were well predicted by the location of luminance peaks (bars) and gradient peaks (edges), but not by energy peaks which (by design) predicted no shift at all. These results encourage models based on a Gaussian-derivative framework, but do not support the idea that human vision uses points of phase alignment to find local, first-order features. Nevertheless, we argue that both approaches are presently incomplete and a better understanding of early vision may combine insights from both. (C)2004 Elsevier Ltd. All rights reserved.

Bars and edges: what makes a feature for human vision?

There have been two main approaches to feature detection in human and computer vision - luminance-based and energy-based. Bars and edges might arise from peaks of luminance and luminance gradient respectively, or bars and edges might be found at peaks of local energy, where local phases are aligned across spatial frequency. This basic issue of definition is important because it guides more detailed models and interpretations of early vision. Which approach better describes the perceived positions of elements in a 3-element contour-alignment task? We used the class of 1-D images defined by Morrone and Burr in which the amplitude spectrum is that of a (partially blurred) square wave and Fourier components in a given image have a common phase. Observers judged whether the centre element (eg ±458 phase) was to the left or right of the flanking pair (eg 0º phase). Lateral offset of the centre element was varied to find the point of subjective alignment from the fitted psychometric function. This point shifted systematically to the left or right according to the sign of the centre phase, increasing with the degree of blur. These shifts were well predicted by the location of luminance peaks and other derivative-based features, but not by energy peaks which (by design) predicted no shift at all. These results on contour alignment agree well with earlier ones from a more explicit feature-marking task, and strongly suggest that human vision does not use local energy peaks to locate basic first-order features. [Supported by the Wellcome Trust (ref: 056093)]

Mach Bands: multi-scale spatial filtering and co-operative coding of edges and bars

Perception of Mach bands may be explained by spatial filtering ('lateral inhibition') that can be approximated by 2nd derivative computation, and several alternative models have been proposed. To distinguish between them, we used a novel set of ‘generalised Gaussian’ images, in which the sharp ramp-plateau junction of the Mach ramp was replaced by smoother transitions. The images ranged from a slightly blurred Mach ramp to a Gaussian edge and beyond, and also included a sine-wave edge. The probability of seeing Mach Bands increased with the (relative) sharpness of the junction, but was largely independent of absolute spatial scale. These data did not fit the predictions of MIRAGE, nor 2nd derivative computation at a single fine scale. In experiment 2, observers used a cursor to mark features on the same set of images. Data on perceived position of Mach bands did not support the local energy model. Perceived width of Mach bands was poorly explained by a single-scale edge detection model, despite its previous success with Mach edges (Wallis & Georgeson, 2009, Vision Research, 49, 1886-1893). A more successful model used separate (odd and even) scale-space filtering for edges and bars, local peak detection to find candidate features, and the MAX operator to compare odd- and even-filter response maps (Georgeson, VSS 2006, Journal of Vision 6(6), 191a). Mach bands are seen when there is a local peak in the even-filter (bar) response map, AND that peak value exceeds corresponding responses in the odd-filter (edge) maps.

Bayesian error bars for regression

Regression problems are concerned with predicting the values of one or more continuous quantities, given the values of a number of input variables. For virtually every application of regression, however, it is also important to have an indication of the uncertainty in the predictions. Such uncertainties are expressed in terms of the error bars, which specify the standard deviation of the distribution of predictions about the mean. Accurate estimate of error bars is of practical importance especially when safety and reliability is an issue. The Bayesian view of regression leads naturally to two contributions to the error bars. The first arises from the intrinsic noise on the target data, while the second comes from the uncertainty in the values of the model parameters which manifests itself in the finite width of the posterior distribution over the space of these parameters. The Hessian matrix which involves the second derivatives of the error function with respect to the weights is needed for implementing the Bayesian formalism in general and estimating the error bars in particular. A study of different methods for evaluating this matrix is given with special emphasis on the outer product approximation method. The contribution of the uncertainty in model parameters to the error bars is a finite data size effect, which becomes negligible as the number of data points in the training set increases. A study of this contribution is given in relation to the distribution of data in input space. It is shown that the addition of data points to the training set can only reduce the local magnitude of the error bars or leave it unchanged. Using the asymptotic limit of an infinite data set, it is shown that the error bars have an approximate relation to the density of data in input space.

Mach bands and multiscale models of spatial vision:the role of first, second, and third derivative operators in encoding bars and edges

Ernst Mach observed that light or dark bands could be seen at abrupt changes of luminance gradient in the absence of peaks or troughs in luminance. Many models of feature detection share the idea that bars, lines, and Mach bands are found at peaks and troughs in the output of even-symmetric spatial filters. Our experiments assessed the appearance of Mach bands (position and width) and the probability of seeing them on a novel set of generalized Gaussian edges. Mach band probability was mainly determined by the shape of the luminance profile and increased with the sharpness of its corners, controlled by a single parameter (n). Doubling or halving the size of the images had no significant effect. Variations in contrast (20%-80%) and duration (50-300 ms) had relatively minor effects. These results rule out the idea that Mach bands depend simply on the amplitude of the second derivative, but a multiscale model, based on Gaussian-smoothed first- and second-derivative filtering, can account accurately for the probability and perceived spatial layout of the bands. A key idea is that Mach band visibility depends on the ratio of second- to first-derivative responses at peaks in the second-derivative scale-space map. This ratio is approximately scale-invariant and increases with the sharpness of the corners of the luminance ramp, as observed. The edges of Mach bands pose a surprisingly difficult challenge for models of edge detection, but a nonlinear third-derivative operation is shown to predict the locations of Mach band edges strikingly well. Mach bands thus shed new light on the role of multiscale filtering systems in feature coding. © 2012 ARVO.

Blobs versus bars:psychophysical evidence supports two types of orientation response in human color vision

The classic hypothesis of Livingstone and Hubel (1984, 1987) proposed two types of color pathways in primate visual cortex based on recordings from single cells: a segregated, modularpathway that signals color but provides little information about shape or form and a second pathway that signals color differences and so defines forms without the need to specify their colors. A major problem has been to reconcile this neurophysiological hypothesis with the behavioral data. A wealth of psychophysical studies has demonstrated that color vision has orientation-tuned responses and little impairment on form related tasks, but these have not revealed any direct evidence for nonoriented mechanisms. Here we use a psychophysical method of subthreshold summation across orthogonal orientations for isoluminant red-green gratings in monocular and dichoptic viewing conditions to differentiate between nonoriented and orientation-tuned responses to color contrast. We reveal nonoriented color responses at low spatial frequencies (0.25-0.375 c/deg) under monocular conditions changing to orientation-tuned responses at higher spatial frequencies (1.5 c/deg) and under binocular conditions. We suggest that two distinct pathways coexist in color vision at the behavioral level, revealed at different spatial scales: one is isotropic, monocular, and best equipped for the representation of surface color, and the other is orientation-tuned, binocular, and selective for shape and form. This advances our understanding of the organization of the neural pathways involved in human color vision and provides a strong link between neurophysiological and behavioral data. © 2013 ARVO.

Vortex shedding from a row of square bars

Interactions of wakes in a flow past a row of square bars, which is placed across a uniform flow, are investigated by numerical simulations and experiments on the tassumption that the flow is two-dimensional and incompressible. At small Reynolds numbers the flow is steady and symmetric with respect not only to streamwise lines through the center of each square bar but also to streamwise centerlines between adjacent square bars. However, the steady symmetric flow becomes unstable at larger Reynolds numbers and make a transition to a steady asymmetric flow with respect to the centerlines between adjacent square bars in some cases or to an oscillatory flow in other cases. It is found that vortices are shed synchronously from adjacent square bars in the same phase or in anti-phase depending upon the distance between the bars when the flow is oscillatory. The origin of the transition to the steady asymmetric flow is identified as a pitchfork bifurcation, while the oscillatory flows with synchronous shedding of vortices are clarified to originate from a Hopf bifurcation. The critical Reynolds numbers of the transitions are evaluated numerically and the bifurcation diagram of the flow is obtained.

Point-wise confidence interval estimation by neural networks: A comparative study based on automotive engine calibration.

In developing neural network techniques for real world applications it is still very rare to see estimates of confidence placed on the neural network predictions. This is a major deficiency, especially in safety-critical systems. In this paper we explore three distinct methods of producing point-wise confidence intervals using neural networks. We compare and contrast Bayesian, Gaussian Process and Predictive error bars evaluated on real data. The problem domain is concerned with the calibration of a real automotive engine management system for both air-fuel ratio determination and on-line ignition timing. This problem requires real-time control and is a good candidate for exploring the use of confidence predictions due to its safety-critical nature.

Mach edges: a critical test of the nonlinear 3rd derivative model for edge-detection

Feature detection is a crucial stage of visual processing. In previous feature-marking experiments we found that peaks in the 3rd derivative of the luminance profile can signify edges where there are no 1st derivative peaks nor 2nd derivative zero-crossings (Wallis and George 'Mach edges' (the edges of Mach bands) were nicely predicted by a new nonlinear model based on 3rd derivative filtering. As a critical test of the model, we now use a new class of stimuli, formed by adding a linear luminance ramp to the blurred triangle waves used previously. The ramp has no effect on the second or higher derivatives, but the nonlinear model predicts a shift from seeing two edges to seeing only one edge as the added ramp gradient increases. In experiment 1, subjects judged whether one or two edges were visible on each trial. In experiment 2, subjects used a cursor to mark perceived edges and bars. The position and polarity of the marked edges were close to model predictions. Both experiments produced the predicted shift from two to one Mach edge, but the shift was less complete than predicted. We conclude that the model is a useful predictor of edge perception, but needs some modification.

Mach edges: a key role for 3rd derivative filters in spatial vision

Edges are key points of information in visual scenes. One important class of models supposes that edges correspond to the steepest parts of the luminance profile, implying that they can be found as peaks and troughs in the response of a gradient (first-derivative) filter, or as zero-crossings (ZCs) in the second-derivative. A variety of multi-scale models are based on this idea. We tested this approach by devising a stimulus that has no local peaks of gradient and no ZCs, at any scale. Our stimulus profile is analogous to the classic Mach-band stimulus, but it is the local luminance gradient (not the absolute luminance) that increases as a linear ramp between two plateaux. The luminance profile is a smoothed triangle wave and is obtained by integrating the gradient profile. Subjects used a cursor to mark the position and polarity of perceived edges. For all the ramp-widths tested, observers marked edges at or close to the corner points in the gradient profile, even though these were not gradient maxima. These new Mach edges correspond to peaks and troughs in the third-derivative. They are analogous to Mach bands - light and dark bars are seen where there are no luminance peaks but there are peaks in the second derivative. Here, peaks in the third derivative were seen as light-to-dark edges, troughs as dark-to-light edges. Thus Mach edges are inconsistent with many standard edge detectors, but are nicely predicted by a new model that uses a (nonlinear) third-derivative operator to find edge points.

The interaction of luminance and texture amplitude in surface depth perception

Previous studies have suggested separate channels for detection of first-order luminance modulations (LM) and second-order modulations of the local amplitude (AM) of a texture. Mixtures of LM and AM with different phase relationships appear very different: in-phase compounds (LM + AM) look like 3-D corrugated surfaces, while out-of-phase compounds (LM - AM) appear flat and/or transparent. This difference may arise because the in-phase compounds are consistent with multiplicative shading, while the out-of-phase compounds are not. We investigated the role of these modulation components in surface depth perception. We used a textured background with thin bars formed by local changes in luminance and/or texture amplitude. These stimuli appear as embossed surfaces with wide and narrow regions. Keeping the AM modulation depth fixed at a suprathreshold level, we determined the amount of luminance contrast required for observers to correctly indicate the width (narrow or wide) of 'raised' regions in the display. Performance (compared to the LM-only case) was facilitated by the presence of AM, but, unexpectedly, performance for LM - AM was as good as for LM + AM. Thus, these results suggest that there is an interaction between first-order and second-order mechanisms during depth perception based on shading cues, but the phase dependence is not yet understood.

The interaction of luminance and texture amplitude in depth perception

Previous studies have suggested separate channels for the detection of first-order luminance (LM) and second-order modulations of the local amplitude (AM) of a texture (Schofield and Georgeson, 1999 Vision Research 39 2697 - 2716; Georgeson and Schofield, 2002 Spatial Vision 16 59). It has also been shown that LM and AM mixtures with different phase relationships are easily separated in identification tasks, and (informally) appear very different with the in-phase compound (LM + AM), producing the most realistic depth percept. We investigated the role of these LM and AM components in depth perception. Stimuli consisted of a noise texture background with thin bars formed as local increments or decrements in luminance and/or noise amplitude. These stimuli appear as embossed surfaces with wide and narrow regions. When luminance and amplitude changes have the same sign and magnitude (LM + AM) the overall modulation is consistent with multiplicative shading, but this is not so when the two modulations have opposite sign (LM - AM). Keeping the AM modulation depth fixed at a suprathreshold level, we determined the amount of luminance contrast required for observers to correctly indicate the width (narrow or wide) of raised regions in the display. Performance (compared to the LM-only case) was facilitated by the presence of AM, but, unexpectedly, performance for LM - AM was even better than for LM + AM. Further tests suggested that this improvement in performance is not due to an increase in the detectability of luminance in the compound stimuli. Thus, contrary to previous findings, these results suggest the possibility of interaction between first-order and second-order mechanisms in depth perception.

Resistance heating of steel conductors of circular cross-section

The object of this thesis is to develop a method for calculating the losses developed in steel conductors of circular cross-section and at temperatures below 100oC, by the direct passage of a sinusoidally alternating current. Three cases are considered. 1. Isolated solid or tubular conductor. 2. Concentric arrangement of tube and solid return conductor. 3. Concentric arrangement of two tubes. These cases find applications in process temperature maintenance of pipelines, resistance heating of bars and design of bus-bars. The problems associated with the non-linearity of steel are examined. Resistance heating of bars and methods of surface heating of pipelines are briefly described. Magnetic-linear solutions based on Maxwell's equations are critically examined and conditions under which various formulae apply investigated. The conditions under which a tube is electrically equivalent to a solid conductor and to a semi-infinite plate are derived. Existing solutions for the calculation of losses in isolated steel conductors of circular cross-section are reviewed, evaluated and compared. Two methods of solution are developed for the three cases considered. The first is based on the magnetic-linear solutions and offers an alternative to the available methods which are not universal. The second solution extends the existing B/H step-function approximation method to small diameter conductors and to tubes in isolation or in a concentric arrangement. A comprehensive experimental investigation is presented for cases 1 and 2 above which confirms the validity of the proposed methods of solution. These are further supported by experimental results reported in the literature. Good agreement is obtained between measured and calculated loss values for surface field strengths beyond the linear part of the d.c. magnetisation characteristic. It is also shown that there is a difference in the electrical behaviour of a small diameter conductor or thin tube under resistance or induction heating conditions.