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.

The spatial distribution of self-employment in India:evidence from semi-parametric geoadditive models

The spatial distribution of self-employment in India: evidence from semiparametric geoadditive models, Regional Studies. The entrepreneurship literature has rarely considered spatial location as a micro-determinant of occupational choice. It has also ignored self-employment in developing countries. Using Bayesian semiparametric geoadditive techniques, this paper models spatial location as a micro-determinant of self-employment choice in India. The empirical results suggest the presence of spatial occupational neighbourhoods and a clear north–south divide in self-employment when the entire sample is considered; however, spatial variation in the non-agriculture sector disappears to a large extent when individual factors that influence self-employment choice are explicitly controlled. The results further suggest non-linear effects of age, education and wealth on self-employment.

Adding constrained discontinuities to Gaussian process models of wind fields

Gaussian Processes provide good prior models for spatial data, but can be too smooth. In many physical situations there are discontinuities along bounding surfaces, for example fronts in near-surface wind fields. We describe a modelling method for such a constrained discontinuity and demonstrate how to infer the model parameters in wind fields with MCMC sampling.

Quantitative risk assessment of groundwater quality utilizing GIS technology and coupled groundwater models

The thesis presents a two-dimensional Risk Assessment Method (RAM) where the assessment of risk to the groundwater resources incorporates both the quantification of the probability of the occurrence of contaminant source terms, as well as the assessment of the resultant impacts. The approach emphasizes the need for a greater dependency on the potential pollution sources, rather than the traditional approach where assessment is based mainly on the intrinsic geo-hydrologic parameters. The risk is calculated using Monte Carlo simulation methods whereby random pollution events were generated to the same distribution as historically occurring events or a priori potential probability distribution. Integrated mathematical models then simulate contaminant concentrations at the predefined monitoring points within the aquifer. The spatial and temporal distributions of the concentrations were calculated from repeated realisations, and the number of times when a user defined concentration magnitude was exceeded is quantified as a risk. The method was setup by integrating MODFLOW-2000, MT3DMS and a FORTRAN coded risk model, and automated, using a DOS batch processing file. GIS software was employed in producing the input files and for the presentation of the results. The functionalities of the method, as well as its sensitivities to the model grid sizes, contaminant loading rates, length of stress periods, and the historical frequencies of occurrence of pollution events were evaluated using hypothetical scenarios and a case study. Chloride-related pollution sources were compiled and used as indicative potential contaminant sources for the case study. At any active model cell, if a random generated number is less than the probability of pollution occurrence, then the risk model will generate synthetic contaminant source term as an input into the transport model. The results of the applications of the method are presented in the form of tables, graphs and spatial maps. Varying the model grid sizes indicates no significant effects on the simulated groundwater head. The simulated frequency of daily occurrence of pollution incidents is also independent of the model dimensions. However, the simulated total contaminant mass generated within the aquifer, and the associated volumetric numerical error appear to increase with the increasing grid sizes. Also, the migration of contaminant plume advances faster with the coarse grid sizes as compared to the finer grid sizes. The number of daily contaminant source terms generated and consequently the total mass of contaminant within the aquifer increases in a non linear proportion to the increasing frequency of occurrence of pollution events. The risk of pollution from a number of sources all occurring by chance together was evaluated, and quantitatively presented as risk maps. This capability to combine the risk to a groundwater feature from numerous potential sources of pollution proved to be a great asset to the method, and a large benefit over the contemporary risk and vulnerability methods.

Mach edges: local features predicted by 3rd derivative spatial filtering

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 (1st derivative) filter, or as zero-crossings in the 2nd derivative (ZCs). We tested those ideas using a stimulus that has no local peaks of gradient and no ZCs, at any scale. The stimulus profile is analogous to the Mach ramp, but it is the luminance gradient (not the absolute luminance) that increases as a linear ramp between two plateaux; the luminance profile is a blurred triangle-wave. For all image-blurs tested, observers marked edges at or close to the corner points in the gradient profile, even though these were not gradient maxima. These Mach edges correspond to peaks and troughs in the 3rd derivative. Thus Mach edges are inconsistent with many standard edge-detection schemes, but are nicely predicted by a recent model that finds edge points with a 2-stage sequence of 1st then 2nd derivative operators, each followed by a half-wave rectifier.

Third-derivative filters predict edge locations in spatial vision

Edge detection is crucial in visual processing. Previous computational and psychophysical models have often used peaks in the gradient or zero-crossings in the 2nd derivative to signal edges. We tested these approaches using a stimulus that has no such features. Its luminance profile was a triangle wave, blurred by a rectangular function. Subjects marked the position and polarity of perceived edges. For all blur widths tested, observers marked edges at or near 3rd derivative maxima, even though these were not 1st derivative maxima or 2nd derivative zero-crossings, at any scale. These results are predicted by a new nonlinear model based on 3rd derivative filtering. As a critical test, we added a ramp of variable slope to the blurred triangle-wave luminance profile. The ramp has no effect on the (linear) 2nd or higher derivatives, but the nonlinear model predicts a shift from seeing two edges to seeing one edge as the ramp gradient increases. Results of two experiments confirmed such a shift, thus supporting the new model. [Supported by the Engineering and Physical Sciences Research Council].

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.

Adding constrained discontinuities to Gaussian process models of wind fields

Gaussian Processes provide good prior models for spatial data, but can be too smooth. In many physical situations there are discontinuities along bounding surfaces, for example fronts in near-surface wind fields. We describe a modelling method for such a constrained discontinuity and demonstrate how to infer the model parameters in wind fields with MCMC sampling.

Modeling geometric rules in object based models:an XML / GML approach

Most object-based approaches to Geographical Information Systems (GIS) have concentrated on the representation of geometric properties of objects in terms of fixed geometry. In our road traffic marking application domain we have a requirement to represent the static locations of the road markings but also enforce the associated regulations, which are typically geometric in nature. For example a give way line of a pedestrian crossing in the UK must be within 1100-3000 mm of the edge of the crossing pattern. In previous studies of the application of spatial rules (often called 'business logic') in GIS emphasis has been placed on the representation of topological constraints and data integrity checks. There is very little GIS literature that describes models for geometric rules, although there are some examples in the Computer Aided Design (CAD) literature. This paper introduces some of the ideas from so called variational CAD models to the GIS application domain, and extends these using a Geography Markup Language (GML) based representation. In our application we have an additional requirement; the geometric rules are often changed and vary from country to country so should be represented in a flexible manner. In this paper we describe an elegant solution to the representation of geometric rules, such as requiring lines to be offset from other objects. The method uses a feature-property model embraced in GML 3.1 and extends the possible relationships in feature collections to permit the application of parameterized geometric constraints to sub features. We show the parametric rule model we have developed and discuss the advantage of using simple parametric expressions in the rule base. We discuss the possibilities and limitations of our approach and relate our data model to GML 3.1. © 2006 Springer-Verlag Berlin Heidelberg.

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.

Spatial and temporal dependencies of cross-orientation suppression in human vision.

A well-known property of orientation-tuned neurons in the visual cortex is that they are suppressed by the superposition of an orthogonal mask. This phenomenon has been explained in terms of physiological constraints (synaptic depression), engineering solutions for components with poor dynamic range (contrast normalization) and fundamental coding strategies for natural images (redundancy reduction). A common but often tacit assumption is that the suppressive process is equally potent at different spatial and temporal scales of analysis. To determine whether it is so, we measured psychophysical cross-orientation masking (XOM) functions for flickering horizontal Gabor stimuli over wide ranges of spatio-temporal frequency and contrast. We found that orthogonal masks raised contrast detection thresholds substantially at low spatial frequencies and high temporal frequencies (high speeds), and that small and unexpected levels of facilitation were evident elsewhere. The data were well fit by a functional model of contrast gain control, where (i) the weight of suppression increased with the ratio of temporal to spatial frequency and (ii) the weight of facilitatory modulation was the same for all conditions, but outcompeted by suppression at higher contrasts. These results (i) provide new constraints for models of primary visual cortex, (ii) associate XOM and facilitation with the transient magno- and sustained parvostreams, respectively, and (iii) reconcile earlier conflicting psychophysical reports on XOM.

Low spatial frequencies are suppressively masked across spatial scale, orientation, field position, and eye of origin

Masking is said to occur when a mask stimulus interferes with the visibility of a target (test) stimulus. One widely held view of this process supposes interactions between mask and test mechanisms (cross-channel masking), and explicit models (e.g., J. M. Foley, 1994) have proposed that the interactions are inhibitory. Unlike a within-channel model, where masking involves the combination of mask and test stimulus within a single mechanism, this cross-channel inhibitory model predicts that the mask should attenuate the perceived contrast of a test stimulus. Another possibility is that masking is due to an increase in noise, in which case, perception of contrast should be unaffected once the signal exceeds detection threshold. We use circular patches and annuli of sine-wave grating in contrast detection and contrast matching experiments to test these hypotheses and investigate interactions across spatial frequency, orientation, field position, and eye of origin. In both types of experiments we found substantial effects of masking that can occur over a factor of 3 in spatial frequency, 45° in orientation, across different field positions and between different eyes. We found the effects to be greatest at the lowest test spatial frequency we used (0.46 c/deg), and when the mask and test differed in all four dimensions simultaneously. This is surprising in light of previous work where it was concluded that suppression from the surround was strictly monocular (C. Chubb, G. Sperling, & J. A. Solomon, 1989). The results confirm that above detection threshold, cross-channel masking involves contrast suppression and not (purely) mask-induced noise. We conclude that cross-channel masking can be a powerful phenomenon, particularly at low test spatial frequencies and when mask and test are presented to different eyes. © 2004 ARVO.

Adaptation and gain pool summation:Alternative models and masking data

Foley [J. Opt. Soc. Am. A 11 (1994) 1710] has proposed an influential psychophysical model of masking in which mask components in a contrast gain pool are raised to an exponent before summation and divisive inhibition. We tested this summation rule in experiments in which contrast detection thresholds were measured for a vertical 1 c/deg (or 2 c/deg) sine-wave component in the presence of a 3 c/deg (or 6 c/deg) mask that had either a single component oriented at -45° or a pair of components oriented at ±45°. Contrary to the predictions of Foley's model 3, we found that for masks of moderate contrast and above, threshold elevation was predicted by linear summation of the mask components in the inhibitory stage of the contrast gain pool. We built this feature into two new models, referred to as the early adaptation model and the hybrid model. In the early adaptation model, contrast adaptation controls a threshold-like nonlinearity on the output of otherwise linear pathways that provide the excitatory and inhibitory inputs to a gain control stage. The hybrid model involves nonlinear and nonadaptable routes to excitatory and inhibitory stages as well as an adaptable linear route. With only six free parameters, both models provide excellent fits to the masking and adaptation data of Foley and Chen [Vision Res. 37 (1997) 2779] but unlike Foley and Chen's model, are able to do so with only one adaptation parameter. However, only the hybrid model is able to capture the features of Foley's (1994) pedestal plus orthogonal fixed mask data. We conclude that (1) linear summation of inhibitory components is a feature of contrast masking, and (2) that the main aftereffect of spatial adaptation on contrast increment thresholds can be assigned to a single site. © 2002 Elsevier Science Ltd. All rights reserved.

Spatially structured cognitve models of semantic information:the implications for computerised databases

Existing theories of semantic cognition propose models of cognitive processing occurring in a conceptual space, where ‘meaning’ is derived from the spatial relationships between concepts’ mapped locations within the space. Information visualisation is a growing area of research within the field of information retrieval, and methods for presenting database contents visually in the form of spatial data management systems (SDMSs) are being developed. This thesis combined these two areas of research to investigate the benefits associated with employing spatial-semantic mapping (documents represented as objects in two- and three-dimensional virtual environments are proximally mapped dependent on the semantic similarity of their content) as a tool for improving retrieval performance and navigational efficiency when browsing for information within such systems. Positive effects associated with the quality of document mapping were observed; improved retrieval performance and browsing behaviour were witnessed when mapping was optimal. It was also shown using a third dimension for virtual environment (VE) presentation provides sufficient additional information regarding the semantic structure of the environment that performance is increased in comparison to using two-dimensions for mapping. A model that describes the relationship between retrieval performance and browsing behaviour was proposed on the basis of findings. Individual differences were not found to have any observable influence on retrieval performance or browsing behaviour when mapping quality was good. The findings from this work have implications for both cognitive modelling of semantic information, and for designing and testing information visualisation systems. These implications are discussed in the conclusions of this work.

Contrast integration over area is extensive: a three-stage model of spatial summation

Classical studies of area summation measure contrast detection thresholds as a function of grating diameter. Unfortunately, (i) this approach is compromised by retinal inhomogeneity and (ii) it potentially confounds summation of signal with summation of internal noise. The Swiss cheese stimulus of T. S. Meese and R. J. Summers (2007) and the closely related Battenberg stimulus of T. S. Meese (2010) were designed to avoid these problems by keeping target diameter constant and modulating interdigitated checks of first-order carrier contrast within the stimulus region. This approach has revealed a contrast integration process with greater potency than the classical model of spatial probability summation. Here, we used Swiss cheese stimuli to investigate the spatial limits of contrast integration over a range of carrier frequencies (1–16 c/deg) and raised plaid modulator frequencies (0.25–32 cycles/check). Subthreshold summation for interdigitated carrier pairs remained strong (~4 to 6 dB) up to 4 to 8 cycles/check. Our computational analysis of these results implied linear signal combination (following square-law transduction) over either (i) 12 carrier cycles or more or (ii) 1.27 deg or more. Our model has three stages of summation: short-range summation within linear receptive fields, medium-range integration to compute contrast energy for multiple patches of the image, and long-range pooling of the contrast integrators by probability summation. Our analysis legitimizes the inclusion of widespread integration of signal (and noise) within hierarchical image processing models. It also confirms the individual differences in the spatial extent of integration that emerge from our approach.