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.

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].

Added luminance ramp alters perceived edge blur and contrast:A critical test for derivative-based models of edge coding

In many models of edge analysis in biological vision, the initial stage is a linear 2nd derivative operation. Such models predict that adding a linear luminance ramp to an edge will have no effect on the edge's appearance, since the ramp has no effect on the 2nd derivative. Our experiments did not support this prediction: adding a negative-going ramp to a positive-going edge (or vice-versa) greatly reduced the perceived blur and contrast of the edge. The effects on a fairly sharp edge were accurately predicted by a nonlinear multi-scale model of edge processing [Georgeson, M. A., May, K. A., Freeman, T. C. A., & Hesse, G. S. (in press). From filters to features: Scale-space analysis of edge and blur coding in human vision. Journal of Vision], in which a half-wave rectifier comes after the 1st derivative filter. But we also found that the ramp affected perceived blur more profoundly when the edge blur was large, and this greater effect was not predicted by the existing model. The model's fit to these data was much improved when the simple half-wave rectifier was replaced by a threshold-like transducer [May, K. A. & Georgeson, M. A. (2007). Blurred edges look faint, and faint edges look sharp: The effect of a gradient threshold in a multi-scale edge coding model. Vision Research, 47, 1705-1720.]. This modified model correctly predicted that the interaction between ramp gradient and edge scale would be much larger for blur perception than for contrast perception. In our model, the ramp narrows an internal representation of the gradient profile, leading to a reduction in perceived blur. This in turn reduces perceived contrast because estimated blur plays a role in the model's estimation of contrast. Interestingly, the model predicts that analogous effects should occur when the width of the window containing the edge is made narrower. This has already been confirmed for blur perception; here, we further support the model by showing a similar effect for contrast perception. © 2007 Elsevier Ltd. All rights reserved.

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.

Perceiving edge blur: Gaussian-derivative filtering and a rectifying nonlinearity

We studied the visual mechanisms that encode edge blur in images. Our previous work suggested that the visual system spatially differentiates the luminance profile twice to create the `signature' of the edge, and then evaluates the spatial scale of this signature profile by applying Gaussian derivative templates of different sizes. The scale of the best-fitting template indicates the blur of the edge. In blur-matching experiments, a staircase procedure was used to adjust the blur of a comparison edge (40% contrast, 0.3 s duration) until it appeared to match the blur of test edges at different contrasts (5% - 40%) and blurs (6 - 32 min of arc). Results showed that lower-contrast edges looked progressively sharper. We also added a linear luminance gradient to blurred test edges. When the added gradient was of opposite polarity to the edge gradient, it made the edge look progressively sharper. Both effects can be explained quantitatively by the action of a half-wave rectifying nonlinearity that sits between the first and second (linear) differentiating stages. This rectifier was introduced to account for a range of other effects on perceived blur (Barbieri-Hesse and Georgeson, 2002 Perception 31 Supplement, 54), but it readily predicts the influence of the negative ramp. The effect of contrast arises because the rectifier has a threshold: it not only suppresses negative values but also small positive values. At low contrasts, more of the gradient profile falls below threshold and its effective spatial scale shrinks in size, leading to perceived sharpening.

The primal sketch revisited: locating and representing edges in human vision via Gaussian-derivative filtering

Marr's work offered guidelines on how to investigate vision (the theory - algorithm - implementation distinction), as well as specific proposals on how vision is done. Many of the latter have inevitably been superseded, but the approach was inspirational and remains so. Marr saw the computational study of vision as tightly linked to psychophysics and neurophysiology, but the last twenty years have seen some weakening of that integration. Because feature detection is a key stage in early human vision, we have returned to basic questions about representation of edges at coarse and fine scales. We describe an explicit model in the spirit of the primal sketch, but tightly constrained by psychophysical data. Results from two tasks (location-marking and blur-matching) point strongly to the central role played by second-derivative operators, as proposed by Marr and Hildreth. Edge location and blur are evaluated by finding the location and scale of the Gaussian-derivative `template' that best matches the second-derivative profile (`signature') of the edge. The system is scale-invariant, and accurately predicts blur-matching data for a wide variety of 1-D and 2-D images. By finding the best-fitting scale, it implements a form of local scale selection and circumvents the knotty problem of integrating filter outputs across scales. [Supported by BBSRC and the Wellcome Trust]

The interrogation and multiplexing of long period grating curvature sensors using a Bragg grating based, derivative spectroscopy technique

A long period grating is interrogated with a fibre Bragg grating using a derivative spectroscopy technique. A quasi-linear relationship between the output of the sensing scheme and the curvature experienced by the long period grating is demonstrated, with a sensitivity of 5.05 m and with an average curvature resolution of 2.9 × 10-2 m-1. In addition, the feasibility of multiplexing an in-line series of long period gratings with this interrogation scheme is demonstrated with two pairs of fibre Bragg gratings and long period gratings. With this arrangement the cross-talk error between channels was less than ± 2.4 × 10-3 m-1.

Synthesis of polymers and copolymers of lactide through titanium initators

The research described in this thesis explored the synthesis tlnd characteristltion of biocompatible and biodegradable polymers of lactide through non-toxic titanium alkoxide nitiators. The research objectives focused on the preparation of polylactides in both solvent and solventless media, to produce materials with a wide range of molecular weights. The polylactides were fully characterised using gel permeation chromatography and 1H and 13C NMR spectroscopy. NMR spectroscopy was carried out in the study the reaction mechanisms. Kinetic studies of the ring opening polymerisation of lactide with titanium alkoxide initiators were also conducted using NMR spectroscopy. The objectives of this research were also focused on the enhancement of the flexibility of the polymer chains by synthesising random and block copolymers of lactide and ε-caprolactone using Ti(0-i-Pr)4 as an initiator, This work involved extensive characterisalion of the synthesised copolymers using gel permeation chromatography and 1H and 13C NMR spectroscopic analysis. Kinetic studies of the ring opening polymerisation of ε-caplrolactone and of the copolymerisation of lactide and ε-caprolactone with Ti(O-i-Pr)4 as an initiator were also carried out. The last section of this work involved the synthesis of block and star-shaped copolymers of lactide and poly(ethylene glycol) [PEG]. The preparation of lactide/PEG block copolymers was carried out by ring opening polymerisation of L-Iactide using Ti(O-i-Pr)4 as an initiator and hydroxyl-terminated PEG's with different numbers of hydroxyl groups as co-initiators both in solution and solventless media. These all-in-one polymersations yielded the synthesis of both lactide homopolymer and lactide/PEG block copolymer. In order to selectively synthesise copolymers of lactide and PEG, the experiment was carried out in two steps. The first step consisted of the synthesis of a titanium macro-initiator by exchanging the iso-propoxide ligands by PEG with different numbers of hydroxyl groups. The second step involved the ring opening polymerisation of lactide using the titanium macrocatalyst that was prepared as an initiator. The polymerisations were carried out in a solventless media. The synthesis of lactide/PEG copolymers using polyethylene glycol with amino terminal groups was also discussed. Extensive characterisation of the lactide block copolymers and macroinitiators was carried out using techniques such as, gel permeation chromatography (GPC), NMR spectroscopy and differential scanning calorimeter (DeS).

The relationship between fracture toughness and microstructure in titanium alloys

Linear Elastic Fracture Mechanics has been used to study the microstructural factors controlling the strength and toughness of two alpha-beta, titanium alloys. Fracture toughness was found to be independent of orientation for alloy Ti/6A1/4-V, but orientation dependent for IMI 700, bend and tension specimens giving similar toughness values. Increasing the solution temperature led to the usual inverse relationship between strength and toughness, with toughness becoming a minimum as the beta transus was approached. The production of a double heat treated microstructure led to a 100% increase in toughness in the high strength alloy and a 20% increase in alloy Ti/6A1/4V, with little decrease in strength. The double heat treated microstruoture was produced by cooling from the beta field into the alpha beta field, followed. by conventional solution treatment and ageing. Forging above the beta transus led to an increase in toughness over alpha beta forging in the high strength alloy, but had little effect on the toughness of Ti/6A1/4V. Light and electron microscopy showed that the increased toughness resulted from the alpha phase being changed from mainly continuous to a discontinuous platelet form in a transformed beta matrix. Void formation occurred at the alpha-beta interface and crack propagation was via the interface or across the platelet depending on which process required the least energy. Varying the solution treatment temperature produced a varying interplatelet spacing and platelet thickness. The finest interplatelet spacing was associated with the highest toughness, since a higher applied stress was required to give the necessary stress concentration to initiate void formation. The thickest alpha platelet size gave the highest toughness which could be interpreted in terms of Krafftt's "process zone size" and the critical crack tip displacement criterion by Hahn and Rosenfield from an analysis by Goodier and Field.

The effect of zinc and titanium on the structure of calcium-sodium phosphate based glass

An array of different structural probes has been used to define the effect of adding Zn and Ti to a sodium-calcium phosphate glass. X-ray absorption spectroscopy at the Zn K-edge suggests that the Zn atoms occupy mixed (4- and 6-fold) sites within the glass matrix. X-ray diffraction reveals a feature at 2.03 angstrom that develops with the addition of Zn and Ti and is consistent with Zn-O and Ti-O near-neighbour distances. Neutron diffraction is used to resolve two distinct P-O distances and highlights the decrease in P center dot center dot center dot P coordination number from 2.0 to 1.7 as the Ti metal concentration rises, which is attributed to the O/P fraction moving away from the metaphosphate value of 3.0 to 3.1 with the addition of Ti. Other correlations, such as those associated with CaO(x) and NaO(x) polyhedra, remain largely unaffected. These results suggest that the network forming P center dot center dot center dot P correlation is most disrupted, with the disorder parameter rising from 0.07 to 0.10 angstrom with the additional modifiers. Zn appears to be introduced into the network as a direct replacement for Ca and causes no structural variation over the composition range studied.

Do ‘passive’ medical titanium surfaces deteriorate in service in the absence of wear?

Globally, more than 1000 tonnes of titanium (Ti) is implanted into patients in the form of biomedical devices on an annual basis. Ti is perceived to be ‘biocompatible’ owing to the presence of a robust passive oxide film (approx. 4 nm thick) at the metal surface. However, surface deterioration can lead to the release of Ti ions, and particles can arise as the result of wear and/or corrosion processes. This surface deterioration can result in peri-implant inflammation, leading to the premature loss of the implanted device or the requirement for surgical revision. Soft tissues surrounding commercially pure cranial anchorage devices (bone-anchored hearing aid) were investigated using synchrotron X-ray micro-fluorescence spectroscopy and X-ray absorption near edge structure. Here, we present the first experimental evidence that minimal load-bearing Ti implants, which are not subjected to macroscopic wear processes, can release Ti debris into the surrounding soft tissue. As such debris has been shown to be pro-inflammatory, we propose that such distributions of Ti are likely to effect to the service life of the device.

Titanium phosphate glass microspheres for bone tissue engineering

We have demonstrated the successful production of titanium phosphate glass microspheres in the size range of ~10-200 µm using an inexpensive, efficient, easily scalable process and assessed their use in bone tissue engineering applications. Glasses of the following compositions were prepared by melt-quench techniques: 0.5P2O5-0.4CaO-(0.1 - x)Na2O-xTiO2, where x = 0.03, 0.05 and 0.07 mol fraction (denoted as Ti3, Ti5 and Ti7 respectively). Several characterization studies such as differential thermal analysis, degradation (performed using a novel time lapse imaging technique) and pH and ion release measurements revealed significant densification of the glass structure with increased incorporation of TiO2 in the glass from 3 to 5 mol.%, although further TiO2 incorporation up to 7 mol.% did not affect the glass structure to the same extent. Cell culture studies performed using MG63 cells over a 7-day period clearly showed the ability of the microspheres to provide a stable surface for cell attachment, growth and proliferation. Taken together, the results confirm that 5 mol.% TiO2 glass microspheres, on account of their relative ease of preparation and favourable biocompatibility, are worthy candidates for use as substrate materials in bone tissue engineering applications.

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.

Structural characterization of titanium-doped Bioglass using isotopic substitution neutron diffraction

Melt quenched silicate glasses containing calcium, phosphorus and alkali metals have the ability to promote bone regeneration and to fuse to living bone. Of these glasses 45S5 Bioglass® is the most widely used being sold in over 35 countries as a bone graft product for medical and dental applications; particulate 45S5 is also incorporated into toothpastes to help remineralize the surface of teeth. Recently it has been suggested that adding titanium dioxide can increase the bioactivity of these materials. This work investigates the structural consequences of incorporating 4 mol% TiO2 into Bioglass® using isotopic substitution (of the Ti) applied to neutron diffraction and X-ray Absorption Near Edge Structure (XANES). We present the first isotopic substitution data applied to melt quench derived Bioglass or its derivatives. Results show that titanium is on average surrounded by 5.2(1) nearest neighbor oxygen atoms. This implies an upper limit of 40% four-fold coordinated titanium and shows that the network connectivity is reduced from 2.11 to 1.97 for small quantities of titanium. Titanium XANES micro-fluorescence confirms the titanium environment is homogenous on the micron length scale within these glasses. Solid state magic angle spinning (MAS) NMR confirms the network connectivity model proposed. Furthermore, the results show the intermediate range order containing Na-O, Ca-O, O-P-O and O-Si-O correlations are unaffected by the addition of small quantities of TiO2 into these systems.