Papers by Mitchell A Buckley
Orientals and cubes, inductively
Advances in Mathematics, 2016
International Journal of Computer Vision, Mar 28, 2012
Image-based 3D reconstruction remains a competitive field of research as state-of-the-art algorit... more Image-based 3D reconstruction remains a competitive field of research as state-of-the-art algorithms continue to improve. This paper presents a voxel-based algorithm that adapts the earliest space-carving methods and utilises a minimal surface technique to obtain a cleaner result. Embedded Voxel Colouring is built in two stages: (a) progressive voxel carving is used to build a volume of embedded surfaces and (b) the volume is processed to obtain a surface that maximises photo-consistency data in the volume. This algorithm combines the strengths of classical carving techniques with those of minimal surface approaches. We require only a single pass through the voxel volume, this significantly reduces computation time and is the key to the speed of our approach. We also specify three requirements for volumetric reconstruction: monotonic carving order, causality of carving and water-tightness.
The Catalan Simplicial Set II
Applied Categorical Structures, 2015
ABSTRACT The Catalan simplicial set C is known to classify skew-monoidal categories in the sense ... more ABSTRACT The Catalan simplicial set C is known to classify skew-monoidal categories in the sense that a map from C to a suitably defined nerve of Cat is precisely a skew-monoidal category [1]. We extend this result to the case of skew monoidales internal to any monoidal bicategory B. We then show that monoidal bicategories themselves are classified by maps from C to a suitably defined nerve of Bicat and extend this result to obtain a definition of skew-monoidal bicategory that aligns with existing theory.
A formal verification of the theory of parity complexes
We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including t... more We formalise, in Coq, the opening sections of Parity Complexes [Street1991] up to and including the all important excision of extremals algorithm. Parity complexes describe the essential combinatorial structure exhibited by simplexes, cubes and globes, that enable the construction of free $\omega$-categories on such objects. The excision of extremals is a recursive algorithm that presents every cell in such a category as a composite of atomic cells, this is the sense in which the $\omega$-category is free. Due to the complicated multi-dimensional nature of this work, the detail of definitions and proofs can be hard to follow and verify. Indeed, some corrections [Street1994] were required some years following the original publication. Our formalisation verifies that all cases of each result operate as stated. In particular, we indicate which portions of the theory can be proved directly from definitions, and which require more subtle and complex arguments. By identifying results that...
Mathematical Proceedings of the Cambridge Philosophical Society, 2014
Journal of Pure and Applied Algebra, 2014
We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibre... more We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2category into 2Cat. Fibred bicategories correspond to trihomomorphisms from a bicategory into Bicat. We describe the Grothendieck construction for each kind of fibration and present a few examples of each. Fibrations in our sense, between bicategories, are closed under composition and are stable under equiv-comma. The free such fibration on a homomorphism is obtained by taking an oplax comma along an identity.
International Journal of Computer Vision, 2012
Image-based 3D reconstruction remains a competitive field of research as state-of-the-art algorit... more Image-based 3D reconstruction remains a competitive field of research as state-of-the-art algorithms continue to improve. This paper presents a voxel-based algorithm that adapts the earliest space-carving methods and utilises a minimal surface technique to obtain a cleaner result. Embedded Voxel Colouring is built in two stages: (a) progressive voxel carving is used to build a volume of embedded surfaces and (b) the volume is processed to obtain a surface that maximises photo-consistency data in the volume. This algorithm combines the strengths of classical carving techniques with those of minimal surface approaches. We require only a single pass through the voxel volume, this significantly reduces computation time and is the key to the speed of our approach. We also specify three requirements for volumetric reconstruction: monotonic carving order, causality of carving and water-tightness.
Breast Cancer Research, 2013
Introduction: While Cumulusa semi-automated method for measuring breast densityis utilised extens... more Introduction: While Cumulusa semi-automated method for measuring breast densityis utilised extensively in research, it is labour-intensive and unsuitable for screening programmes that require an efficient and valid measure on which to base screening recommendations. We develop an automated method to measure breast density (AutoDensity) and compare it to Cumulus in terms of association with breast cancer risk and breast cancer screening outcomes. Methods: AutoDensity automatically identifies the breast area in the mammogram and classifies breast density in a similar way to Cumulus, through a fast, stand-alone Windows or Linux program. Our sample comprised 985 women with screen-detected cancers, 367 women with interval cancers and 4,975 controls (women who did not have cancer), sampled from first and subsequent screening rounds of a film mammography screening programme. To test the validity of AutoDensity, we compared the effect estimates using AutoDensity with those using Cumulus from logistic regression models that tested the association between breast density and breast cancer risk, risk of small and large screen-detected cancers and interval cancers, and screening programme sensitivity (the proportion of cancers that are screen-detected). As a secondary analysis, we report on correlation between AutoDensity and Cumulus measures. Results: AutoDensity performed similarly to Cumulus in all associations tested. For example, using AutoDensity, the odds ratios for women in the highest decile of breast density compared to women in the lowest quintile for invasive breast cancer, interval cancers, large and small screen-detected cancers were 3.2 (95% CI 2.5 to 4.1), 4.7 (95% CI 3.0 to 7.4), 6.4 (95% CI 3.7 to 11.1) and 2.2 (95% CI 1.6 to 3.0) respectively. For Cumulus the corresponding odds ratios were: 2.4 (95% CI 1.9 to 3.1), 4.1 (95% CI 2.6 to 6.3), 6.6 (95% CI 3.7 to 11.7) and 1.3 (95% CI 0.9 to 1.8). Correlation between Cumulus and AutoDensity measures was 0.63 (P < 0.001). Conclusions: Based on the similarity of the effect estimates for AutoDensity and Cumulus in models of breast density and breast cancer risk and screening outcomes, we conclude that AutoDensity is a valid automated method for measuring breast density from digitised film mammograms.
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Papers by Mitchell A Buckley