Collingwood on religious atonement

R. G. Collingwood’s philosophical analysis of religious atonement as a dialectical process of mortal repentance and divine forgiveness is explained and criticized. Collingwood’s Christian concept of atonement, in which Christ TeX the Atonement (and also TeX the Incarnation), is subject in turn to another kind of dialectic, in which some of Collingwood’s leading ideas are first surveyed, and then tested against objections in a philosophical evaluation of their virtues and defects, strengths and weaknesses. Collingwood’s efforts to synthesize objective and subjective aspects of atonement, and his proposal to solve the soteriological problem as to why God becomes flesh, as a dogma of some Christian belief systems, is finally exposed in adversarial exposition as inadequately supported by one of his main arguments, designated here as Collingwood’s Dilemma. The dilemma is that sin is either forgiven or unforgiven by God. If God forgives sin, then God’s justice is lax, whereas if God does not forgive sin, then, also contrary to divine nature, God lacks perfect loving compassion. The dilemma is supposed to drive philosophy toward a concept of atonement in which the sacrifice of Christ is required in order to absolve God of the lax judgment objection. God forgives sin only when the price of sin is paid, in this case, by the suffering and crucifixion of God’s avatar. The dilemma can be resolved in another way than Collingwood considers, undermining his motivation for synthesizing objective and subjective facets of the concept of atonement for the sake of avoiding inconsistency. Collingwood is philosophically important because he asks all the right questions about religious atonement, and points toward reasonable answers, even if he does not always deliver original philosophically satisfactory solutions.

Automatic X-ray landmark detection and shape segmentation via data-driven joint estimation of image displacements

In this paper, we propose a new method for fully-automatic landmark detection and shape segmentation in X-ray images. To detect landmarks, we estimate the displacements from some randomly sampled image patches to the (unknown) landmark positions, and then we integrate these predictions via a voting scheme. Our key contribution is a new algorithm for estimating these displacements. Different from other methods where each image patch independently predicts its displacement, we jointly estimate the displacements from all patches together in a data driven way, by considering not only the training data but also geometric constraints on the test image. The displacements estimation is formulated as a convex optimization problem that can be solved efficiently. Finally, we use the sparse shape composition model as the a priori information to regularize the landmark positions and thus generate the segmented shape contour. We validate our method on X-ray image datasets of three different anatomical structures: complete femur, proximal femur and pelvis. Experiments show that our method is accurate and robust in landmark detection, and, combined with the shape model, gives a better or comparable performance in shape segmentation compared to state-of-the art methods. Finally, a preliminary study using CT data shows the extensibility of our method to 3D data.

Localization and Segmentation of 3D Intervertebral Discs in MR Images by Data Driven Estimation

This paper addresses the problem of fully-automatic localization and segmentation of 3D intervertebral discs (IVDs) from MR images. Our method contains two steps, where we first localize the center of each IVD, and then segment IVDs by classifying image pixels around each disc center as foreground (disc) or background. The disc localization is done by estimating the image displacements from a set of randomly sampled 3D image patches to the disc center. The image displacements are estimated by jointly optimizing the training and test displacement values in a data-driven way, where we take into consideration both the training data and the geometric constraint on the test image. After the disc centers are localized, we segment the discs by classifying image pixels around disc centers as background or foreground. The classification is done in a similar data-driven approach as we used for localization, but in this segmentation case we are aiming to estimate the foreground/background probability of each pixel instead of the image displacements. In addition, an extra neighborhood smooth constraint is introduced to enforce the local smoothness of the label field. Our method is validated on 3D T2-weighted turbo spin echo MR images of 35 patients from two different studies. Experiments show that compared to state of the art, our method achieves better or comparable results. Specifically, we achieve for localization a mean error of 1.6-2.0 mm, and for segmentation a mean Dice metric of 85%-88% and a mean surface distance of 1.3-1.4 mm.

Do relocated villages experience more forest cover change? Resettlements, shifting cultivation and forests in the Lao PDR

This study explores the relationships between forest cover change and the village resettlement and land planning policies implemented in Laos, which have led to the relocation of remote and dispersed populations into clustered villages with easier access to state services and market facilities. We used the Global Forest Cover Change (2000–2012) and the most recent Lao Agricultural Census (2011) datasets to assess forest cover change in resettled and non-resettled villages throughout the country. We also reviewed a set of six case studies and performed an original case study in two villages of Luang Prabang province with 55 households, inquiring about relocation, land losses and intensification options. Our results show that resettled villages have greater baseline forest cover and total forest loss than most villages in Laos but not significant forest loss relative to that baseline. Resettled villages are consistently associated with forested areas, minority groups, and intermediate accessibility. The case studies highlight that resettlement coupled with land use planning does not necessarily lead to the abandonment of shifting cultivation or affect forest loss but lead to a re-spatialization of land use. This includes clustering of forest clearings, which might lead to fallow shortening and land degradation while limited intensification options exist in the resettled villages. This study provides a contribution to studying relationships between migration, forest cover change, livelihood strategies, land governance and agricultural practices in tropical forest environments.

Fast Update of Conditional Simulation Ensembles

Gaussian random field (GRF) conditional simulation is a key ingredient in many spatial statistics problems for computing Monte-Carlo estimators and quantifying uncertainties on non-linear functionals of GRFs conditional on data. Conditional simulations are known to often be computer intensive, especially when appealing to matrix decomposition approaches with a large number of simulation points. This work studies settings where conditioning observations are assimilated batch sequentially, with one point or a batch of points at each stage. Assuming that conditional simulations have been performed at a previous stage, the goal is to take advantage of already available sample paths and by-products to produce updated conditional simulations at mini- mal cost. Explicit formulae are provided, which allow updating an ensemble of sample paths conditioned on n ≥ 0 observations to an ensemble conditioned on n + q observations, for arbitrary q ≥ 1. Compared to direct approaches, the proposed formulae proveto substantially reduce computational complexity. Moreover, these formulae explicitly exhibit how the q new observations are updating the old sample paths. Detailed complexity calculations highlighting the benefits of this approach with respect to state-of-the-art algorithms are provided and are complemented by numerical experiments.

On degeneracy and invariances of random fields paths with applications in Gaussian process modelling

We study pathwise invariances and degeneracies of random fields with motivating applications in Gaussian process modelling. The key idea is that a number of structural properties one may wish to impose a priori on functions boil down to degeneracy properties under well-chosen linear operators. We first show in a second order set-up that almost sure degeneracy of random field paths under some class of linear operators defined in terms of signed measures can be controlled through the two first moments. A special focus is then put on the Gaussian case, where these results are revisited and extended to further linear operators thanks to state-of-the-art representations. Several degeneracy properties are tackled, including random fields with symmetric paths, centred paths, harmonic paths, or sparse paths. The proposed approach delivers a number of promising results and perspectives in Gaussian process modelling. In a first numerical experiment, it is shown that dedicated kernels can be used to infer an axis of symmetry. Our second numerical experiment deals with conditional simulations of a solution to the heat equation, and it is found that adapted kernels notably enable improved predictions of non-linear functionals of the field such as its maximum.

Differentiating the Multipoint Expected Improvement for Optimal Batch Design

This work deals with parallel optimization of expensive objective functions which are modelled as sample realizations of Gaussian processes. The study is formalized as a Bayesian optimization problem, or continuous multi-armed bandit problem, where a batch of q > 0 arms is pulled in parallel at each iteration. Several algorithms have been developed for choosing batches by trading off exploitation and exploration. As of today, the maximum Expected Improvement (EI) and Upper Confidence Bound (UCB) selection rules appear as the most prominent approaches for batch selection. Here, we build upon recent work on the multipoint Expected Improvement criterion, for which an analytic expansion relying on Tallis’ formula was recently established. The computational burden of this selection rule being still an issue in application, we derive a closed-form expression for the gradient of the multipoint Expected Improvement, which aims at facilitating its maximization using gradient-based ascent algorithms. Substantial computational savings are shown in application. In addition, our algorithms are tested numerically and compared to state-of-the-art UCB-based batchsequential algorithms. Combining starting designs relying on UCB with gradient-based EI local optimization finally appears as a sound option for batch design in distributed Gaussian Process optimization.