On some entropy functionals derived from R\'enyi information divergence

We consider the maximum entropy problems associated with R\'enyi $Q$-entropy, subject to two kinds of constraints on expected values. The constraints considered are a constraint on the standard expectation, and a constraint on the generalized expectation as encountered in nonextensive statistics. The optimum maximum entropy probability distributions, which can exhibit a power-law behaviour, are derived and characterized. The R\'enyi entropy of the optimum distributions can be viewed as a function of the constraint. This defines two families of entropy functionals in the space of possible expected values. General properties of these functionals, including nonnegativity, minimum, convexity, are documented. Their relationships as well as numerical aspects are also discussed. Finally, we work out some specific cases for the reference measure $Q(x)$ and recover in a limit case some well-known entropies

On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles

In this paper we revisit the Bialynicki-Birula & Mycielski uncertainty principle and its cases of equality. This Shannon entropic version of the well-known Heisenberg uncertainty principle can be used when dealing with variables that admit no variance. In this paper, we extend this uncertainty principle to Renyi entropies. We recall that in both Shannon and Renyi cases, and for a given dimension n, the only case of equality occurs for Gaussian random vectors. We show that as n grows, however, the bound is also asymptotically attained in the cases of n-dimensional Student-t and Student-r distributions. A complete analytical study is performed in a special case of a Student-t distribution. We also show numerically that this effect exists for the particular case of a n-dimensional Cauchy variable, whatever the Renyi entropy considered, extending the results of Abe and illustrating the analytical asymptotic study of the student-t case. In the Student-r case, we show numerically that the same behavior occurs for uniformly distributed vectors. These particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a "Gaussianization" of the vector when the dimension increases

Ethical implications of AI in robotic surgical training: A Delphi consensus statement

CONTEXT: As the role of AI in healthcare continues to expand there is increasing awareness of the potential pitfalls of AI and the need for guidance to avoid them. OBJECTIVES: To provide ethical guidance on developing narrow AI applications for surgical training curricula. We define standardised approaches to developing AI driven applications in surgical training that address current recognised ethical implications of utilising AI on surgical data. We aim to describe an ethical approach based on the current evidence, understanding of AI and available technologies, by seeking consensus from an expert committee. EVIDENCE ACQUISITION: The project was carried out in 3 phases: (1) A steering group was formed to review the literature and summarize current evidence. (2) A larger expert panel convened and discussed the ethical implications of AI application based on the current evidence. A survey was created, with input from panel members. (3) Thirdly, panel-based consensus findings were determined using an online Delphi process to formulate guidance. 30 experts in AI implementation and/or training including clinicians, academics and industry contributed. The Delphi process underwent 3 rounds. Additions to the second and third-round surveys were formulated based on the answers and comments from previous rounds. Consensus opinion was defined as ≥ 80% agreement. EVIDENCE SYNTHESIS: There was 100% response from all 3 rounds. The resulting formulated guidance showed good internal consistency, with a Cronbach alpha of >0.8. There was 100% consensus that there is currently a lack of guidance on the utilisation of AI in the setting of robotic surgical training. Consensus was reached in multiple areas, including: 1. Data protection and privacy; 2. Reproducibility and transparency; 3. Predictive analytics; 4. Inherent biases; 5. Areas of training most likely to benefit from AI. CONCLUSIONS: Using the Delphi methodology, we achieved international consensus among experts to develop and reach content validation for guidance on ethical implications of AI in surgical training. Providing an ethical foundation for launching narrow AI applications in surgical training. This guidance will require further validation. PATIENT SUMMARY: As the role of AI in healthcare continues to expand there is increasing awareness of the potential pitfalls of AI and the need for guidance to avoid them.In this paper we provide guidance on ethical implications of AI in surgical training

Entropic graphs for image registration

Given 2D or 3D images gathered via multiple sensors located at different positions, the multi-sensor image registration problem is to align the images so that they have an identical pose in a common coordinate system. Image registration methods depend crucially upon a robust image similarity measure to guide the image alignment. This thesis concerns itself with a new class of such similarity measures. The launching point of this thesis is the entropic graph based estimate of Renyi's alpha-entropy developed by Ma for image registration. This thesis extends this initial work to develop other entropic graph-based divergence measures to be used with advanced higher dimensional features. A detailed analysis of entropic graphs is followed by a demonstration of their performance advantages relative to conventional similarity measures. This thesis introduces techniques to extend image registration to higher dimension feature spaces using Renyi's generalized alpha-entropy. The alpha-entropy is estimated directly through continuous quasi-additive power-weighted graphs such as the minimal spanning tree (MST) and k-Nearest Neighbor graph (kNN). Entropic graph methods are further used to approximate similarity measures like the alpha-mutual information, non-linear correlation coefficient, alpha-Jensen divergence, Henze-Penrose affinity and Geometric-Arithmetic mean affinity. Entropic-graph similarity measures are applied to problems in breast Ultrasound image registration for cancer management, geo-stationary satellite registration, feature clustering and classification and for atlas based multi-image registration. This last work is a novel and significant application of divergence estimation for registering several images simultaneously. These similarity measures offer robust registration benefits in a multisensor environment. Higher dimensional features used for this work include basis functions like multidimensional wavelets, independent component analysis (ICA) and discrete cosine transforms.Ph.D.Applied SciencesBiomedical engineeringHealth and Environmental SciencesMedical imagingUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/124899/2/3163898.pd

SimVenture -- A Start-Up Business Simulation

"SimVenture is a business start-up simulation. It focus-es on the first 3 years of a business and deals with issues that an entrepreneur who starts a business from scratch would face. SimVenture offers a tool called Scenarios, which al-lows the user to build different business situations. Partici-pants will be able to experience different starting points including business start up, cash flow crisis and growing pains - representing varied parts of a business. It is a software simulation that can be used in-class or individually with written direction from the tutor.

Image registration methods in high-dimensional space

Quantitative evaluation of similarity between feature densities of images is an important step in several computer vision and data-mining applications such as registration of two or more images and retrieval and clustering of images. Previously we had introduced a new class of similarity measures based on entropic graphs to estimate RÈnyi's Α-entropy, Α-Jensen difference divergence, Α-mutual information, and other divergence measures for image registration. Entropic graphs such as the minimum spanning tree (MST) and k-Nearest neighbor (kNN) graph allow the estimation of such similarity measures in higher dimensional feature spaces. A major drawback of histogram-based estimates of such measures is that they cannot be reliably constructed in higher dimensional feature spaces. In this article, we shall briefly extrapolate upon the use of entropic graph based divergence measures mentioned above. Additionally, we shall present estimates of other divergence viz the Geometric-Arithmetic mean divergence and Henze–Penrose affinity. We shall present the application of these measures for pairwise image registration using features derived from independent component analysis of the images. An extension of pairwise image registration is to simultaneously register multiple images, a challenging problem that arises while constructing atlases of organs in medical imaging. Using entropic graph methods we show the feasibility of such simultaneous registration using graph based higher dimensional estimates of entropy measures. Finally we present a new nonlinear correlation measure that is invariant to nonlinear transformations of the underlying feature space and can be reliably constructed in higher dimensions. We present an image clustering experiment to demonstrate the robustness of this measure to nonlinear transformations and contrast it with the clustering performance of the linear correlation coefficient. © 2007 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 16, 130–145, 2006Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/50687/1/20079_ftp.pd

Image registration in high dimensional feature space

Image registration is a difficult task especially when spurrious image intensity differences and spatial variations between the two images are present. To robustify image registration algorithms to such spurrious variations it can be useful to employ an image registration matching criteria on higher dimensional feature spaces. This paper will present an overview of our recent work on image registration using high dimensional image features and entropic graph matching criteria. New entropic graph estimates of information divergence measures will be presented. We will demonstrate the advantage of our approach for ultrasound breast image registration