The Persistent Homology of Dual Digital Image Constructions

To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital adjacencies) can give different results for the same image. The two constructions are almost dual to each other, and we use this relationship to extend and modify the cubical complexes to become dual filtered cell complexes. We derive a general relationship between the persistent homology of two dual filtered cell complexes, and also establish how various modifications to a filtered complex change the persistence diagram. Applying these results to images, we derive a method to transform the persistence diagram computed using one type of cubical complex into a persistence diagram for the other construction. This means software for computing persistent homology from images can now be easily adapted to produce results for either of the two cubical complex constructions without additional low-level code implementation

The impact of changes in resolution on the persistent homology of images

Digital images enable quantitative analysis of material properties at micro and macro length scales, but choosing an appropriate resolution when acquiring the image is challenging. A high resolution means longer image acquisition and larger data requirements for a given sample, but if the resolution is too low, significant information may be lost. This paper studies the impact of changes in resolution on persistent homology, a tool from topological data analysis that provides a signature of structure in an image across all length scales. Given prior information about a function, the geometry of an object, or its density distribution at a given resolution, we provide methods to select the coarsest resolution yielding results within an acceptable tolerance. We present numerical case studies for an illustrative synthetic example and samples from porous materials where the theoretical bounds are unknown

Association for Women in Mathematics Series

To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital adjacencies) can give different results for the same image. The two constructions are almost dual to each other, and we use this relationship to extend and modify the cubical complexes to become dual filtered cell complexes. We derive a general relationship between the persistent homology of two dual filtered cell complexes, and also establish how various modifications to a filtered complex change the persistence diagram. Applying these results to images, we derive a method to transform the persistence diagram computed using one type of cubical complex into a persistence diagram for the other construction. This means software for computing persistent homology from images can now be easily adapted to produce results for either of the two cubical complex constructions without additional low-level code implementation

Effects of isometric resistance training on resting blood pressure: Individual participant data-analysis

Background: Previous meta-analyses based on aggregate group-level data report antihypertensive effects of isometric resistance training (IRT). However, individual participant data meta-analyses provide more robust effect size estimates and permit examination of demographic and clinical variables on IRT effectiveness. Methods: We conducted a systematic search and individual participant data (IPD) analysis, using both a one-step and two-step approach, of controlled trials investigating at least 3 weeks of IRT on resting systolic, diastolic and mean arterial blood pressure. Results: Anonymized individual participant data were provided from 12 studies (14 intervention group comparisons) involving 326 participants (52.7% medicated for hypertension); 191 assigned to IRT and 135 controls, 25.2% of participants had diagnosed coronary artery disease. IRT intensity varied (8–30% MVC) and training duration ranged from 3 to 12 weeks. The IPD (one-step) meta-analysis showed a significant treatment effect for the exercise group participants experiencing a reduction in resting SBP of −6.22 mmHg (95% CI −7.75 to −4.68; P < 0.00001); DBP of −2.78 mmHg (95% CI −3.92 to −1.65; P = 0.002); and mean arterial blood pressure (MAP) of −4.12 mmHg (95% CI −5.39 to −2.85; P < 0.00001). The two-step approach yielded similar results for change in SBP −7.35 mmHg (−8.95 to −5.75; P < 0.00001), DBP MD −3.29 mmHg (95% CI −5.12 to −1.46; P = 0.0004) and MAP MD −4.63 mmHg (95% CI −6.18 to −3.09: P < 0.00001). Sub-analysis revealed that neither clinical, medication, nor demographic participant characteristics, or exercise program features, modified the IRT treatment effect. Conclusion: This individual patient analysis confirms a clinically meaningful and statistically significant effect of IRT on resting SBP, DBP and mean arterial blood pressure