Connection between the Riemann integrability of a multi-valued function and of its convex hull

The detection of spherical targets in workpiece shape clustering and fruit classification tasks is challenging. Spherical targets produce low detection accuracy in complex fields, and single-feature processing cannot accurately recognize spheres. Therefore, a novel spherical descriptor (SD) for contour fitting and convex hull processing is proposed. The SD achieves image de-noising by combining flooding processing and morphological operations. The number of polygon-fitted edges is obtained by convex hull processing based on contour extraction and fitting, and two RGB images of the same group of objects are obtained from different directions. The two fitted edges of the same target object obtained at two RGB images are extracted to form a two-dimensional array. The target object is defined as a sphere if the two values of the array are greater than a custom threshold. The first classification result is obtained by an improved K-NN algorithm. Circle detection is then performed on the results using improved Hough circle detection. We abbreviate it as a new Hough transform sphere descriptor (HSD). Experiments demonstrate that recognition of spherical objects is obtained with 98.8% accuracy. Therefore, experimental results show that our method is compared with other latest methods, HSD has higher identification accuracy than other methods.

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The construction of local convex hull on the task of pattern recognition

Procedia Computer Science ◽

10.1016/j.procs.2021.04.157 ◽

2021 ◽

Vol 186 ◽

pp. 360-365

Author(s):

Kamilov Mirzoyan ◽

Hudayberdiev Mirzaakbar ◽

Khamroev Alisher

Keyword(s):

Pattern Recognition ◽

Convex Hull ◽

Local Convex

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Enhanced CD4+ and CD8+ T cell infiltrate within convex hull defined pancreatic islet borders as autoimmune diabetes progresses

Scientific Reports ◽

10.1038/s41598-021-96327-2 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Alexander J. Dwyer ◽

Jacob M. Ritz ◽

Jason S. Mitchell ◽

Tijana Martinov ◽

Mohannad Alkhatib ◽

...

Keyword(s):

T Cells ◽

T Cell ◽

Convex Hull ◽

Pancreatic Islets ◽

Immune Cell ◽

Nod Mice ◽

Cell Infiltration ◽

Immune Cell Infiltration ◽

Analytical Strategy ◽

Islet Mass

AbstractThe notion that T cell insulitis increases as type 1 diabetes (T1D) develops is unsurprising, however, the quantitative analysis of CD4+ and CD8+ T cells within the islet mass is complex and limited with standard approaches. Optical microscopy is an important and widely used method to evaluate immune cell infiltration into pancreatic islets of Langerhans for the study of disease progression or therapeutic efficacy in murine T1D. However, the accuracy of this approach is often limited by subjective and potentially biased qualitative assessment of immune cell subsets. In addition, attempts at quantitative measurements require significant time for manual analysis and often involve sophisticated and expensive imaging software. In this study, we developed and illustrate here a streamlined analytical strategy for the rapid, automated and unbiased investigation of islet area and immune cell infiltration within (insulitis) and around (peri-insulitis) pancreatic islets. To this end, we demonstrate swift and accurate detection of islet borders by modeling cross-sectional islet areas with convex polygons (convex hulls) surrounding islet-associated insulin-producing β cell and glucagon-producing α cell fluorescent signals. To accomplish this, we used a macro produced with the freeware software ImageJ equipped with the Fiji Is Just ImageJ (FIJI) image processing package. Our image analysis procedure allows for direct quantification and statistical determination of islet area and infiltration in a reproducible manner, with location-specific data that more accurately reflect islet areas as insulitis proceeds throughout T1D. Using this approach, we quantified the islet area infiltrated with CD4+ and CD8+ T cells allowing statistical comparison between different age groups of non-obese diabetic (NOD) mice progressing towards T1D. We found significantly more CD4+ and CD8+ T cells infiltrating the convex hull-defined islet mass of 13-week-old non-diabetic and 17-week-old diabetic NOD mice compared to 4-week-old NOD mice. We also determined a significant and measurable loss of islet mass in mice that developed T1D. This approach will be helpful for the location-dependent quantitative calculation of islet mass and cellular infiltration during T1D pathogenesis and can be combined with other markers of inflammation or activation in future studies.

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Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals

Journal of Applied Analysis ◽

10.1515/jaa-2021-2052 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Lung-Hui Chen

Keyword(s):

Fourier Transform ◽

Entire Function ◽

Convex Hull ◽

Scattering Theory ◽

Phase Retrieval ◽

Indicator Function ◽

Band Limited ◽

The Fourier Transform ◽

Complex Valued ◽

Spectral Invariant

Abstract In this paper, we discuss how to partially determine the Fourier transform F ⁢ ( z ) = ∫ - 1 1 f ⁢ ( t ) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ ( z ) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ ( z ) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}} . Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F ⁢ ( z ) {F(z)} . Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.

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