Quantification of Choriocapillaris with Phansalkar Local Thresholding: Pitfalls to Avoid
Zhongdi ChuYuxuan ChengQinqin ZhangHao ZhouYining DaiYingying ShiGiovanni GregoriPhilip J. RosenfeldRuikang K. Wang
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Inflection point
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Inflection points are important characteristic points of plane curves with speical meanings in engineering,but have little been systematically and completely described in mathematics,and this leads to many confusions in analysis and calculations.Thus this paper first points out some problems and confusions in study of inflection points through a survey of the historical researches,then gives a mathematical definition of inflection point,and induces several properties of plane curves at their inflection points.The study of this paper can provide a complete and unite resolution in researches of inflection points as well as a scientific criterion for analysis and calculation of inflection points.
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We start with a geometric definition: A point of inflection is a point of a smooth curve at which the tangent not only touches the curve but also crosses it . Fig. 1 illustrates the definition and shows a curve, its point of inflection and the inflectional tangent (i.e. the tangent at the point of inflection) which crosses the curve. We can see that the point of inflection occurs where the curve stops bending to the left and starts bending to the right. This leads to a more descriptive definition of a point of inflection as a point where the direction of bending of a smooth curve changes sense.
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Based on the definitions of the strict lower convex curve,the strict lower concave curve and the inflection curve,this paper first makes a summary of the necessary condition for a point in a curve being an inflection point,and then proves the ways to determine whether the point whose second derivative is zero when determined by second derivative,third derivative and nth derivative and the point whose second derivative does not exist are both inflection points,and finally makes a conclusion that a point in a curve cannot be both an inflection point and an extreme point.
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When the second derivative y″(x) of function of a curve given by parameter equation has different signs on two sides of a certain parameter value t0,the corresponding point determined may be a false inflection point.The reason why the false inflection point is erroneously claimed as an inflection point is that the different signs of y″(x) on the two sides of t0 is regarded as the different signs of y″(x) on the two sides of x0.The solution is that the characteristic of the tangent line at any inflection point can be utilized to discriminate the true or false inflection point after getting possible inflection points.By using the curvature formula in differential geometry,the curved degree,the curved direction and the inflection point of a curve can be identified together.
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A localized thresholding method based on boundary detection is proposed. The performance of global thresholding is likely to degrade when the gray level of the background in an image fluctuates and non-background regions with relatively large area exists. The proposed approach, localized thresholding based on boundary (LOTBOB), estimates the regions which are likely to contain the interesting objects by presuming that the appearance of boundaries indicates the existence of objects with a high possibility. Thresholding is then limited in these regions to avoid global thresholding. Experiments on real-world images show that LOTBOB is effective with relatively small and clearly-edged objects.
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We present a variation on classic beam thresholding techniques that is up to an order of magnitude faster than the traditional method, at the same performance level. We also present a new thresholding technique, global thresholding, which, combined with the new beam thresholding, gives an additional factor of two improvement, and a novel technique, multiple pass parsing, that can be combined with the others to yield yet another 50% improvement. We use a new search algorithm to simultaneously optimize the thresholding parameters of the various algorithms.
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Inflection point suggests that there are critical points in the history of an industry or an individual company that signal permanent and enduring change. The chapter describes inflection's applicability to retailing, and what signifies an inflection point as it relates to the retail business. Accelerators and disruptors describe the notion of disruptive change in the marketplace. Inflection points represent a significant and permanent shift in the retail market, and those changes can be the result of companies’ responses to accelerators and disruptors. The inflection point model would help the readers understand that the market movement can be affected if a significant response is made to accelerators or disruptors occurring in the market.
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