Mathematical Modeling of Eyebrow Curvature

Abstract

The aim of the study is to describe a mathematical model for analyzing eyebrow curvature that can be applied broadly to curvilinear facial features. A total of 100 digital images (50 men, 50 women) were obtained from standardized headshots of medical professionals. Images were analyzed in ImageJ by plotting either 8 or 15 points along the inferior-most row of contiguous brow cilia. A best-fit curve was automatically fit to these points in Microsoft Excel. The second derivative of the second-degree polynomial and a fourth-degree polynomial were used to evaluate brow curvature. Both techniques were subsequently compared with each other. A second-degree polynomial and fourth-degree polynomial were fit to all eyebrows. Plotting 15 points yielded greater goodness-of-fit than plotting 8 points along the inferior brow and allowed for more sensitive measurement of curvature across all images. A fourth-degree polynomial function provided a closer fit to the eyebrow than a second-degree polynomial function. This method provides a simple and reliable tool for quantitative analysis of eyebrow curvature from images. Fifteen-point plots and a fourth-degree polynomial curve provide a greater goodness-of-fit. The authors believe the described technique can be applied to other curvilinear facial features and will facilitate the analysis of standardized images.

The Arched Rainbow Brow in Thyroid Eye Disease

“…The methodology to measure the eyebrow curvature has been previously reported. [13][14][15] Each image had a similar size and pixel count and was placed in ImageJ (National Institutes of Health, Bethesda, Maryland). Images were then fit with 15 points along the most inferior row of contiguous brow cilia as a rough correction for various commonplace brow hygiene practices such as plucking and waxing.…”

Quantitative analysis of the eyelid curvature in patients with blepharoptosis

“…Danesh et al proposed multiple vertical lines parallel to MRD1 that measure eyelid height at multiple points medial and lateral to MRD1 [17]. Bezier curves, widely used in computer graphics, define a curve by a mathematical formula and have been suggested to analyze eyelid curvature [18][19][20]. However, a universally accepted method hasn't yet emerged.…”