The Use of Fractal Analysis and Photometry to Estimate the Accuracy of Bulbar Redness Grading Scales

Abstract

purpose. To use physical attributes of redness to determine the accuracy of four bulbar redness grading scales, and to cross-calibrate the scales based on these physical measures.

methods. Two image-processing metrics, fractal dimension (D) and percentage of pixel coverage (% PC), as well as photometric chromaticity were selected as physical measures, to describe and compare grades of bulbar redness among the McMonnies/Chapman-Davies scale, the Efron Scale, the Institute for Eye Research scale, and a validated scale developed at the Centre for Contact Lens Research. Two sets of images were prepared by using image processing: The first included multiple segments covering the largest possible region of interest (ROI) within the bulbar conjunctiva in the original images; the second contained modified scale images that were matched in size and resolution across scales, and a single, equally-sized ROI. To measure photometric chromaticity, the original scale images were displayed on a computer monitor, and multiple conjunctival segments were analyzed. Pearson correlation coefficients between each set of image metrics and the reference image grades were calculated to determine the accuracy of the scales.

results. Correlations were high between reference image grades and all sets of objective metrics (all Pearson’s r ≥ 0.88, P ≤ 0.05); each physical attribute pointed to a different scale as being most accurate. Independent of the physical attribute used, there were wide discrepancies between scale grades, with almost no overlap when cross-calibrating and comparing the scales.

conclusions. Despite the generally strong linear associations between the physical characteristics of reference images in each scale, the scales themselves are not inherently accurate and are too different to allow for cross-calibration.

Red eye, clinically known as bulbar hyperemia, is an increased dilation of blood vessels in the bulbar conjunctiva that gives the eye its red appearance and is a prominent sign of ocular irritation. The recognition of change in redness is crucial for clinicians in management of the ocular surface, particularly in contact lens research and practice. Commonly, redness is estimated in a patient’s eye by subjectively comparing it to references that represent different levels of severity for the condition and assist in monitoring changes over time. The references are descriptive, ^{1 2} illustrative, ^{3 4 5 6} or computer generated ^{7 8} and are presented in the form of grading scales. The subjectivity in grading is a criticism linked to the use of grading scales, and weak repeatability for inter- and intraobserver assessments is of particular concern for clinical practice. ^{9 10} Aside from variability introduced by observer use, grading scales have been criticized for technical difficulties, ^{10 11} such as unequal steps, references not capable of covering the whole range of the scale, or biased depiction of references for different levels of severity. Hence, it has been recommended that the different grading scales not be interchanged. ^{8 12 13 14}  

Repeatability has been the main focus of most research studies of traditional grading, with respect to differences between observers, ^{3 11 12 15} between grading scales, ^{8 12} or between levels of observer training, ^{6 16 17} or compared with novel objective techniques measuring the physical attributes of redness. ^{9 14 18 19 20 21 22 23} The physical attributes to describe conjunctival redness have included various quantitative ^{13 20 21 22 24} (e.g., number of vessels or percentage of vessel coverage) and colorimetric (e.g., chromaticity levels or red intensity ratios) ^{9 18 19 23} variables that were determined using either digital image processing or photometric techniques. However, we found only three studies in which the physical attributes of the scales, per se, were analyzed. ^{13 14 19}  

An interval or ratio scale level has been recommended for grading scales, ²² since it ensures uniform separation of reference images across the scale range; that is, a change from 10 to 20 on a 100-point scale represents the same difference as a change from 70 to 80 on the same scale. The extent of blood vessel coverage (% PC, an objective measure of redness) has been used to examine scales and compare them, but not specifically to investigate the separation of the steps of the scales. ^{13 14 19}  

In this study, we introduce fractal analysis, a new technique for analysis of grading scales, and compare it to % PC ^{13 14 19} and photometric chromaticity. ²⁵ Fractal analysis has been shown to be a powerful objective technique for detecting changes in various biological systems. ^{26 27} It describes the complexity of the object or pattern by estimating the degree of branching of the vascular tree in the respective biological system. ²⁸ Fractals found in nature are so-called random fractals, objects that are scale invariant over a finite range, which means that they look the same under different degrees of magnification or scale (e.g., the branches of a tree). They are quantized by a fractal dimension, D, describing the degree of branching. In a two-dimensional photograph of vascular branches in the eye, the fractal dimension D can take on any decimalized value between 0 and 2. Figure 1shows simulated examples of vascular branching of the bulbar conjunctiva and the range of the expected fractal dimension, D. With respect to the eye, fractal analysis has been used to simulate corneal neovascularization ²⁹ and has been successfully applied to investigate the vessel structure in normal and diseased retinas. ^{28 30 31}  

The purpose of this study was to estimate the accuracy of the grading scales by comparing the distribution and separation of the reference images of illustrative redness grading scales to objective physical attributes of redness defined by fractal analysis and photometric chromaticity and to use these measures to cross-calibrate the scales. 

Methods

Grading Scale Images

The bulbar redness reference images of four grading scales were analyzed: The McMonnies/Chapman-Davies scale ³ (MC-D), the Institute for Eye Research scale ⁴ (IER, previously known as CCLRU scale), the Efron scale, ⁵ and a validated bulbar redness scale ⁶ (VBR) developed at the Centre for Contact Lens Research. The images in these four grading scales were generated by using different procedures and instrumentation, and differ in size, resolution, and the display of the area of interest (Fig. 2) . 

The highest-resolution reference images provided by the producers of each grading scale were used in the study. The MC-D scale images were scanned from the original hardcopy of the scale, and all others were digital in their original format. To perform fractal analysis, we saved each image in the tagged image file format (TIFF); therefore, the reference images for the Efron scale had to be converted from high-resolution joint photographic experts group (JPEG) images into the TIFF format. Table 1shows the original file types and the resolution and size of the original images. 

Image Processing and Fractal Analysis

The public domain Java image-processing program ImageJ 1.38x ³² was used for the analysis of the color photographs or illustrations in the grading scales. Initially, the signal-to-noise ratio (SNR) was determined for each color channel, to select the best channel for further analysis. The selected channel was preprocessed to maximize the SNR and was then binarized before fractal analysis. ^{33 34}  

The image representing the lowest level of severity for each of the scales was used to determine the 8-bit color component (i.e., red, green, or blue) that provided the highest SNR (in decibels). The image was chosen by analyzing the SNR of each color component for the largest rectangle that included only the conjunctiva in each scale (see the Scale Version 2 section for details). An image has a high SNR if the contrast of the object is large relative to the image noise; the noise in an image is characterized by the SD of its brightness differences. ³⁴ In this case, the numerator of the SNR represents the conjunctival blood vessels, and the denominator represents the noise in the background of the image. The background noise is therefore determined for areas of the image that do not include vascular detail. 

To determine the SNR, the procedure suggested by Young et al. ³⁴ was followed. Since the signal (i.e., the blood vessels) is red, the red channel contained minimal target information ¹⁹ and the whole image was used to determine the background noise. The SNR for each color component of each scale was calculated with the following equation:  

\[\mathrm{SNR}{=}20\mathrm{log}_{10}\ {_\ast}\ \frac{(a_{\mathrm{max}}{-}a_{\mathrm{min}})}{s_{\mathrm{n}}}\ ,\]

where a _max and a _min represent the maximum and minimum brightnesses within the whole image (brightness range) and s _n represents the SD of the brightness. Table 2displays the data for each parameter and the SNR. For each of the scales, the green component provided the highest SNR and was consequently the image used for preprocessing and fractal analysis. 

Because of the acquisition differences between the scales, two versions of the grading scales were generated for fractal analysis. The first version used the original reference images for each scale and used multiple segments of the region of interest (ROI) with the intent of covering the largest area possible of the conjunctiva (scale version 1). The size and number of segments required to achieve this varied between scales. The intention of this scale was to mimic clinical subjective grading, where a global estimate of the conjunctiva is commonly preferred to rating a single, prominent vessel. ^{6 8} The second version modified the reference images before fractal analysis to match them in size and resolution across scales, so that for each reference image a single, rectangular, and equally sized ROI was analyzed (scale version 2). For both scale versions, the preprocessing settings were identical for all scales. The procedure used to generate scale versions 1 and 2 is illustrated in Figure 3for the grade 50 reference image of the VBR scale. 

Scale Version 1: Greatest Conjunctival Area Coverage.

The cornea, eyelids, and lashes were excluded in the green component of the reference images to outline the overall region of interest (Figs. 3ai 3aii ). A median filter was used to reduce the background noise. The filter settings were determined incrementally to obtain the highest correlation (for all scales) between scale grades and the objective fractal analysis measures. Next, the image background was subtracted to account for eyeball curvature (Fig. 3aiii) . ²⁵ Finally, a Sobel edge-detection algorithm ³² was applied to highlight vessel edges as well as small capillaries (Fig. 3aiv) . For each scale, 8 to 10 equally sized segments of the ROI were selected to cover the largest area possible on the conjunctiva (Fig. 3av) . Although the size and number of the ROI segments was different between scales, the same segments were used across all steps within a scale. To complete preprocessing of the images, each segment was binarized by using an automated thresholding procedure (Fig. 3avi)where the pixel color (black or white) for foreground (blood vessels) and background was automatically assigned by ImageJ. For two of the images (Efron grades 3 and 4) the background and foreground pixel colors were reversed, and the images were therefore inverted for consistency. 

Scale Version 2: Size-Matching of Reference Images.

To allow fractal analysis of a single, equally sized ROI for each scale image, the resolution and size of the scale images were adjusted to account for the differences between scales shown in Table 1 . First, the original images were matched in resolution with respect to the scale with the lowest resolution (Efron, 72 dpi; Table 1 , Fig. 3bi ). Next, the largest possible ROI that could be fitted within the overall bulbar conjunctiva was determined for each scale image (Fig. 3bii) , and their sizes were compared to determine the ROI which was smallest for any of these images (MC-D; 250 × 156 pixels). This ROI was chosen as reference size and proportionally scaled versions of this reference size (i.e., having the same ratio of horizontal to vertical pixels, ∼1.6025:1) were fitted in the overall conjunctival outlines of the images in the three other scales. To allow the same x/y dimensions for all the scales, the ROI was cropped out of the image in its original size (e.g., 400 × 250 for VBR grade 50) and down-scaled to the reference size of 250 × 156 pixels (Fig. 3biii)and, in the case of the Efron scale, was rotated counterclockwise by 90°. 

The background noise of the size and resolution matched green 8-bit image was removed using a median filter. The settings of the median filter were determined separately for scale version 2 (using the same method described in scale version 1), and the background was subtracted from each image (Fig. 3biv) . The Sobel edge-detection filter was not applied to the image, as it added significant noise and did not enhance the target. The reference images were then binarized as described for scale version 1 (Fig. 3bv) . 

Fractal Analysis.

The ImageJ plug-in FracLac (ver. 2.5 Rel. 1b5i) ^{32 35} was used for fractal analysis. It employs box-counting algorithms to determine the fractal dimension of an object. During this procedure, a series of grids of boxes with decreasing box size is placed over the ROI, and the number of boxes containing pixels with detail is counted. The fractal dimension is then expressed as the slope of the regression line for the log–log plot of box size and count. ³⁵  

The following standardized settings in FracLac were developed according to the recommendations of the FracLac user manual. ³⁵ The size of the series of grids was set to decrease linearly from a maximum box size of 45% of the horizontal ROI size to a minimum size of 1 pixel. Ten global scans were performed for each ROI, with randomly selected starting grid locations to improve the accuracy of the box-counting dimension. The following outcome measures were derived by FracLac and used to describe redness in terms of vascular branching (four different fractal dimensions) and area of vascular coverage (% PC):

•  
  Averaged D (D̄): fractal dimension which is averaged over all 10 global scan locations.
•  
  Slope-corrected D (D _sc): same as D̄, but corrected for periods of no change for the log-log plot of box size and count.
•  
  Most-efficient covering D (D _e): same as D̄, but for each grid size the box-count that required the lowest number of boxes was used.
•  
  Slope-corrected most-efficient covering D (D _sce): combination of all the above.
•  
  % PC: the ratio of the number of black foreground pixels (representing vessels) to the overall number of pixels in the ROI.

Photometric Measurements

A spectrophotometer (Spectrascan650; Photoresearch Inc, Chatsworth, VA) was used to measure chromaticity (to estimate the amount of redness in the grading scale references). The photometer was mounted on a tripod and positioned at a fixed distance of 30 cm from an LCD computer monitor (Flatron L1511S; LG, Seoul, Korea; Fig. 4 ). Experimental settings (room illumination, screen brightness and color, and photometer position) were standardized before the beginning of and controlled throughout the measurements. ImageJ v. 1.38x was used to display the unmodified reference images on the screen and to specify the associated, identical ROIs as established for scale version 1 (Fig. 3av) . To keep measurement settings stable, only the images on the screen and not the photometer itself were moved, and the position of each ROI was adjusted with respect to the fixation target in the eyepiece of the photometer. In a preliminary experiment, it was shown that these measurements were highly repeatable and unaffected by factors such as brightness fading of the screen or inaccurate positioning of the ROI. ²³  

In a 30-second sequence five repeated measures of the same ROI were taken, and the chromaticity values u′ and u* were subsequently averaged to estimate redness. ⁶ The quantities u′ and u* are described by the Commission International de L’Eclairage (CIE) in the CIE (u′, v′) system and the CIE L*u*v* space. ^{36 37} In the CIE (u′, v′) system a chromaticity diagram with axes u′ and v′ is used to describe all possible colors, and the human perception of color differences in this diagram is approximately uniform across the whole diagram. Whereas u′ and v′ determine the chromaticity of a color, the u* and v* of the CIE L*u*v* space includes a third dimension, lightness (L*), the perception of luminance (L), to describe the color. ^{6 37 38}  

Data Analysis

To determine fractal dimensions and % PC for the largest conjunctival area possible of each reference image, the results for the individual segmented ROIs of scale version 1 and the photometric measures were averaged to represent a global estimate of the conjunctival redness. 

The Pearson’s product moment correlation coefficient (Pearson’s r) was used to estimate the strength of linear association between scale steps and physical attributes of the scales (D, % PC, and photometric chromaticity). 

Results

Preprocessing of the grading scale reference images resulted in a sequence of binarized images that either consisted of individual segment ROIs (scale version 1) or single ROIs (scale version 2) for each of the scales. The changes in severity across the scale range for these binarized images are shown in Figures 5a (scale version 1, one segment ROI per scale) and 5b(scale version 2). 

Independent of the measure (D, % PC, or photometric chromaticity), strong linear associations between grading scale steps and associated physical attributes were found, as expressed by Pearson’s r of at least 0.88 (all P ≤ 0.05) for any of the scales (Table 3) . Correlation levels between scale steps and fractal dimensions as well as % PC for each grading scale are given, subdivided into scale versions 1 and 2, in Table 3 . The table also shows the correlation levels of the combinations of scale steps and chromaticity measures u′ and u*. Pearson correlations for any combination of physical attributes were at least r = 0.89 (all P < 0.05). 

A graphic display of the relation between scale grades and physical attributes of the scales is shown in Figure 6 . To allow display and comparison of all grading scales in the same graph, we translated the VBR scale from its original 100-point format relative to the other scales. The results for D _sce are used to illustrate the relation between scale grade and fractal dimension for the images based on greatest conjunctival area covered (scale version 1, Fig. 6a ) and after size-matching of the scales (scale version 2, Fig. 6b ). The relation between scale grade and % PC for the greatest conjunctival area coverage and after size-matching of the images is shown in Figures 6c and 6d , respectively. Figure 6edisplays the relation between photometric chromaticity (shown as u′) and the associated scale grade for each grading scale. 

Discussion

The purpose of this study was to estimate the accuracy of the grading scales by comparing the distribution and separation of the reference images of illustrative redness grading scales to objective physical attributes of redness defined by fractal analysis and photometric chromaticity, and to use these measures to cross-calibrate the scales. 

One limitation of the study was that scale reference images were provided at different resolutions (72–300 dpi). As with all image-processing techniques, the highest spatial resolution possible is advantageous, particularly when small spatial details form the object of interest. However, even at the lowest resolution used in this experiment, there was a systematic relationship between grading scale steps and the estimated fractal dimension. If we were to attempt to glean recommendations from these resolution data, our results suggest that images acquired only at the common resolution (72 dpi) of a screen display were sufficient to quantify redness reasonably, based on fractal dimension. Most common clinical digital image-acquisition instrumentation would provide much higher resolution. 

Accuracy of the Grading Scales

The use of fractal analysis to assess changes in vascular branching is an emerging strategy in clinical research. ^{27 28 31} This is the first study in which fractal analysis was used to evaluate vascular structures in the conjunctiva, as well as to compare differences in the physical attributes between grading scale images. The strong correlations (all r ≥ 0.89; P ≤ 0.05) between physical measures to describe conjunctival redness (% PC ^{13 14 19 22} and photometric chromaticity ²⁵ ) and fractal dimensions indicate that fractal analysis is capable of describing changes in severity of bulbar redness. 

The preprocessed scale images showing the changes in severity across the whole scale range are displayed in Figure 5 . The physical attributes derived from these images (% PC and D) correlated highly with the grading scale steps (Table 3) . The types of fractal dimensions (D̄, D _sc, D _e, and D _sce) calculated by FracLac showed only minimal differences in the raw data for any scale for the same preprocessing procedure, which resulted in very small variations of the Pearson correlation coefficients (Table 3 ; differences within a scale ±0.01). Therefore the slope-corrected, most efficient covering fractal dimension (D _sce), which eliminates periods of no change in the data by using the lowest number of boxes, was selected to illustrate the results of fractal analysis for scale versions 1 (Fig. 6a)and 2 (Fig. 6b) . 

The results of this study showed high levels of linear association between grading scale reference levels and physical attributes (Table 3 ; range of 0.88 ≤ r ≤ 1.0) for all grading scales. A Pearson correlation of 1.0 represents a perfectly linear association between scale grades and physical attributes; grading scales that exhibit this feature may be characterized at least as interval. For the purposes of this study, we decided to define the accuracy of a grading scale by the level of linear association it exhibited between scale steps and associated physical attribute (Pearson’s r, Table 3 ). Thus, the most accurate grading scale would be the scale for which this correlation was the highest, and the least accurate scale the one with the lowest Pearson correlation level. 

In our study, a Pearson correlation of r = 1.0 between scale reference grades and one physical attribute was found for each grading scale except for the MC-D scale. However, each physical attribute extracted from the images pointed to a different scale as being most accurate. If the accuracy of a scale were defined only on the amount of vessel coverage across the scale range, the IER scale would be the most accurate (Table 3 , % PC). Based on fractal dimension (Table 3 , all types of D), the Efron and the VBR scales were more accurate. With the third physical attribute of redness in our study, photometric chromaticity, the VBR scale showed the highest linear association with the reference grades and therefore might be described as the most accurate scale. 

The high accuracy of the VBR scale with respect to chromaticity is not surprising, since this scale was developed based on a combination of subjective estimates and photometric chromaticity measures. ⁶ The reference images for the Efron scale were painted, perhaps focusing on highlighting certain features and simultaneously avoiding confounding artifacts. ³⁹ Our study supports this intention, as we determined consistently strong associations with degree of vascular branching and the amount of vessel coverage, whereas the linear association between photometric chromaticity changes and scale steps was lowest of all scales. 

These results show that estimating the accuracy of a grading scale is closely related to the technique and the physical attributes used. Overall, fractal dimension, % PC, and photometric chromaticity all were capable of detecting changes in the severity of redness. The high linear associations between scale steps and physical attributes indicate that the anticipated change in the scales could be determined with the physical attributes used, and that all scales thus may be considered accurate. Superior or inferior accuracy of a scale, however, should be defined solely by indicating the physical attribute that was used. Based on our results, it seems that each scale describes one characteristic of redness best: The VBR and MC-D scales best describe redness in terms of photometric chromaticity, the Efron scale with respect to changes of vascular branching (D), and the IER scale with respect to changes in the area that is covered (% PC). The consistently high correlation levels for the VBR scale (all r ≥ 0.97), however, suggest that this scale is the least affected by the selection of the physical attributes or the preprocessing procedure. 

Comparison and Cross-Calibration between the Grading Scales

Various studies have suggested that grading scales should not be used interchangeably. ^{8 12 13 14} However, the physical attributes of each grading scale image could be used in an attempt to cross-calibrate the grading scales. An illustration of this is Figure 6 , in which physical attributes of each image are plotted against their associated nominal scale grades. 

Figure 6shows that there are large differences in the physical attributes of each scale for equivalent grades highlighting that cross-calibration across all grades is impossible. The differences in size and resolution (among many others) between the scale images complicate the selection of the physical measure to cross-calibrate the scales. 

Equivalency between grading scales and the physical attributes of the conjunctival images occurs at points where the scales coincide. For fractal dimension, there was little convergence of the scales; indeed, at step 1 of the scales, the fractal dimension ranged between 0.38 and 1.35. % PC also showed little convergence across scales as evidenced in Figure 6 , containing graphed data for scale version 1. A contributory factor to the large spread of values may be the different sizes of the ROIs. In this study, fractal dimensions appeared to be affected by the size of the ROI (the larger the ROI, the higher the number of box counts and the fractal dimension) and because each scale’s images have inherently different sizes, ability to cross-calibrate scales is confounded. This problem is ameliorated by using boxes that have the same size, as evidenced in Figures 6b and 6d(scale version 2). The regions where there is greatest convergence are approximately at a grade of 2.2. 

Figure 6eshows that three intersections between scales were found when chromaticity was used. Chromaticity was measured on the original, unaltered images when displayed on a computer screen under identical illumination and monitor settings. Except for adjusting magnification levels on the screen to account for the differences in size and resolution of the original images (Table 1) , no further alterations of the images were required. Because of the least observer intervention involved in this procedure, the selection of photometric chromaticity seems to be the most logical solution for cross-calibrating the scales. As can be seen in Figure 6e , however, there was a wide range of scale grades for the same photometric chromaticity measure. As an example, the image representing grade 4 in the IER scale corresponds to a u′ of 0.24. For the other scales, this chromaticity would correspond to interpolated grades of approximately 1.3 (VBR), 2.3 (MC-D), and 3.4 (Efron), showing that the same level of photometric chromaticity represents a range of approximately three scale steps for all scales. It seems that chromaticity is very different between the scales, and that, except for the VBR scale, chromatic changes between scale steps are not linear across the full-scale range. 

An unfortunate conclusion that we are inevitably left with is that, despite the generally strong linear associations between the physical characteristics of reference images in each scale, the scales themselves are not inherently accurate. What allows the switching between scales by clinicians is the huge nonlinear compensations that are made by those using the various scales: We propose that this compensation is accomplished by a novel mechanism that we refer to as clinical scale constancy and is similar to perceptual mechanisms such as color constancy, ⁴⁰ which allow relatively invariant perceptions despite differences in physical attributes of the image. This effect is exactly what occurred in our study: The images are physically different but, for example, grade 0 is perceived to be low (not red) regardless of the scale used, similar to a dark object appearing the darkest of its surroundings, regardless of strong or weak illumination. ⁴⁰ This observation of our ability to rescale clinical attributes despite their physical content has profound implications for developing measuring tools and suggests that current measurement theories ^{41 42} are inadequate because certain measurement tenets are less important and soft/weak metrology ^{43 44} should take these looser constraints into account when defining what constitutes measurement. In addition, teaching clinical judgments is also influenced by understanding that under certain conditions we are able to ignore absolute physical attributes while using relative (within scale) characteristics to reach appropriate clinical conclusions about redness (and presumably other aspects of ocular appearance). 

Conclusion

The first goal of this study was to determine the accuracy of bulbar redness grading scales based on their correlation levels between scale grades and three physical measures (D, % PC, and photometric chromaticity). Based on our results, all scales can be considered accurate according to the criteria we specified; however, it seems that each scale describes one characteristic of redness best: the VBR and MC-D scales best describe redness in terms of photometric chromaticity, the Efron scale with respect to changes of vascular branching (D), and the IER scale with respect to changes in the area that is covered (% PC). 

The second intent of this series of analyses was to cross-calibrate the scales so that they could be compared via the physical measures. We have shown that an objective cross-calibration between scales would be technically possible, but as is apparent because of the wide discrepancies and the relatively low overlap, it cannot be recommended. Differences between acquisition methods, image quality, and physically obtained measures show that the differences between grading scales are too severe to allow for cross-calibration. Among many things, this highlights the need for standardization, inasmuch as scales are proposed with few physical similarities, bringing into question whether the numbers that represent the steps on the scale (and therefore the numbers derived using the scales) are measurements at all. 

 Supported by a Canada Foundation for Innovation (CFI) Equipment Grant.

 Submitted for publication October 9, 2007; revised December 4, 2007; accepted February 28, 2008.

 Disclosure: M. M. Schulze, None; N. Hutchings, None; T. L. Simpson, None

 The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked “advertisement” in accordance with 18 U.S.C. §1734 solely to indicate this fact.

 

Figure 1.

 

Simulated fractal dimensions (D) representing different degrees of vascular branching on the conjunctiva.

Figure 2.

 

The bulbar redness grading scales analyzed. ^{3 4 5 6}

Table 1.

 

Original File Types for the Grading Scale Reference Images and Their Associated Image Size and Resolution

	Original File type	Size (px × px)	Resolution (dpi)
VBR	TIFF	1524 × 1012	300
IER	TIFF	700 × 525	100
Efron	JPEG	1628 × 1399	72
MC-D	TIFF	372 × 271	100

Table 2.

 

Grayscale Brightness Values, Standard Deviation and Signal-to-Noise Ratio for Each Grading Scale and Each 8-bit RGB Component

	Red							Green							Blue
		s n	a max	a min	SNR	s n	a max	a min	SNR	s n	a max	a min	SNR
VBR	12	255	190	14.7	3	247	114	32.9	4	242	102	30.9
IER	13	255	117	20.5	2	251	103	37.4	3	253	76	35.4
Efron	6	255	147	25.1	1	255	20	47.4	2	255	45	40.4
MC-D	24	255	121	14.9	7	242	40	29.2	10	210	42	24.5

Figure 3.

 

Image preprocessing steps used for generating scale versions 1 (a, top) and 2 (b, bottom) are illustrated by using the grade 50 reference image of the VBR scale. (i) Original image; (ii) defined ROI; (a iii) noise reduction and background subtraction; (a iv) Sobel edge detection; (a v) specification of multiple segments; (a vi) segment binarization; (b iii) matched resolution and size for ROI; (b iv) noise reduction and background subtraction; and (b v) ROI binarization.

Figure 4.

 

Standardized photometric setup. The spectrophotometer was placed on a tripod 30 cm away from the computer screen.

Figure 5.

 

Images resulting from fractal analysis using scale versions 1 (a, top) and 2 (b, bottom). For scale version 1, a single segment of the ROI is illustrated across the reference levels for each scale. For scale version 2, the whole ROI (fixed size and resolution) is illustrated across the reference levels for each scale. The ROI in the Efron scale is rotated by 90° counterclockwise, to match the orientation of the other scale references. Within a scale, each image represents a scale step. The order of the scales is presented to be consistent with Figure 2 .

Table 3.

 

Pearson Correlation Coefficients between Scale Grades and Their Associated Physical Attributes

	VBR			IER			Efron			MC-D
		1	2	1	2	1	2	1	2
D̄	0.97	0.98	0.93	0.96	0.99	1.00	0.88	0.95
D sc	0.97	0.98	0.93	0.96	0.99	1.00	0.88	0.95
D e	0.98	0.98	0.93	0.95	0.99	0.99	0.88	0.94
D sce	0.97	0.98	0.92	0.94	0.99	0.99	0.88	0.95
% PC	0.98	0.98	0.97	1.00	0.99	0.97	0.95	0.97
u′	0.99			0.95			0.94			0.97
u*	1.00			0.96			0.96			0.98

 1 and 2 represent the scale version.

Figure 6.

 

Graphs showing the relationships between scale grades and D _sce for scale versions 1 (a) and 2 (b), between scale grades and % PC for scale versions 1 (c) and 2 (d), and between scale grades and CIE chromaticity, u′ (e).

The authors thank Charles McMonnies, Nathan Efron, and the International Association of Contact Lens Educators (IACLE) for providing high-resolution copies of the original reference images.