4.2 The Effect of Image Display Size upon Delineation Accuracy

4.2.1 Objectives

This experiment aims to determine the following:

It is desired to express the results concerning both of these objectives in terms of percentages of wound area in order to aid appreciation of the sizes of errors concerned. In accordance with the stated objectives, the data collected throughout this experiment will be used in the fitting of two regression models, one for bias and the other for variation, both of them expressed as functions of display scale.

 

4.2.2 Development of Regression Model for Bias

The proposed model for bias is developed here with reference to a practical delineation error model which is shown in Figure 4.6(a). This model seeks to express the bias error as a consequence of the discretisation of the image plane into finite size pixels. In Figure 4.6(a), delineation of the wound boundary is shown as a finite-thickness line centred at the wound edge. Here, the drawing implement is modelled as a mouse pointer drawing a line of 1 pixel thickness. It is the thickness of this line (and its length, i.e. the perimeter-length of the shape) in relation to the actual wound size that governs the bias error, the error being the area enclosed between the outer edge of the thickened delineating line and the centreline. Thus it is possible to intuitively write an equation expressing this notion and attempt to validate it. A first approximation to a working model for bias is given by (4.2.1).

(a)

(b)

Figure 4.6 (a) Production of biased area measurements with thickened delineating line; (b) Biased area function as a result of delineation from (a)

(4.2.1)

where B0=P0d is the bias coefficient

P0 is the perimeter-length of the boundary

d is the delineation translation error constant

l is the linear magnification factor.

 

This equation has two components: The first is a proportional term in l2 having slope A0 which equates to the true area of the wound given the area magnification factor, l2. The second is a component proportional to l which equates to the absolute size of the bias error, and is a product of perimeter length P0 and a constant offset, d. In relation to the scaled true-area component, the bias component diminishes with increasing scale, and hence in general the curve will tend to appear quite linear, since one expects A0 >> B0. To aid appreciation of the nature of the bias component it is possible to express – at any scale – bias as a fraction of the scaled area, by dividing both sides of (4.2.1) either by the area component, l2A0, of (4.2.1) or simply by l2. Equation (4.2.2) is the result of dividing through by l2 which yields the expected prototype plot shown in Figure 4.6(b) – a constant term plus inverse-square root term decaying from a highly biased position at low levels of the predictor variable, l2. As the predictor variable increases towards infinity, the response curve asymptotically approaches the horizontal line drawn at the true value of the variable being measured. This plot is included here to aid appreciation of the expected nature of the bias function and of the coefficients A0 and B0. The experimental results for the bias analysis will be shown in a form similar to Figure 4.6(b).

(4.2.2)

Although (4.2.1) is not linear in l, its normalised form (4.2.2) may be considered linear in a variable 1/l. This form of presentation would show the fractional bias as a function of scale. Applying the linearised model to the area measurement data yields residuals which are not particularly normal, have increasing variance and skewed spacing of the predictor levels. In contrast, it has been found that working with a basic datum related to the square root of the area produces residuals which better satisfy the assumptions of normality and constant variance. As part of the validation of (4.2.1), consider the bias in area measurements that would arise from delineating scaled versions of a circle having radius r0 with a constant offset d. This is visualised in Figure 4.6(a) and the effect is modelled by (4.2.3):

(4.2.3)

Using the standard formula Acircle=pr2 and substituting (4.2.3) for r yields:

(4.2.4)

Identifying the individual terms, the equation may be expressed as:

(4.2.5)

where area A0=pr02 and circumference C0=2pr0=P0.

 

Note the similarity between the expressions for delineated area in (4.2.1) and (4.2.5). Equation (4.2.5) has an additional term d 2 which is negligible when d << r.

Equation (4.2.5) is an exact relationship for the delineated area when the object is a circle and therefore the intuitive model of (4.2.1) is inexact for a circle because of the omission of the constant d 2, which is small, however, when d << r, which is generally the case. The final step in determining a bias model expressed in radial terms is to generalise (4.2.3) to allow it to apply to non-circular objects. Equation (4.2.6) defines the ‘average radius’ r of a general shape having area A:

(4.2.6)

The ‘average radius’ r is thus, equivalently, the radius of a circle having area equal to A.

For generalised shapes, (4.2.7) defines the scale-invariant shape factor F, for a simply connected object [Wahl, 1987], where P is the perimeter length of the object and A its area. When the shape is a circle, F has a value of unity, for non-circular shapes F > 1.

(4.2.7)

thus

If C is the circumference of a circle with area A, and P is the perimeter-length of a general shape also with area A and F is its scale-independent shape-factor, then equating the expressions for area yields:

(4.2.8)

Thus the perimeter of the shape, which partly governs the area bias, is given by:

(4.2.9)

Substituting for Pshape from (4.2.9) into (4.2.1) yields:

(4.2.10)

Where the factor Fshape provides the enlargement of the bias term above that expected from a circle of equal area by virtue of the shape’s elongated perimeter length in relation to its area. Applying this notion to the radial model of (4.2.3) and thus generalising it to apply to non-circular shapes gives a linear equation in l:

(4.2.11)

Thus the simple linear regression model for noisy observations of r(l) becomes:

(4.2.12)

where

ei ~ NID(0,s 2)

For each set of results, this equation may be used to obtain estimates b0 and b1 respectively of the regression parameters b0 and b1. From (4.2.11) the following identities may be established:

(4.2.13)

from

b0=F

where

L=Ö F

(4.2.14)

from

b1=r0

From these two identities one can obtain the following (biased*) estimates:

(4.2.15)

the true (unbiased) area

(4.2.16)

the delineation translation error constant.

* The use of the term ‘biased’ here refers to the fact that a square-transformation of an unbiased estimate introduces a small bias into the estimate of the transformed variable by virtue of its ‘skewing’ effect upon the distribution of the transformed variable. However, it is the case that when the value of the estimate is much greater than its standard error, the magnitude of the skew becomes negligible, as in this case. Using the expression for average radius, r(l) from (4.2.11), the equivalent expression for delineated area becomes:

(4.2.17)

where A0=pr02 and B0=2pr0Ld

pFd2 is the error introduced by the approximation.

Hence, the fractional area bias error may be expressed as:

(4.2.18)

In terms of the estimated regression parameters, (4.2.18) becomes:

(4.2.19)

The standard errors of the regression parameters, s{b0} and s{b1}, are obtained from the regression analysis and as such are automatically produced by standard statistical packages — see Neter et al. (1996) for the requisite formulae. In the present case the parameters of interest are not b0 and b1 directly, but rather functions of them, viz.: equations (4.2.15), (4.2.16) and (4.2.19). When functions of random variables are non-linear in form, the standard error of the function result cannot be exactly expressed in terms of the standard errors of its arguments. Generally, the standard error, S, of any combination of independent estimates (m1, m2,..., mN) with respective standard errors (S1, S2,..., SN) may be first-order approximated by:

(4.2.20)

In each specific applied case of the general function, f, from (4.2.20), the appropriateness of the approximation should be considered. Applying (4.2.20) to the estimation of unbiased area A0 (4.2.15) the standard error is approximated by:

(4.2.21)

provided

s{b1}<<b1

Although b0 and b1 are not completely independent, the dependence should be negligible when the total sample size of the regression data is large (>50). Thus the standard error of the estimate of BF from (4.2.19) should yield:

(4.2.22)

provided

s{b1}<<b1

Finally, (4.2.19) may be expressed as a function of the magnified unbiased area, so that the direct relationship between fractional bias, BF, and A0 may be realised:

(4.2.23)

 

4.2.3 Development of Regression Model for Variance

The proposed model for variation of area measurements as a function of the area magnification factor, a, is

(4.2.24)

where

a =l2

k is a coefficient taking into account the sum variation due to the particular wound image, delineator and equipment used.

Equation (4.2.24) may be linearised by taking a logarithmic transform of both sides, which allows the use of the simple linear regression model. Applying this transformation to (4.2.24) yields:

(4.2.25)

where b0=log(k) and b1=n

It should be noted that under the logarithmic transformation of (4.2.25) the expected response curves for variance and standard deviation are linearly related, so that:

(4.2.26)

The fractional precision is given by the standard deviation of area measurements expressed as a fraction of the expected scaled area at scale a . Thus using the hypothetical relationship between s 2 and a from (4.2.24), one obtains:

(4.2.27)

If magnification of an image has no effect upon the fractional precision, PF, then in reality (4.2.27) must be a constant expression, independent of a. This requires that the parameter n=2, signifying that the standard deviation of area measurements is proportional to the area. In terms of the regression analysis this translates to a null hypothesis that b1=2. This hypothesis is considered supported when the 95% confidence interval for b1 overlaps 2. If the fractional precision improves with increasing magnification of the image, then one expects b1<2.

Taking logarithms of both sides of (4.2.27) yields:

(4.2.28)

where b0*=˝log(k)-log(A0) and b1*= n/2-1

Comparing (4.2.26) with (4.2.28) it is clear that they are linearly related, so that:

(4.2.29)

(4.2.30)

These are the model parameters expressing the fractional precision of area measurements as a function of the area-magnification parameter a. Since A0 is the only unknown variable in this transformation, the effect of the uncertainty introduced into the fitted precision model by using an estimate of A0 is to add a small amount of variability into the intercept parameter (i.e. the logarithm of k). The slope parameter (exponent) is unaffected by such additional error.

When precision is expressed as a direct function of the scaled area of a wound (aA0), a meaningful comparison can be made between the precision curves estimated for different wounds produced by different delineators. This is possible because the common factor for comparison of delineation performance is the actual display size of the wound.

 

4.2.4 Experimental Set-up and Procedure

Two volunteer delineators were asked to use a standard computer mouse to make repeated delineation of three different wounds, with each wound displayed at seven different sizes. The delineation software produced for the first experiment (§4.1) was used to perform the delineation task. The set of linear scale ratios was chosen to be L =(1.00, 1.33, 1.66, 2.00, 2.33, 2.66, 3.00) and these values were used to scale both the horizontal and vertical dimensions of each of the three wound images. Thus for each wound set the ratio of the largest area to the smallest area is 9:1, since the area scale is the square of the linear scale. The largest image (area scale 9) in each set was scaled from its original digitised image so that its dimensions fitted a standard 800´ 600 pixel display. The smaller scaled versions were then produced with appropriate sizing relative to the largest version.

Three different wound images were selected from a library of wound images. The first image was image 1 from the first delineation experiment and was selected because it has a relatively well-defined boundary, is fairly homogenous and does not lead to large differences of opinion concerning the wound’s area (cf. §4.1.4.3). The second image, a ‘phantom’ wound modelled on the first image, was created using a mask defining the wound (cf. §4.2.4.1). The third wound image, image 6 from the first experiment, was selected because the wound boundary was subjectively less well defined than the first image. The reason for including such an image is to identify whether image quality/clarity affects bias and precision.

Both delineators were asked to delineate all seven versions of the three images once per day for ten days and record the result as a bitmapped image of edge pixels, as in the first experiment. The area of each delineation was thus calculated by counting pixels both inside and on the pixelated boundary.

 

4.2.4.1 Creation of Phantom Wound Image

To create the phantom image a wound template was first obtained from the 25 delineations produced for image 1 from the first delineation experiment. The mask of the median boundary of the associated 25 wound masks was produced and contained 8779 pixels. Secondly, samples of the colour pixels from image 1 were taken both from inside and outside of the wound in order to more realistically colour the phantom image. Finally, the coloured phantom image (size: 2562 pixels) was blurred by application of an isometric Gaussian smoothing filter with s =1.5 pixels and the smoothed image was used to create the 7 scaled versions required for the experiment by bilinear interpolation. The mask image was also scaled to coincide with the smallest phantom image (a =1) and contained 5358 wound pixels. This is the true value of A0 for this image.

The notion of using the phantom wound image is that it should enable the measurement of delineator performance on a wound shape without the influence of ambiguous boundary segments.

 

4.2.5 Bias Model Results

The average radius data ri was calculated from the measured area data with (4.2.6). Using the regression model of (4.2.12), the average radius data for each set of measurements was regressed upon the linear scale variable l to provide parameter estimates b0 and b1. From these regression parameters, the values of A0 and BF0 were calculated using (4.2.15) and (4.2.19) respectively. The mean value of L for each delineator sample was calculated from the delineated images and used to estimate d. Table 4.5 contains the estimates of A0 and BF0 together with the estimates of L and d. The area-related parameters A0 and BF0 are used to plot the fractional bias against unbiased magnified area as shown in Figure 4.7(a)-(d) for all sets of results. The relationship between fractional bias and unbiased area used to plot these graphs is given by equation (4.2.19).

 

Image

Delineator

A0± s{A0}

BF0±s{BF0}

s{L}

(95% CI) d

1

1

5230, 27

0.062, 0.011

1.060, 00.001

00.77, 1.60

2

5198, 25

0.054, 0.010

1.068, 00.001

00.64, 1.42

2

1

5346, 15

0.012, 0.006

1.052, <0.001

00.25, 0.70

2

5351, 13

0.051, 0.005

1.054, 00.001

00.80, 1.20

3

1

6179, 36

0.015, 0.012

1.116, 00.001

-0.19, 0.78

2

5548, 57

0.036, 0.022

1.128, 00.002

-0.13, 1.47

Table 4.5 Parameter estimates and associated standard errors for bias analysis

 

Figure 4.8 plots 95% confidence intervals of the estimated area constant A0 for images 1 and 2. Figure 4.9 plots 95% confidence intervals of fractional bias constant BF0 for all three images. The area estimates from both delineators for images 1 and 2, which are essentially of the same wound, are combined produce the following higher-confidence estimates of A0:

 

Area of image 1 :

(5230 + 5198)/2 

= 5214 ± 18 pixels

Area of image 2 :

(5346 + 5351)/2 

= 5349 ± 10 pixels

Difference

 

=  135 ± 21 pixels

 

(a) Delineator 1

(b) Delineator 2

(c) Delineator 1

(d) Delineator 2

Figure 4.7 Transformed fractional bias plots showing bias as a function of unbiased magnified area for (a) delineator 1 and images 1 and 2, (b) delineator 2 and images 1 and 2, (c) delineator 1 and image 3, and (d) delineator 2 and image 3.

 

Figure 4.8 Confidence intervals for estimates of unbiased wound area A0 for images 1 and 2.

Figure 4.9 Confidence intervals and estimates of fractional bias constants BF0 for all three images.

 

4.2.6 Variance Model Results

For each wound/delineator the variance estimate of the area measurements at each level of scale was calculated. Using (4.2.23) the logarithm of the estimated area variance at each level of area scale was regressed against the logarithm of the area-scaling factor. The estimated regression parameters b0 and b1 are shown in Table 4.6 with estimated standard errors. The fitted linear regression curves obtained with these estimates are plotted in Figure 4.10(a,c,e). Each graph shows a comparison between the results obtained from the data produced by the two delineators. The linear equations were transformed by (4.2.24) to produce fractional variance results. Figure 4.10 (b,d,f) shows precision plotted as a function of wound area for each of the three images. Again, each graph compares the curves produced from the two delineators’ results. The b1 linear regression parameter determines the specific form of the variance/scale relationship. The estimates of b1 obtained from the results of both delineators’ measurements for each image are plotted in Figure 4.11 along with the appropriate 95% confidence intervals.

Image

Delineator

b0=log k

b1=n

Ök

Ök/A0

1

1

3.86, 0.11

1.79, 0.18

85

0.016

2

4.36, 0.17

0.87, 0.27

151

0.029

2

1

3.42, 0.08

1.63, 0.12

51

0.010

2

3.67, 0.13

1.47, 0.20

68

0.013

3

1

4.29, 0.17

1.46, 0.27

140

0.023

2

4.27, 0.18

1.96, 0.29

136

0.025

Table 4.6 Parameter Estimates and Transformed Coefficients for Variance Regression Analysis

 

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4.10 (a, c, e) Plotted regression of area variance on area magnification factor a for wound images (1,2,3) respectively (equation 4.2.25). Each graph compares the regression lines produced from the results of both delineators, (b, d, f) plot the corresponding transformed functions (equation 4.2.27) for images (1,2,3).

Figure 4.11 Confidence intervals for area variance power constant for each image and for both delineators

 

To make a comparison between the fractional precision levels for all three images, the estimated area constant A0 is used to determine two levels of the predictor variable a required in each bias function plotted in Figure 4.7 that sets the unbiased area at (a) 10,000 pixels and (b) 40,000 pixels. This involves using the regression function to predict the value of the response variable log(s2) at two new a levels and transform them into approximate predictions of fractional precision. The results of the predictions for fractional precision of area measurements at the two specified sizes of wound area are shown in Table 4.7.

 

Image

Delineator

A0

PF (%)

A(a)=10000 pixels

PF (%)

A(a)=40000 pixels

1

1

5214

1.5%

1.3%

2

5214

2.0%

0.9%

2

1

5349

0.9%

0.7%

2

5349

1.1%

0.8%

3

1

6179

2.0%

1.4%

2

5548

2.4%

2.4%

Table 4.7 Fractional precision estimates at specific areas

 

4.2.7 Discussion

A. Bias Analysis

Area Constant, A0

The area constant, A0, for the first two images, 1 and its phantom, 2, are all estimated with ±1% accuracy. Also, the estimates of A0 for each image are comparable between the two delineators so that the average of the two areas for each wound provide a high-confidence estimate of A0, respectively 5214±18 pixels and 5349±10 pixels which is subsequently required for the variance analysis. The difference between these values of 136±21 pixels is not expected and efforts to trace the source of this disparity have failed. The mask area (5358 pixels) was subsequently found to be larger than the scaled version of the median boundary mask by a similar amount, so the disparity may be explained as a lapse of procedure.

The estimates of area constant A0 for image 2 (phantom) from both delineators were consistent with the scaled true value of the mask used to generate the phantom. This is a practical validation of the model for bias at least when using a phantom image.

Estimation of translation constant, d

The column for d in Table 4.5 records the fact that four of the six estimates of the translation constant produced results about twice as large as expected, given the prior discussion in §4.2.2. The two estimates which support the default value of d =˝ are obtained from image 3 and have the widest confidence intervals, suggesting that since they are in agreement to some extent with the first four estimates of d that there is no contradiction in concluding that d >˝. Since this value is greater than originally supposed, it does not invalidate the notion of the ˝ pixel width translation being a factor in producing bias, but it is necessary to accept that the explanation is incomplete. Thus, it is proposed that an extra translation component exists that increases the bias. One hypothesis for producing such an increased shift is to consider the influence of the display monitor’s g -factor upon the apparent positioning of the edge of the wound. This would certainly seem to shift the supposed centre of an edge whose contrast extends over several pixel-widths. However under the model of (4.2.1) the constant serves to affect only the bias (l) term. In other words, a translation effect that is proportional to the width of the edge will serve to magnify the area – affecting the l2 term in (4.2.1)) – and will not affect the bias term. The enlargement of the translation constant could be explained if human visual perception somehow normalises the apparent width of an edge and subsequently assesses the edge of the wound as being offset by the displacement due to the monitor’s g -factor. To be sure, it would be necessary to conduct a further study attempting to measure this effect uniquely. Regardless of this however, the magnitude of the bias effect upon area measurements has been successfully measured.

With reference to the bias curves shown in Figure 4.7, it can be seen that both delineators produce comparable results for bias when delineating image 1, with the bias contributing about 6% (i.e. reading 0.06 from the graph) to the area measurement at the lowest scale tested and 2% of the area measured at the highest scale. Comparing the bias results for images 1 and 2 clearly shows two things:

(1) Delineator 1 recorded a reduction in bias when delineating the phantom image (2), with the level of bias introduced being roughly half that of the bias from image 1, while delineator 2 recorded no such improvement.

(2) There is great similarity between delineator 2’s curves for images 1 and 2 and delineator 1’s curve for image 1, suggesting that the bias level is consistent between the two delineators, except for delineator 1’s attempt on image 1 which is much more consistent with delineator 1’s attempt on image 3.

 

B. Precision Analysis

The regression curves for variance in Figure 4.10(a) and (b) for image 1 show that delineator 1 is more precise at lower scales than delineator 2 but becomes less precise at higher scales. This comparison between the two delineators is not consistent across the three images and the disparity in performance between the delineators on images 2 and 3 is much less than that on image 1. The entries for the exponent parameter n (delineator 2) in Table 4.2 (also displayed in Figure 4.11) show that the value of 0.87 is the lowest recorded. However, the size of the confidence intervals for all six measurements of n imply that there may well be no difference of any practical size over both delineators and the three images, with a mean estimated value of 1.53±0.09.

The range of the area magnification factor l2 translates to an approximate range of 5000 pixels to 50000 pixels for the areas of the wounds measured. This translates again, approximately, to an actual display size of 5cm2 to 50cm2 (calculated from a 15" monitor, 800 ´ 600 pixel mode).

One factor that could affect variability is the time taken to delineate a wound. Thus it is possible to conjecture, for instance, that if one delineator takes roughly the same amount of time to delineate a wound, regardless of its size, then one could expect a steeper precision curve in comparison with a delineator who applied equal diligence to the delineation process - hence requiring more time to delineate the longer boundaries of larger wounds - regardless of scale.

The results for image 3 are less precise than the results for images 1 and 2 (the better-defined images). The precision curve for delineator 2 (Figure 4.10(f)) shows no predictable change in precision over the experimental range, with the value of n @ 2. This delineator delineated a different boundary to delineator 1, choosing a rather difficult to delineate boundary section, which could account for unpredictable results in terms of variances.

 

4.2.8 Conclusions

 

 

[Ü Chapter 4 : Part 1]


Title Page

4 Manual Delineation

Appendix A

Contents

5 Parameter Setting

Appendix B

1 Introduction

6 Results

Appendix C

2 Literature Survey

7 Discussion

Colour Plates

3 Algorithm Development

8 Conclusion

References