5

Parameter Setting Experiments


 

5.1 Introduction and Preliminary Notes

The two finite element active contour models described in Chapter 3 – the Ground-Attract (GA) and Tangent-Normal (TN) algorithms – each have two internal parameters which govern the weights of the elastic and stiffness energy terms of the active contour model. The levels at which these parameters are set will affect each algorithm’s performance. Due to the differences in the formulation of the internal energy terms for the two algorithms, the behaviour of the contours will not be the same but will share some similar characteristics. The energy component weights of the two minimax algorithms (MX and MG) are implicitly set and so require no such concerns. However, in addition to the setting of internal parameters, the image itself is subject to regularisation (smoothing or low-pass filtering) to encourage uniqueness of the boundary solution under differing starting contours. Regularisation of the image (external energy function) is a necessary component of all four algorithms previously described and it is necessary to determine, for each algorithm, the optimal filter scale parameters for the scale descent algorithm described in §3.1. The purpose of the experiments described in this chapter is to determine the effects of varying these parameters upon the area enclosed by the contour over a suitably wide range, given that the objective of the application of these algorithms is ultimately to reliably and accurately measure the wound area. This chapter considers the effect of the following variables that have the ability to affect the final solution of the active contour models:

All three variables have the potential to influence the bias and precision of the measured area of a wound. It should be considered that each of the variables is either essentially a random variable (initial position – see below) or an unknown parameter which must be set at some appropriate level in order to produce desirable results. Certain parameters are left unaltered through the course of these experiments, viz.: number of contour elements (set at 32) and the step size reduction scheme (unit time or space steps, reducing in single octave steps until converged at 1/32).

 

5.1.1 A Note on Varying the Initial Contour Position

A spatial discretisation of the generalised parametric contour v(s) into a piecewise continuous form allows one to specify changes to the contour in a finite number of degrees of freedom. Clearly, in each image there is a region in the vicinity of the wound boundary within which a suitably regularised contour should converge to an approximation of the boundary. This corresponds to an ‘energy well’ surrounding a local minimum in the active contour model’s energy functional. Ambiguous contours will affect the consistency of such a solution. However, there will be possible initialisations of the contour where the contour covers part of the image that contains edges other than those belonging to the wound which will lead to a boundary solution which does not uniquely apply to one object in the image. Defining this domain for any image would be a difficult task. The domain would be unique to each image and would be affected by the regularisation parameters. Rather than attempting to generate arbitrary initial contours that may produce hybrid solutions, the approach taken here is to use a set of manual delineations to initialise the active contour model. These delineations should not place the contour over a boundary segment other than those which belong to the wound, although ambiguity of the exact wound boundary will clearly lead to initialisations in different energy wells thus implying differing results.

 

5.1.2 Calculation of Contour Area

In the following experiments the basic datum is the area enclosed by a piece-wise parametric contour. All algorithms use the cubic B-Spline basis functions to represent the contour. For a closed contour composed of N such elements, the enclosed area may be calculated by:

(5.1)

The functions and derivative functions can be easily computed given the cubic coefficients defining each element, which may be calculated from the control vertices (see Bartels et al., 1987). The integral in (5.1) is of a quintic equation (arising here from a cubic function multiplied by a quadratic) which, given the curve coefficients, may be calculated exactly by a Newton-Cotes closed formula of degree five requiring five equally spaced abscisses ("Bode’s rule", [Press et al., 1992]).

Note that areas calculated by (5.1) do not suffer from the biasing implicit in calculating areas from counting a finite number of pixels (see §4.2.2).

 

5.1.3 Calculation of Area Precision

It is considered more informative to present and discuss results when expressed in the form of fractional precision rather than variance. This is because variance gives no information of the relative size of the variation of area measurements about their mean value in comparison with the size of that mean value, i.e. it is assumed, within reasonable limits, that magnifying a wound image either by altering the imaging geometry or by resampling the captured image gives rise to a proportionate change in the spread of those measurements. Therefore, variation in area measurements is expressed in terms of fractional precision:

 

(5.2)

Note that this equation does not take account of the bias implicit in the estimator for standard deviation. In the following sections the estimates are based on samples of 25 measurements, so that the bias constant for standard deviation estimates is 0.99.

 

5.2 Contour Regularisation Parameters

This section describes the effect of varying the internal energy weights of the GA and TN algorithms upon the mean and variance of the area enclosed by the contour. The same 10 images used for the manual delineation experiments of Chapter 4 were used for this experiment and the 25 manually defined contours produced for each image were used to seed the runs with both algorithms. The image gradient was computed by applying the derivative-of-Gaussian gradient operator to the green-band component of each colour image and used to define the external potential energy given by equation (3.3). It is recognised that the samples of 25 delineations for the ten wounds are composites of biased sets of results (§4.1). Thus the manual precision figure calculated by applying (5.2) to each sample of 25 area measurements contains both intra-delineator variance and inter-delineator variance. Thus (5.2) is a measure of root-mean-square (rms) error. The alternative is to consider bias and precision separately and compare the bias and precision results of each algorithm with the manual biases and precisions. It will be seen in the next section that selecting regularisation parameters for the active contour models described in Chapter 3 is a somewhat complicated issue and the possible combinations of contour regularisation, image regularisation and the need for scale-descent necessitates that a division be made between the two types of regularisation. The approach taken is to first set the contour regularisation parameters to minimise variance (improve precision) and subsequently to study the effect of scale descent upon both bias and precision.

 

5.2.1 Range and Resolution of Parameter Levels

The combinations of the individual parameter settings used for this experiment, a, b and the filter scales used to generate gradient potentials at different levels of smoothing, are given by Table 5.1. The external energy weighting parameter was kept constant at 1.0. The choice of parameters tessellates the normalised regularisation parameter plane in a symmetrical pattern, where the normalised parameters may be defined by l1=a/(a+b+1), l2=b/(a+b+1) and l3=1-l1-l2.

Parameter

Range

Step Size

#Levels

Tension, a0

1/64..64

1 octave

13

Stiffness, b0

1/64..64

1 octave

13

Gradient Filter Scale, s

2..16

½ octave

7

Table 5.1 Specification of parameter settings

 

Both algorithms were tested on all possible combinations of the three parameters listed in Table 5.1. The analysis is mainly based upon the variance of area measurements, and hence measurement precision, for the following two reasons: (a) the algorithms are generally stable in the absence of external forces and thus are not expected to collapse if the contour is over-regularised, (b) the mean area is expected to vary with filter scale as discussed in §3.1. In the case of the TN algorithm, the stiffness term has a small component that causes the contour to collapse at very high levels of b. The scale descent procedure is intended to reduce the bias introduced into the area measurement whilst keeping the area variance under control.

The experiments are real world applications of the algorithms to wound images, which are very varied by their nature, thus it is expected that the results of parameter variation will differ from image to image. The results presented here are a representative summary of the general features of the total sum of results obtained.

 

5.2.2 GA Algorithm: Parameter-Varying Results

Running the algorithm at each level of scale listed in Table 5.1 and calculating the estimated area variance of the 25 contours at each particular combination of (a,b) yields a variance map for each level. Instead of observing the variances directly, a more instructive appraisal is obtained by considering the fractional precision of area measurements using the relationship between area variance and precision as given by equation (5.2). The representative features of the precision maps are noted and considered in the following discussion. All precision maps are plotted against the logarithm of the internal parameters, i.e. PF (5.2) is plotted as a function of (log2(a),log2(b)) as shown in Figure 5.1 on the next page.

 

A. Area Variance Reduced by Increased Image Regularisation

Figure 5.1 shows a representative example of this effect at four levels of image smoothing. As the image smoothing is increased, most images give rise to a variance map with a floor that is relatively smooth and flat over the lower parts of the range of each internal parameter. Only the two most poorly defined images did not exhibit this behaviour. At low scales this floor tends to rise and become less smooth, although at the outer-limits of this floor a small depression may be evident where the variance is optimally reduced by the action of the increasing contour regularisation as evidenced by Figure 5.1(c). The overall effect of variance reduction is due to the image smoothing removing noise and merging together local gradient contours.

(a)

(b)

(c)

(d)

Figure 5.1 (a)-(d) Precision maps at gradient filter scales s = 8, 5.7, 4, and 2.8 pixels respectively.

 

B. Variance Increased by Increased Contour Regularisation

The floor of each precision map in Figure 5.1(a)-(c) is surrounded by steeply sloping sides, where the area variance increases towards the level of the manually-defined area variance, as the internal parameters are increased. At all filter-scales tested, no images show a significant deviation from the manual results (i.e. mean and variance) when the internal regularisation is very high. The precision begins to rise when the internal parameters rise above a level which is image-dependent – on some images this is as low as 1 and on some as high as 16. This level also varies to a lesser extent with filter scale: at higher scales the image forces are weaker and this is manifested by the precision rising at lower levels of the internal parameters. At low filter scales this effect can break down (Figure 5.1(d)) and lead to a band of parameter levels where the variance actually increases above the area variance of the manually defined contours. In this case a few of the contours that define the smallest areas follow the same wound-boundary as the majority of delineations, but at some point deviate inside the wound and come under the influence of a different edge segment. As the internal regularisation is increased through intermediate levels of the internal parameters, the areas of these contours decrease (in contrast to the majority of contours) before increasing again at high levels of the parameters. This large bias in a few measurements causes the variance to increase.

 

C. Over-Regularising the Image Can Cause Variance to Increase

Figure 5.2 Increased variance at high filter scale and low contour regularisation

At the highest filter tested scales two effects become evident on some images that may be regarded as symptoms of over-regularising the image: (a) the floor of the precision map may begin to rise sharply at the very lowest levels of contour regularisation (see Figure 5.2) and (b) the smooth flat floor may start to rise and become increasingly rippled. In the former case the sharp rise in variance at the weakest levels of internal regularisation is caused by the edge of the wound merging with the edge of the limb or another nearby strongly contrasted object as a result of excessive image smoothing. This explanation also partly answers the case for (b). In addition to this the peak attractive forces of an edge are weakened, not only by suppression of high frequency components of the edge as a result of smoothing, but also by merging of all nearby gradients. In any image of a leg ulcer the background is rarely flat and without contrasted features. These weakened forces compete at different parts of the edge of the wound and attempt to attract parts of a nearby contour.

 

D. Variance Reduced by Increased Image Regularisation and Decreased Contour Regularisation

The floor of the precision map may sometimes begin to slope as the image regularisation is increased, with the precision gradually improving towards the zero-parameter point (a=b=0) so that the map becomes a continuous rise as the effect of edges merging due to the blurring. Figure 5.3 shows an example of this at two different levels of image regularisation. The effect occurs when epithelialisation tissue is present to a substantial degree at the edge of the wound. This tissue surrounds all or part of the wound and is contrasted both at its interface with the surrounding ‘healthy’ and with the wound and can be ambiguously interpreted by a human observer leading to varied initialisations. At higher levels of image smoothing the edges on both sides of the epithelialisation tissue begin to merge with both edges moving towards each other until one edge is created in the middle. This middle point, depending upon the amount of epithelial tissue, can represent a substantial displacement from initial contour positions on both of its sides. Thus as the internal parameters are weakened the contour continuously converge. This explains why the precision map may exhibit a continuous slope at high scales rather than a flattened floor.

(a)

(b)

Figure 5.3 Continuously decreasing variance at low contour regularisation with no floor (a) pronounced effect at high scale, (b) insignificant effect at low scale

 

5.2.3 TN Algorithm: Parameter-Varying Results

The experiments are repeated in an identical fashion for the TN algorithm using the same initial contours and ranges of parameters. The salient features of the many experimental trials are presented and discussed in this section.

 

A. Area Variance Decreased by Increased Image Regularisation

The intended effect of image smoothing is to converge multiple contour initialisations to the same position regardless of the algorithm. In contrast to the GA algorithm, however, high levels of internal regularisation do not constrain the final solution to be near to the initial solution. Thus the characteristic ‘floor surrounded by steep sides’ topology of the GA precision map is not shared by this algorithm. Figure 5.4 (a)-(d) shows that the algorithm is generally less sensitive to the setting of the internal parameters than the previous algorithm. In addition to this, the algorithm tends to produce lower levels of variance, being much more able to converge multiple initialisations towards a unique position. This may be observed by comparing the floor of the precision maps in Figure 5.4 to those of Figure 5.3.

(a)

(b)

(c)

(d)

Figure 5.4 Example area precision maps for TN algorithm at four levels of filter scale : (a) s = 11.3, (b) s = 8, (c) s = 5.7, (d) s = 4

 

B. Over-Regularising the Image Can Cause Variance to Increase

Similar to case ‘C’ for the GA algorithm (§5.2.2), the TN algorithm becomes sensitive to over-regularisation of the image, especially when the contour is only weakly regularised. Thus similar performance is obtained on images where the edge of the wound merges with a nearby non-wound edge, but only when the image is highly smoothed and contour is weakly smoothed. As stated above, this algorithm has considerable flexibility regarding setting the internal parameters and thus avoiding setting the parameter below a certain threshold is not likely to compromise performance.

 

C. Excessive Contour Stiffness Can Cause Partial Collapse

Recalling the discussion concerning the contractile effect of the bending forces, it is apparent that when the b parameter is set very high, the contour begins to detach itself from the wound outline and may then shrink to an elliptical shape. The shrinkage is caused by the fact that the bending energy is minimised when the contour is a point, similar to the case for the standard elastic energy. However, the high level of b also forces the contour to become rather rigid and the forces promoting this rigidity are much stronger in relation to their contractile component (the intended effect of the bending energy term is to impart rigidity to the contour – the shrinkage is more of a side effect). When the contour becomes too rigid, the external forces are overcome by the bending forces and the contour begins to detach itself from the edge of the wound. The effect of such over-regularisation upon the ability of the final contour to represent the wound boundary is dependent upon the shape of the wound. Figure 5.5 (a) and (b) show two example wound images overlaid with final contours obtained with a=1, b =64, and s =8. In Figure 5.5(a) the whole contour is partially detached (on the right-hand edge of the wound). If the wound is not particularly elliptical the area can shrink and the result can become a rigid ellipse held in equilibrium by two or three points of high external forces around the contour. An example of this is Figure 5.5(b) where the stiffness of the contour has forced it to detach from both left and right-hand ends and shrink. The effect of excessive contour stiffness on the precision of contour area measurements is observable in Figure 5.4: as b increases through the upper-most part of its range, the area precision shows a sharp increase at all levels of scale and at all levels of the elasticity parameter.

(a)

(b)

Figure 5.5 Examples of the effect of excessive contour stiffness: (a) Minimal effect on an elliptical-shaped wound, (b) shrinking effect on wound with points of high curvature.

 

5.2.4 Setting Contour Regularisation Parameters

In order to set explicitly the contour regularisation parameters it is necessary to specify a criterion function that when minimised yields an optimal set of parameters with respect to that criterion. A possible criterion function is to minimise the average precision over the set of ten wounds. This minimisation is then performed for the seven levels of scale parameter s defined in Table 5.1.

(5.3)

This series of calculations is performed for both algorithms and yields somewhat varying choices for different scales. Minimising the average precision over the set of wounds equates to selecting the minimum precision point from an averaged precision map. Instead of selecting explicitly the values a and b that completely minimise (5.3) it is possible to manually inspect the averaged precision maps for each algorithm at each scale. Selecting the absolute minimum from (5.3) is more sensitive to random error than would be the case if a multiple regression analysis were performed (Note: the complexity of the precision maps requires a moderately large number of parameters to model them). The averaged precision maps for the GA algorithm at all scales are topologically very similar, whilst the averaged precision maps for the TN algorithm display increasing levels of noise at low scales. Figure 5.6 shows two example averaged precision maps. In all the maps inspected it has been established that the parameter set a=0.4 and b=0.4 yields averaged precision values that are a close approximation of the absolute minimum values suggested by applying (5.3).

(a)

(b)

Figure 5.6 Example averaged precision maps: (a) GA and (b) TN algorithms at an intermediate level of scale (s =8)

 

5.2.5 Summary

 

Ground-Attract Algorithm

Varying both internal parameters appears to have the same effect upon the precision of area measurements, the variances forming smooth concentric contours spread over the parameter plane. The judicious initialisation of the contour near to the wound boundary represents input from a higher level process, which is maintained as part of the final solution with a weight determined by the contour regularisation parameters. This property is an explanation for the well-behaved precision surfaces in Figure 5.2.

 

Tangent-Normal Algorithm

This algorithm appears to be more able to reduce area variance than the previous algorithm, although it suffers from the similar problem of losing precision when the image is over-regularised. The flexibility of the contour under a wide range of parameter levels enables contour elements to deform by stretching and translating and, under the control of the stiffness parameter, bend to match any nearby contour. The GA algorithm constrains all three of these degrees of freedom and this may explain why its precision performance is generally poorer.

 

5.3 Scale-Descent Parameters

The previous section has appraised the general performance in terms of area precision of two algorithms as the contour and image parameters are varied. Precision is naturally improved by blurring the image and the algorithm is stabilised by a small amount of contour smoothing. In this section attention is turned to the subject of recovering the bias introduced by image regularisation, whilst preserving as much as possible of the gain in performance due to improved precision. The mechanism for this recovery is the scale descent algorithm of §3.1. Additionally, the two minimax algorithms, MX and MG, are considered in this section. The objective of this section is to determine a set of parameters for each algorithm that allows optimal performance. The definitions of the performance criteria are considered next.

 

5.3.1 Performance Measures

Experimental error may be expressed in terms of (a) a function of mean area and (b) a function of variance, corresponding to measures related to bias and precision of area measurements respectively. The view taken here is that the mean manually delineated area is the most representative single quantity defining the area of a wound. A similarity measure to compare the (related) distributions of manual area measurements and corresponding active contour area measurements needs to account for the effect of wound size upon bias and precision measurements. One such measure is simply the square error of the active contour model results with respect to the manual mean area measurement. This measure of comparative performance may be decomposed into a bias and a spread term thus:

(5.4)

where: ai are the active contour model area measurements;

 and  are the mean areas of the N measurements for a particular active contour model algorithm and manual delineations respectively;

and s2acm is the variance estimate of the algorithm’s area measurements.

This measure does not account for the uncertainty invested in  due to the variance of the manual results themselves. An alternative suggestion is to assign a weight to the error arising between the mean of a set of manual measurements and the associated set of measurements produced by running a scale descent algorithm with one of the four active contour models. The problem is that of combining bias errors and precision errors into one representative quantity, since ultimately a comparative performance measure that is used as a basis for setting some parameter level must be a scalar quantity. Is a bias error less important than a precision error? A bias between a set of manually delineated areas and a set of areas obtained by running an algorithm can simply be mainly due to the combined uncertainty between the two means. A hypothesis test between the means will serve to disregard all biases below a certain precision-dependent threshold. Consequently, the criterion used here is the ‘safe‘ option of using unweighted errors so that equation (5.4) is used as the performance measure for the purposes of setting parameters. This does not however, preclude an assessment of the final results in an in-depth manner.

 

5.3.2 Scale-Descent Results

For each of the four algorithms the scale-descent algorithm is initialised at five scales from s =4 to s =16 in half-octave steps. The scale descent decreases the scale in half-octave steps to a minimum scale of s =2. This process is applied to the ten wound images used for the manual delineation trial. The scale descent algorithm is run 25 times from each starting scale (using the manual delineations) so that the means and variances of measured areas can be estimated. This section discusses the behaviour of the algorithms during scale-descent. The overall behaviours of the four algorithms under the scale descent algorithm of §3.1 are substantially similar. Thus the effects of scale descent can be discussed in a general manner without presenting detailed analyses of each algorithm in turn. The particular differences noted between the algorithms are more associated with the levels of bias and precision errors rather than how the magnitude and ratio of these errors tend to change as the image is de-blurred. This section presents examples cases that are common to all the algorithms. In order to show the variation in both bias and precision and the relationship between them bias is plotted against precision for each scale. Joining the bias/precision co-ordinates obtained at each level of scale produces a locus, which is referred to as the ‘accuracy locus’. For each wound image and each algorithm, five such loci are obtained.

 

A. Example case of Reduced-Variance with Recoverable Bias by Scale Descent

The image for which the accuracy loci are plotted in Figure 5.7 is an example of a case where the sample manual delineations are mutually biased. This has occurred because epithelial tissue surrounding the partially healed wound has been ambiguously interpreted consistently by one delineator and twice by other delineators. This ambiguity leads to a delineation precision of 6% of the mean delineated area. Applying the GA algorithm with little regularisation of the image fails to substantially reduce this figure because the initial (delineated) contours correspond to more than one energy well and associated minimum of the contour’s energy minimum. Successive increases in initial regularisation of the image introduce increasing biases, but the scale descent procedure allows the biases to be removed, at least to a consistent level of 1-2 %. An increase of this order is expected since the manual mean represents area measurements of more than one distinct wound boundary. The particular examples in Figure 5.7 are taken from the results produced by the GA and MX algorithms.

(a)

(b)

Figure 5.7 Improved precision after scale descent with recovery of large initial bias: (a) GA Algorithm, (b) MX Algorithm. The dashed line joins the starting points for scale-descents starting at different scales (ss).

 

B. Example of Permanent Bias Introduced by Over-Regularisation of the Image

Figure 5.8 Scale descent accuracy loci of area measurements for one image. In this case the image is over-regularised for scales ss>8.

Figure 5.6 shows the scale-descent accuracy loci for an image where the loci follow the classical pattern at low levels of regularisation: increased bias at the initial (highest) scale which is reduced to a low level as the scale descent proceeds with the intended effect of providing area measurements with lower variation. This is the case for image regularisation scales up to s =8 pixels. However, as the scale is increased beyond this level the loci take on a vastly different form culminating in very large biases, approx. 20%, with no decrease in variation. In this case the image is affected by undesirable illumination effects which, whilst causing no problem of ambiguity to a delineator, causes increasingly large boundary displacement at higher scales. In the example shown in Figure 5.8 the bias is accompanied by a large increase in variance, something which is not typical of the general performance of the algorithms. An additional observation from Figure 5.8 is that the ss=16 scale descent locus passes close to the (zero bias, zero variance) point at an intermediate level of image smoothing. This is an unusual occurrence of the RMS error being minimised at a high level of scale.

 

5.3.3 Optimal Scale Parameter Determination

For each algorithm in turn, the RMS error expressing the total error between manual and algorithmic area measurements is calculated for the ten wound images using equation (5.2). Since the error is expressed in percentage terms it is possible to combine them into a single representative value of the algorithmic performance for each setting of the scale descent algorithm, i.e. start scale and end scale, denoted ss and se respectively. The approach employed here is to take the arithmetic mean of the RMS errors for each wound that arise at each possible pairing of start and end scales, . Thus the objective function to be minimised is:

(5.5)

where  is the RMS error for a particular wound.

It is most unlikely that the scale descent pairing yielded by (5.4) will be optimal for each image. Thus, selecting one static pairing of start and end scales will introduce a loss into the overall RMS error, signifying the maximum amount of improvement that could be obtained by an image-dependent scale adjustment procedure. The average loss is defined by:

(5.6)

Table 5.2 contains the results of calculating the RMS error using equation (5.2) for the GA, TN, MX and MG algorithms respectively and lists the start and end scales at which the RMS values are obtained for each algorithm. The average loss computed by (5.6) is also listed and this is subtracted from the overall RMS error value to give an indication of the minimum level of RMS error that could be obtained by selecting appropriate start and end scales for each wound. Figure 5.9 (a)-(d) shows the RMS error and losses for each algorithm on a case-by-case image basis. The darker part of each bar represents the loss for that image introduced by using the fixed scale-descent procedure. These graphs show that the maximum RMS error could be limited to 10% for the GA algorithm, 8% for the MG algorithm and 6% for the TN and MX algorithms. A possible adaptation of the scale selection procedure is to consider the size of the wound. In the present case a good approximation to the size of the wound is provided by the manual delineation procedure. Two issues must be considered when attempting to base the start scale selection upon the wound size:

 

Active Contour Model

[%]

[pixels]

[%]

Predicted

 [%]

GA

4.8

(8,2)

1.0

3.8

TN

3.9

(8,2)

1.6

2.3

MX

4.8

(5.7,2)

1.9

2.9

MG

4.8

(4,2)

1.7

3.1

Table 5.2 RMS errors and scale descent parameters for the four active contour models.

(a) GA Algorithm

(b) TN Algorithm

(c) MX Algorithm

(d) MG Algorithm

Figure 5.9 Case-by-case analyses of the contribution to overall RMS errors for each active contour model. The total height of each column shows the RMS error associated with each wound image measurement. The darkened part of the bar shows the amount by which the error could be reduced if the optimal scale could be selected specifically for each wound.

 

5.4 Summary

This chapter has discussed the effect that regularising both the contour and gradient (potential) image has upon the performance of the active contour models described in Chapter 3. The regularisation parameters have been set with regard to precision (contour regularisation) and RMS error (image regularisation). The particular mix of bias and precision that constitutes this error has not been evaluated – this is a subject considered in the following chapter, where the performance of the algorithms given the specifications for the regularisation parameters is tested upon more images. The MX and MG algorithms do not produce lower levels of RMS error than the GA and TN algorithms indicating that, given the varied set of wounds tested, the exact settings of the contour regularisation parameters are not critical. For the TN algorithm, Figure 5.6 shows that, on average, the precision at all levels of scale employed in the study is quite stable over a wide range of contour regularisation parameters. Due to the nature of its energy formulation the precision of measurements produced by the GA algorithm is progressively degraded as the contour regularisation is increased. There is an unavoidable reduction in the accuracy of measurements produced by all four algorithms as a consequence of choosing working parameters for the scale descent procedure – the ‘loss’ in accuracy, expressed in terms of RMS error, is shown in Figure 5.9.

[Ü Manual Delineation] ¨ [Þ Results]


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