Fuzzy histograms, weak fuzzification and accumulation of periodic quantities
Abstract
The influence of the scale of a fuzzy membership function used to fuzzify a histogram is analysed. It is shown that for a class of fuzzifying functions it is possible to indicate the limit for fuzzification, at which the mode of the histogram equals the mean of the data accumulated in it. The fuzzification functions for which this appears are: the quadratic function for aperiodic histograms and the cosine square function for periodic ones. The scaled and clipped versions of these functions can be used to control the degree of fuzzification belonging to the interval [0,1]. While the quadratic function is related to the widely known Huber-type clipped mean or the kernel function derived from the Epanechnikov kernel, the clipped cosine square seems to be less known. The indications for using strong or weak fuzzification, according to the value of the fuzzification degree, are justified by examples in two applications: classic Hough transform-based image registration and novel accumulation-based line detection. Typically, the weak fuzzification is recommended. The images used are related to simulation images from teleradiotherapy and to mammographic images.
Keywords
Fuzzy histogram Accumulation Scale Mode to mean transition Limit fuzzification Periodic histogram Line detection Mammograms Image registration1 Introduction
One of the robust methods of estimating the measured quantity on the basis of a series of measurements, in the presence of errors, is the voting organized as forming a histogram and finding its mode. The histogram can be viewed as an empirical approximation of the probability density function and in this convention the mode is an approximation of its mode (or simply a maximum). This is the source of robustness of this method against noise as well as incomplete or partly erroneous data. For the voting to be possible the values are quantized and the histogram is an array. Its dimensionality corresponds to the dimensionality of the measured quantity. Each measurement is a vote for a specific value of the quantity of interest.
In the scope of computational methods, if the specified quantity can be calculated from the data in multiple ways, each such result can be treated as a measurement. This is particularly the case in image processing at low level, where the pixel data are a source of redundant but partly misleading information on the objects present in the image. Within the domain of image processing, the most widely known method using the analysis of a histogram is the Hough transform (HT), which has been developed into numerous versions applied to detection of several classes of shapes (see, for example, [1, 2, 3, 4, 5, 6] for the early works and [7, 8, 9] for the first surveys). The key feature of the method making it a robust image analysis tool is that data are accumulated in the array called the accumulator array, which is the histogram of the occurrences of specific values of the parameters of the detected shape (e.g., straight line, quadric, shape given by a template) or relation (e.g., image registration parameters). The presence of a single mode or multiple modes in this histogram is the evidence of the presence of the instances of the object sought. Recently, the methods of the detection of shapes which are described neither by parametric equations nor by a template have been developed (see [10, 11, 12] for the methods using Fourier descriptors and [13, 14, 15] for the accumulation in the image domain). The methods related to HT are now frequently called the evidence accumulation [9] or evidence gathering [10, 12] methods.
One of the important issues in the methods where a histogram is used is the choice of its resolution. The problem underlying this issue, as stated explicitly in [16], is the uncertainty–precision duality: the higher the histogram resolution is, the more precise the result of detection of the mode can be; however, together with resolution increase, the certainty that this mode exists and that it corresponds to the correct result decreases. This problem has been analysed by numerous authors. In [17, 18, 19, 20] the approaches related to establishing proper resolution have been proposed, including the non-uniform, multiresolution and adaptive divisions of the domain of the solution. In [21] it has been proposed to distribute each vote into more than one element of the histogram. In [22] the fuzzy set theory has been directly used and the fuzzy Hough transform was introduced explicitly.
The paper by Strauss [16] seems to be the most complete work on fuzzy histograms related to the Hough transform up till now. It presents a solution to the problem of finding a compromise between uncertainty and precision, with the issues of image thresholding, quantization of the parameter space and data localization uncertainty, and enhanced peak detection in the parameter space, taken into consideration. The membership functions used in fuzzification of the subsequent stages of the method are derived from the basic relationships pertaining to the way in which the pixels form an image of a line. A large number of citations on fuzzy Hough transform are given in that paper.
The methods used in this paper have a strong relationship to robust statistics and to kernel density estimation. The relation goes along the line of the use of functions treated in this paper as fuzzy membership functions in histogram fuzzification. Among them, there is the clipped square function used in the fuzzification of a non-periodic histogram. On the grounds of robust statistics, this function is related to the skipped mean function, traced back by [23, 24] to the fundamental works of Huber (collected in the book [25]). In the domain of kernel density estimation, this function is the kernel function derived from the Epanechnikov kernel (see [26, 27], also [28], as cited by [29]; the references go back to [30]). The clipped square function, as related to the skipped mean, picks the mean value of a number of histogram elements immediately neighbouring the given element, while skipping the values from the remaining elements. This widely known property is readily applicable in aperiodic histograms. It will be shown that in the case of periodic histograms, the function which has the analogous property is the cosine square function; the use of its scaled down and clipped version seems to be less generally acknowledged.
An overview of the applications of robust methods in computer vision problems can be found in [29].
There also exists an obvious analogy of histogram fuzzification to simple low-pass filtering, which is the reason why the users of fuzzy histograms frequently apply the Gaussian function as the membership function. It can be easily shown that this function has similar properties to those of the functions mentioned before.
In all the functions used here as the fuzzy membership functions in histogram fuzzification there is a single parameter related to the width of the support, or scale. In [29] (immediately preceding and following the formula (4.4.34), Sect. 4.4.3) a discussion on the question of scale can be found. This discussion covers several estimates of the scale and can be summarized in two questions: (1) How to find a good estimate of the scale if little is known on the structure of the data, i.e., the share of outliers and the noise? and (2) what coefficient should be used to tune the actual scale to be used in calculations with respect to this estimate? The scope of the present paper is less extensive than that of the paper by Strauss [16] and that of the cited works on robust statistics in computer vision. In this paper, a very simple yet useful approach to the problem of scale is proposed. The considerations will be restricted to accumulation-based, HT-related methods, which are the main domain of interest to the author. The cases of an aperiodic and a periodic histogram will be studied. The problem will be investigated not from the perspective of an estimate of scale, but rather from the perspective of the upper limit for fuzzification. This limit will be treated as the basis for simple indications concerning the choice of scale. These indications will be verified against two test problems: the image registration with the classical HT method according to [31, 32, 33, 34] and the novel line detection method according to [13, 14, 15].
The paper is organized as follows. Section 2 recalls the explanation why it is useful to solve the precision–certainty duality with fuzzy voting rather than by changing the histogram resolution. In this context the observation on the existence of the limit for fuzzification is made. This section is based on easily understandable artificial examples. In Sect. 3 the properties of a fuzzified histogram are briefly presented. The piecewise continuous transition between the mode of a histogram and the mean of the data accumulated in it is described. This constitutes the basis for explaining the notion of the limit fuzzification and the degree of fuzzification. The case of periodic data and a periodic histogram is treated in Sect. 4. The results analogous to those of an aperiodic histogram are discussed. Section 5 briefly recapitulates the obtained results and presents indications for the choice of scale in the fuzzifying function, i.e., the use of weak and strong fuzzification. In Sect. 6 these indications are confirmed on the basis of experiments with two HT-related accumulation methods. Conclusions come in Sect. 7.
2 Introductory example: fuzzification of a histogram
Results of 12 measurements \(\check{x}_j\) of the variable x
j | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\check{x}_j\) | 0.6 | 1.1 | 1.4 | 3.8 | 7.2 | 12.2 | 12.8 | 13.9 | 14.1 | 15.4 | 15.9 | 18.8 |
\(\check{x}_j/2\) | 0.3 | 0.55 | 0.7 | 1.9 | 3.6 | 6.1 | 6.4 | 6.95 | 7.05 | 7.7 | 7.95 | 9.4 |
\(\check{x}_j/4\) | 0.15 | 0.275 | 0.35 | 0.95 | 1.8 | 3.05 | 3.2 | 3.475 | 3.525 | 3.85 | 3.975 | 4.7 |
In consequence of the preceding considerations, the measured value should be estimated with the mean value of the measurements for j = 6, ..., 11, which leads to the estimation\(\hat{x}=14.05 \approx 14.\) The mean of all measurements is 9.75, the median is 12, and neither of these values conform to the estimation \(\hat{x}.\) Clearly, the mode of the histogram corresponding to x = 1 does not represent the proper result of the measurement. This discredits the use of the histogram and the accumulation principle as a robust measure of the result of measurements \(\check{x}_j.\)
At this point the basic question of this paper can be asked: what is the influence of the scale parameter, or in other words, the width of the support of the fuzzifying function, on the fuzzified histogram? To what extent the histogram can be fuzzified and is there a limit of a reasonable fuzzification? The commonsense intuition is enough to say that for a ‘very narrow’ support there is no fuzzification at all and for a ‘very wide’ the histogram tends to become uniform. It is easy to state that ‘very narrow’ is such that there is only the central zero point inside the support and the fuzzy histogram is equal to the crisp one.
In the next section the reason for such a behaviour of the fuzzified histogram will be shown. The considerations for the case of an aperiodic histogram will be analogical to those for the skipped mean and the Epanechnikov kernel, as stated in Sect. 1. In the case of a periodic histogram this analogy will not be so close, however. Hence, both cases will be presented, so the reader less acquainted with the basics of the domains of robust statistics and kernel density estimation can follow them easily. For the same reason the proofs of the presented properties will be given, despite the fact that they are quite simple.
3 Aperiodic histogram
3.1 Quadratic fuzzifying function
It can be demonstrated that the fuzzified histogram has the following property:
The straightforward proof of the property is given in Appendix.
3.2 Other fuzzifying functions
Now let us consider a more general case of symmetric functions depending on scale, having a single maximum. For such function (and its derivative, used in (38)) the following property holds:
The proof of the property is given in Appendix.
By virtue of Property 2, the fuzzification with any fuzzifying function fulfilling the preconditions of this property has its limit, at which the mode of the histogram stabilizes at its mean. For each such function, the appropriate expression for the fuzzification degree can be derived in analogy to the expression (14). Coming back to the example illustrated in Fig. 3, the limit scale at which the fuzzification degree becomes one is s≈ 9.
Remark 1 (to Properties 1, 2 and formula (14)) In the case of a crisp histogram, when the resolution increases, consequently the certainty of the solution consisting in finding the mode decreases. This is because the average number of votes in the histogram elements decreases. However, in the case of a fuzzy histogram, the mode always tends to the mean of the histogram when the degree of fuzzification grows. Moreover, if the resolution is larger then the precision of indexing is larger, so the mean of the histogram is closer to the mean of the data.
Remark 2 (to Properties 1, 2, formula (14) and Fig. 3) In cases like that presented in Sect. 2, illustrated in Fig. 3, the useful range of the fuzzification degree is closer to zero than to one. The dependence of this range on the histogram resolution is negligible, provided the resolution is sufficient for the indices to represent the values of the data.
The results obtained can be easily extended to a multidimensional case.
4 Periodic histogram
Analogous to the quadratic function in the case of an aperiodic histogram, for a periodic histogram the cosine square function is the fuzzifying function that leads to the convergence of the mode of the fuzzified histogram to some characteristic value for the set of the accumulated data. This characteristic value in the case of periodic data can be called the dominating value. If the period is 2π, this value can be strictly treated as the mean value. In the case of an arbitrary period, the same way of understanding the dominating value as the mean value can be adopted. As in the case of an aperiodic histogram, the considerations are related to the data quantized according to the indexing function. The following properties hold:
The joint proof of the Properties 3 and 4 is given in Appendix.
What seems the most conspicuous is that in the case of periodic data the dominating value falls between indices 2 and 3, while in the case of aperiodic data it was between 1 and 2. This is reasonable, however, because now the value H(0) is opposite to H(2): if they were equal, the fuzzified histogram containing only these two values would be strictly a circle. The reader is encouraged to investigate this contradiction by doubling the phase of histogram elements and treating them as vectors, according to Property 4. The net effect of the elements H(0) and H(2) is 1 at i = 1, so after adding H(3) = 1 the result falls between i = 1 and 2, and the intensity of the result is\(\sqrt{2}.\)
The form of the equations (25) indicates that there is an analogy between a periodic quantity with intensity A and phase β having the period T, to a vector with modulus A and phase τβ = (2π/T)β. This analogy, described in 1972 by Mardia [35], was recalled by Zwiggelaar et al. in 1999 [36]. There exists a difficulty in some calculations on the directions of elongated objects in the images, for which the sense (in the meaning of the sense of a vector) is not defined, so the period of the angle which characterizes the direction is π. Therefore, the angle of a line can have two values differing by π, and there exists the problem of phase wrapping, which makes it arduous to calculate such an otherwise straightforwardly defined function on a set of data like the mean value. The same applies to the gradient if its sense is not important in the given application. An attempt to find the mean direction without using the vector analogy of the periodic quantity has been described and successfully used in [37]. Some heuristics must have been used in that paper to avoid the ambiguity resulting from the phase wrapping. Zwiggelaar et al. [36] used the transformation of angles into vectors according to [35] to calculate the mean direction, thus obtaining simple, well-defined formulae. In the case considered in [36] the angles alone were important, not the line intensities, so unit vectors were used. The formulae (25) make it possible to treat periodic quantities having arbitrary moduli.
It should be remarked that the use of the vector analogy is not the only way of overcoming the difficulty with the angles of elongated structures. The feature ALOE [38] is calculated as the standard deviation of the histogram of directions instead of the directions themselves, so the quantities of which the mean is calculated are not periodic.
In the periodic case, the property of similarity of other functions to the function (28) which could be analogical to the case of the Property 2 for the aperiodic case does not exist.
As has been said in Sect. 1, the considerations on the quadratic function as the fuzzification function and the use of its clipped version (12) is only a kind of reformulation, on the grounds of histogram fuzzification, of the notions already known from the domains of kernel density estimation (Epanechnikov kernel) and robust statistics (Huber-type skipped mean). However, to the best of author’s knowledge, it seems that the use of the clipped cosine square function (28) to the fuzzification of a periodic histogram, and the resulting definition of the fuzzification degree, has not been presented up till now.
5 Implications of the transition between the mode and the mean
In the preceding parts of the paper it has been demonstrated that for any crisp histogram, a piecewise continuous transition can be made between its mode and the mean of the data accumulated in the histogram. This is done by fuzzifying the histogram with the fuzzification function (12) for the case of an aperiodic histogram and with the function (28) for the periodic case, with the scale s changing so that the degree of fuzzification, according to (14) or (30), respectively, changes from zero to one. The fuzzy histogram with the real argument or with the integer argument can be considered (see Figs. 4, 5). The piecewise continuity consists in that the mode of the fuzzy histogram with a real argument changes smoothly between the jumps, and these jumps appear when the maximum of the histogram with the integer argument moves from one index to another.
A similar transition can be made with fuzzifying functions depending on the scale, other than (12) or (28), but then the definition of the degree of fuzzification has to be modified accordingly.
Another implication, important for the algorithms which incorporate the process of estimating the result on the basis of a set of data, is as follows. It can be observed in the continuous way how the change of the degree of fuzzification influences the result and an optimum in respect of some measure of quality of the result can be found. If and only if the optimum is not far from the limit fuzzification, which can take place when the data do not contain many outliers, the accumulation of the data followed by finding the mode of the histogram can be replaced by the calculation of the mean value of the data, which is simpler and requires less memory.
Examples of the application of weak and strong fuzzification will be presented in the next section.
6 Examples: the choice of scale
6.1 Image registration with the HT
The smallest subset of the set of feature pixels necessary to find all the transformation parameters is called the elemental subset [29]. In the case of four parameters, it contains m = 4 pixels: two from each image. By taking all the possible elemental subsets, accumulating the calculated transformation parameters in a four-dimensional accumulator, and finding its maximum, the registering transformation parameters can be found. To reduce the number of calculations, some randomly chosen part of the set of all the elemental subsets can be taken into account [39]. Further, only pairs of pixels in each image farther from each other than a specified distance can be considered. This and very similar versions of the HT have been described in [31, 32, 33] and compared with other versions in [34]. The reviews of image registration techniques can be found in [40, 41, 42, 43].
The robustness of the registration method has been tested with the use of example feature images derived from real-life images received from a medical application, which will be described further. The images were corrupted in such a way that a specified share ζ of the feature pixels was moved from their original position to a new position chosen randomly from the whole image (except other feature pixels), with the uniform probability density. These moved features are the outliers, while the remaining ones are inliers. The parameter ζ can be called the outlier share or noise share.
It should be stressed that with the described method of introducing noise to the data, the outlying data are not inserted directly to the accumulator, but to the algorithm as a whole. Therefore, the whole test concerns primarily the image registration algorithm as a whole. However, the contamination of the input with strong noise implies the contamination of the data in the histogram in an adequately strong way.
The figures to be registered will be the edges of the anatomical structures selected from the images used in the quality assessment of teleradiotherapy—the treatment of patients with cancer by irradiation with external beams of ionizing radiation. The actual geometry in a therapeutical session is recorded in a portal image. The planned geometry is recorded in a simulation image, before the therapy begins. The simulation image should be registered with each of the portal images, made during each of the therapeutical sessions. The simulation image is an X-ray of high quality. The portal image is produced by the therapeutical beam of the ionizing radiation and is inherently of low contrast as different tissues, like bones and muscles, attenuate the radiation very similarly. In both images, the traces of anatomical structures of the patient, the shape of the therapeutic beam, and possibly the shields used to precisely shape the irradiation field are visible. The quality assessment of the therapy by registration of portal and simulation images has been presented for example in [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. What is most important is that generally the registration is performed using the edges found in the images. From all the edges, the significant fragments belonging to the patient’s anatomical structures and irradiation fields are chosen semi-automatically or fully automatically. In the present study it was vital to have full information on the transformation between the images and to have the possibility to precisely evaluate the transformation error. Therefore, the pairs of images to be registered have been derived from single simulation images containing the edges of patient’s bony structures, by transforming these images in a known way.
Ranges of the transformation parameters and resolutions of the accumulator array
| T_{x} = α_{1} | T_{y} = α_{2} | φ = α_{3} | c = α_{4} |
---|---|---|---|---|
min | − 20 pix | − 20 pix | − 10° | 0.90 |
max | 20 pix | 20 pix | 10° | 1.10 |
N | 81 | 81 | 201 | 81 |
Δ | 0.5 pix | 0.5 pix | 0.01° | 0.0025 |
The number of elemental subsets has been reduced by choosing one-fourth of feature pixels in each image at random and specifying the minimum distance of pixels in a pair as approximately half the characteristic dimension of the undistorted figure. With the images used, each having about 1,000 feature pixels, these conditions together with the limits imposed by the bounds on the parameters in the accumulator restrict the number of elemental subsets actually used in the accumulation to about 200,000 for the strongest noise and about 4,000,000 for no noise, which is from 0.0002 to 0.85 of their total number.
The images of the skull, Fig. 9, were registered for 152 combinations of the fuzzification degree d_{f} and the share of outlying feature pixels ζ: d_{f} from 0.0 to 0.8, at 0, 0.02, 0.05, 0.10, and further changing by 0.1, and ζ from 0.0 to 0.9 changing by 0.05. Note that for this image both scale factors were set to one (Table 2). This has been done to make it possible to solve the registration problem for this image in only three dimensions, which made the calculations quicker. The longest time was necessary to fuzzify the accumulator for very large values of d_{f}.
- 1.
The more the noise, the larger the errors.
- 2.
For a small share of noise, the errors grow slowly with the growth of the fuzzification degree.
- 3.
For large shares of noise, without fuzzification the errors are large. They have a deep minimum for a weak fuzzification, at d_{f} = 0.05 and 0.1. For stronger fuzzification, they grow similarly as for small noise share, but quicker.
- 4.
This is less important, but there is an upper limit of registration error for very large noise and strong fuzzification, related to the fact that the result of registration tends to the average of the data, and with large noise it tends to the middle of the histogram, which in this case corresponds to an identity transformation.
The 287 pairs of values used were: d_{f} from 0.0 to 0.3, changing by 0.05, and ζ from 0.5 to 0.9 changing by 0.01.
The results have been shown in Fig. 12b. The errors still have a clear minimum for d_{f} = 0.05 and 0.1. A closer analysis reveals that for d_{f} = 0.10, for example, at ζ = 0.80 it is ɛ = 3.61 pixels, then with the increase of ζ the error goes down, and further grows and exceeds a reasonable limit for ζ = 0.84 with ɛ = 9.22 pixels. The range of acceptable errors for these degrees of fuzzification reaches the share of noise ζ_{max} = 0.80, and this value seems to constitute the measure of global robustness of the fuzzified Hough transform registration, in the considered case.
The observations made with the above experiments make it possible to draw the following conclusion of general applicability. If the data collected in a histogram contain significant noise, then the weak fuzzification of this histogram makes it possible to localize its mode in a robust way. If the noise is small or absent, the weak fuzzification has no harmful influence on the mode. The weak fuzzification is such a fuzzification which makes the values of the neighbouring elements in the histogram significantly cooperative in forming peaks.
6.2 Accumulation-based line detection
The issue of fuzzification at various degrees has been investigated for the accumulation-based line detection algorithm described in [13, 14, 15]. The algorithm has been developed primarily for the application of detecting blood vessels in mammographic images, so the robustness to various kinds of irregularities in the images was the basic challenge. Therefore the principle of data accumulation and the analysis of modes in fuzzified histograms has been used at various levels in the algorithm. Here, only the basic information on the method will be recalled; all the details can be found in the papers cited above.
The detection of elongated structures is founded on the search for loci of image intensity gradients cooperatively conforming to a model of a line, which is the ridge in the image intensity graph. The evidence on the existence of such loci is accumulated in the accumulator array congruent with the image domain, keeping for each pixel the information on line intensity and direction for a range of widths, limited by the lower limit width w_{w} and upper limit width w_{u}. Then the accumulator is searched through for maxima, and those which exceed the intensity f_{l} times the intensity in the global maximum are retained, where f_{l} is the lineness threshold coefficient. From these maxima the central lines of the elongated structures are tracked across pixels and widths, as ridges in the line intensity. The tracking stops if the intensity falls below f_{a} times the average ridge intensity in the current line, where f_{a} is the average accumulated value threshold coefficient.
During accumulation, the result of (36) is subjected to a number of fuzzifications: in location, to compensate for inter-pixel locations of the central point p_{c}; in line length, to enhance line continuity; in line width, to enhance direction invariance; finally, in line angle ψ, as described further. The details are described in [13, 14]. Here it should be said that in the fuzzification in location and in line width, a weak fuzzification with the fuzzification function similar to that shown in Fig. 6 is applied. For the fuzzification in line length there is no reason to use weak fuzzification, as it has been designed to improve the line continuity in a strong manner and the support of the fuzzification function should be related mainly to the line width.
The lineness is accumulated in the central point p_{c} of the current pair p_{1},p_{2}. As all the possible pairs are analysed (reduced by some reasonable conditions), each pixel of the image (except its border pixels) is equivalent to the central point of some pair for many times. In the present paper this accumulation is of special interest. The lineness is characterized by the intensity l and angle \(\psi = \measuredangle({\mathbf{Ox}}, {\mathbf{V}}).\) In each pixel, for each width, a histogram \(H_{\rm l}(\tilde{i}(\psi))\) of l versus ψ is formed and fuzzified to obtain H_{fl}(ξ(ψ)) with a fuzzifying function (28). The resulting line intensity in the pixel is calculated as the difference between the histogram extrema, like in (27), with the phase corresponding to the maximum. The fuzzification degree d_{f} according to (30) can be set to a chosen value from an interval [0,1].
An experiment has been performed which consisted in the detection of lines in a series of phantom images with the Gaussian noise. The images are described further. The fuzzification degree d_{f} has been set to a series of seven values in [0,1] at intervals partly conforming to a geometric series. Other parameters were: lower and upper limits for line width: w_{u} = 3 and w_{w} = 15, the lineness threshold coefficient f_{l} = 0.25 and the average accumulated value threshold coefficient f_{a} = 0.25. In the histogram H_{fl}(ξ(ψ)) the angle [0,π] has been quantified into 90 intervals.
Signal to noise ratio SNR, ratio of the noise energy to signal energy σ^{2}/A^{2}, and standard deviation of noise σ, for the mean square amplitude A = 5.4235
SNR | 9 | 6 | 3 | 0 | − 3 | − 6 | − 9 |
---|---|---|---|---|---|---|---|
σ^{2}/A^{2} | 0.13 | 0.25 | 0.50 | 1.00 | 2.00 | 3.98 | 7.94 |
σ | 1.92 | 2.72 | 3.84 | 5.42 | 7.66 | 10.82 | 15.29 |
The values of the true positive rate (TPR) and false positive rate (FPR) received for the acceptable values of SNR and d_{f} used in the experiment
d_{f} | SNR | |||
---|---|---|---|---|
∞ | − 3 | |||
TPR | FPR | TPR | FPR | |
0 | 0.872 | 0.062 | 0.907 | 0.078 |
1 | 0.939 | 0.065 | 0.889 | 0.058 |
The results indicate that the use of the limit fuzzification is equally acceptable as the use of weaker fuzzification. Therefore, it is justified to replace the accumulation with the calculation of the mean value. Implementing the calculations in the form of finding the mean of vectors according to [35], as described in Sect. 4, instead of forming and analysing the histograms of the line intensities versus direction at each pixel, results in a dramatic reduction of the memory requirements. Instead of a 90-element histogram for angular resolution of 2°, only three elements are necessary: for the modulus and for the cosine and sine of the phase (phase could be stored in one element as angle, but then the number of computations would be larger). The time of operation is roughly the same for both algorithms.
The possibility of applying the strong or limit fuzzification in the considered stage of the algorithm can be explained by the fact that in the previous stages the weak fuzzification was already used. This could have significantly reduced the number of the outlying votes in the final accumulation.
The algorithm based on summation has been actually utilized from the early stage of development of the accumulation-based line detection method, but the present study has made this approach better validated.
An example of results of line detection in the phantom image obtained with the described algorithm has been shown in Fig. 17. For presentation, the result for the image with the largest acceptable noise has been chosen to show the possibilities of the method in a relatively difficult case.
7 Conclusion
The problem of the influence of the scale of a fuzzy membership function used to fuzzify a histogram was analysed. This problem arises, among others, in the context of the fuzzy image processing methods related to the Hough transform and other accumulation-based methods. The questions concern the resolution of the accumulator, precision of results, and their certainty. Both the cases of aperiodic and periodic histograms were analysed.
The membership functions used in fuzzy voting are usually symmetrical and have a single maximum, located at the histogram element for which the vote would be cast if the voting were crisp. If a parameter of scale is introduced into such a function, then for large scales the function tends to a quadratic function. It has been stated that for a scaled and clipped quadratic function it is possible to precisely define the notion of the degree of fuzzification, belonging to the real interval [0,1]. For other functions the notion can be treated as approximate. Thus, it is possible to indicate an upper limit for fuzzification and thus to qualify the fuzzification as, for example, weak or strong, using this limit as a reference.
It has been demonstrated that in the case of periodic histograms, the fuzzifying function which makes the histogram mode conform to the mean value is the cosine square function with the period equal to the period of the quantity of interest. By analogy to the aperiodic case, this fuzzification can be treated as the limit fuzzification. The scaled and clipped cosine square function can then be used to fuzzify a periodic histogram with the given degree of fuzzification.
While the quadratic function is related to the classical Huber-type skipped mean, used in robust statistics, and to the kernel function derived from the Epanechnikov kernel, used in kernel density estimation, the use of the clipped cosine square function seems to be less known, at least on the grounds of the fuzzy accumulation methods.
The conclusion pertaining both to aperiodic and periodic cases is that if there are outliers among the accumulated values, then weak fuzzification is recommended, and if the influence of outliers is not significant, then strong fuzzification can be used, which at the limit is equivalent to the simple calculation of the mean value. The conclusion concerning the mean value is trivial, but the one concerning the weak fuzzification is not easy to trace in the literature on accumulation-based image processing. In practice, the weak accumulation is meant as such that the values of the neighbouring elements in the accumulator can strongly interact in forming common peaks. In the considered examples this took place for the values of the degree of fuzzification close to 0.1.
The examples of the use of fuzzy histograms were the classical HT image registration method and the recently proposed evidence accumulation-based line detection algorithm. It has been demonstrated how the proper choice of the degree of fuzzification can improve the robustness of the algorithms against strong noise. The images used in the experiments were taken from medical applications. The image registration was performed on test images derived from simulation images used in teleradiotherapy. The line detection algorithm was tested with images related to mammograms.
8 Originality and contribution
The concepts known from the domains of robust statistics and kernel density estimation have been adapted to the fuzzification of aperiodic histograms. The concept of limit fuzzification has been described and the notion of the degree of fuzzification has been introduced for the scaled and clipped square fuzzy membership functions, corresponding to the Huber-type skipped mean, and to the kernel function derived from the Epanechnikov kernel. The results have been extended to the case of periodic histograms by using the scaled and clipped cosine square as the membership function. The choice of the scale has been made departing from an upper limit for fuzzification rather than from an estimate of scale made using the information on noise and outliers in the data. Simple indications for using strong or weak fuzzification, controlled by the value of the fuzzification degree, have been given. These indications have been justified by examples in two image processing problems: image registration with the Hough transform-based method and line detection with the accumulation-based method, using images related to medical applications, containing strong noise.
Footnotes
Notes
Acknowledgments
The research was financed by the Ministry of Education and Science as the research project no. 3 T11C 050 29 in 2005–2008.
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