the incidence of malignant melanoma in fair - skinned patients has increased dramatically in most parts of the world over the past few decades . because the prognosis of melanoma depends almost entirely on tumor thickness , early detection of thin melanoma is important for the survival of patients [ 1 , 2 ] .
the diagnostic accuracy of the clinical examination of pigmented skin lesions , however , is still rather poor .
literature results arise the evidence that the ability of general practitioners to early diagnose cmm with the naked eye is very low ;   the ability of dermatologists to early diagnose cmm with the naked eye ranges from 50% to 75% ;   there is a high rate of false positive ( causing unneeded surgical excision ) .   in the last decade
dermoscopy is a noninvasive technique that enables the clinician to perform direct microscopic examination of diagnostic features , not seen by the naked eye , in pigmented skin lesions .
this technique is more accurate than naked eye examination for the diagnosis of cutaneous melanoma , in suspicious skin lesions when performed in the clinical setting .
several rather successful computer programs have been implemented to the aim of an automatic analysis of melanocytic lesions and their discrimination between naevi and melanomas ( see , e.g. , [ 48 ] ; see also [ 9 , 10 ] for a comparison between automatic and human performance ) .
most of them keep into account the traditional abcde parameters used by dermatologists : asymmetry ( of boundary , texture , and color ) , boundary ( irregularity and dishomogeneity ) , color ( presence of several colors ) , dimension , and evolution .
in particular , asymmetry is generally based on quantitative comparison of the two parts into which a lesion image is split by its principal axes . here
we have developed a new method for comparing in a qualitative , yet precise way the two parts of a lesion at the sides of a splitting line .
the mathematical tool for comparison is the theory of size functions , applied to three features : boundary shape , mass , and color distribution .
for each splitting line of a pencil we get an asymmetry measure , so forming a map ( two for each of the three features ) . some characteristic numbers of the six maps are finally fed to a support vector machine . a classification experiment has been led on data set of 977 lesions with very good results .
the whole research is a follow - up of the adam project of the european union .
we are well aware that  qualitative measure  reads like an oxymoron ; of course , we mean that we compute a precise , objective , repeatable measure of the difference between the two half images ; yet , this difference is of a qualitative kind , in that it is not bound to geometric deformations , superimpositions or the like .
size functions ( sfs ) are modular invariants of whatever signal the user is interested in ; in the present case , the concerned features are boundary shape , mass , and color distribution .
they depend on two inputs : an object ( e.g. , a lesion boundary ) and a real map , called measuring function defined on it ( e.g. , distance from the center of mass ) .
essentially , the sf registers the behavior of the measuring function by using morse theory ( see ) .
sfs are  qualitative  not only in that they are topological in nature , but also in that a  similarity  based on them depends on the user 's choice of a measuring function and of a distance between sfs adapted to the context .
let us recall the definition of an sf , adapted from the more general setting of , where measuring functions are allowed a multidimensional range .
consider a continuous real - valued function  : m  r , defined on a subset m of a euclidean space .
the   size function of the pair ( m ,  ) is a function (m, ) : r 
n  { }. for each pair ( x , y )  r , consider the set mx = { p  m : (p )  x}. the value (m,)(x , y ) is defined to be the number of the connected components of my which contain at least one point in mx .
the discrete version of the theory substitutes the subsets of the plane with a graph g = ( v , e ) , the function  : m  r with a function  : v  r and the concept of topological connectedness with the usual connectedness notion for graphs .
figure 1 shows the size function obtained from a curve with the ordinate as measuring function .
sfs have a standard structure , the one of superimposed triangles already apparent in figure 1 .
this has an important outcome , in that the relevant information can be condensed in the vertices of those triangles .
comparison of two images ( as far as the criterion intrinsic to the measuring function is concerned ) can then be carried out by comparing the sets of these points .
several distances can be defined on the set of sfs ; one which is very successful is the matching distance ( see figure 2 ) .
unfortunately , there do not exist archetypal naevi or melanomas , so the task is harder than for classical classification problems .
statistical learning comes into play ; vectors of characteristic numbers are the input of a support vector machine .
the first processing step is segmentation , that is , the isolation of the skin lesion from its background .
this is carried out with well - tested methods depending on several parameters , most of which have been fixed by experiment .
this is notoriously a serious problem in the processing of dermatological images , and has been solved by the operations of erosion and dilation coming from mathematical morphology .
the experience of dermatologists suggests that a major criterion for suspecting malignancy is the asymmetry of various aspects of the lesion . we have followed this suggestion by splitting each lesion in two halves by a straight line passing through the center of mass .
comparison of the two halves is then performed by computing the distance between their size functions .
this represents a definite progress with respect to classical methods for detecting asymmetry ; these detected only geometrical asymmetry , while distances of size functions determine also qualitative asymmetry .
we repeat the splitting for 45 equally spaced radial lines , so getting distance as a function of angle ( see figure 4 ) . from this curve
the software extracts a set of characteristic numbers : min , max , average , min plus the value at 90 from min , integral , first moment , variation , min derivative , max derivative , integral of absolute value of derivative , and variation of absolute value of derivative .
a support vector machine with a third - order kernel is fed with these numbers , computed for each measuring function .
actually , the vectors also contain three more parameters : area , perimeter , and a bumpiness measure coming from the sf of the whole lesion , with distance from center of mass as the measuring function .
an initial set of experiments had been carried out with 90 lines instead of 45 , but the hit ratio was just slightly higher , while almost doubling computing time .
we have used six measuring functions to distil the structure of boundary , mass distribution , and color distribution , respectively .
the third sums distances of colors ( in rgb space ) of consecutive pixels along segments orthogonal to the splitting line .
our initial experiments used just these three measuring functions . adding their three opposite functions improved the hit ratios of 2 to 5 percentage points .
the acquisition setup consists of an leica 650  m stereomicroscope and a sony 3ccd-930 color video camera .
the illumination of the stereomicroscope consists of a 12  v/50  w halogen lamp that creates a bundle of light perpendicular to the area of interest .
the digital images have been archived by means of the dbdermo mips software package ( dell'eva - burroni , siena ) . over half of the data set used in the present research ,
had already been the subject of a formal study of clinical diagnostic validation using also the local population - based cancer registry ( i.e. , registro tumori romagna ) to cross - check for possible false negatives , published on .
the data set comes from the daily practice of one of us ( stanganelli ) ; of course , only  interesting  naevi had been acquired .
all melanomas and several naevi have been subjected to histological test ; all remaining naevi have been subjected to follow - up .
we have selected 977 images of melanocytic lesions ( melanomas and naevi ) acquired in epiluminescence microscopy with a fixed 16-fold magnification .
the data set contains 50 melanomas ( 28 of them with thickness less than 0.75  mm ) and 927 naevi .
every second image was assigned to the training set ( melanomas were listed consecutively ) . in tests
r1 and r2 , a training set of 25 melanomas and 500 naevi was randomized from the data set .
the test set was formed by the complement ( the remaining 25 melanomas and 427 naevi ) .
a fourth test ( s ) was performed without cross - validation , with the whole data set both as training and test set ; we interpret the not much higher scores of test s as a proof of stability . in table 1
we report , for each of tests h , r1 , r2 , and s , the specificity and sensitivity of what we judge to be the best performances . as a further information , in test s a 100% specificity
was attained only at cost of 4% sensitivity , but the decrease of specificity to 93.64% yielt a jump to 70% sensitivity .
our system is not intended to be provided to the public as a yes / no diagnostic tool ; it yields a risk index in the following way .
two classifiers , one tuned at high sensitivity , the other at fairly good specificity , give their response ; if they agree to classify the lesion as a naevus ( resp . , a melanoma ) then a low ( resp . , high ) risk is stated ; if they disagree , the output is of middle risk . a comparison has been done between the output of this compound classifier and the judgement of an expert dermatologist , who had classified the lesions as sure melanomas , sure naevi and uncertain .
the percentages reported in table 2 refer to the fractions of the three classes ( as classified by the human expert ) labeled by the machine with the three risk levels .
as stressed in , there are quite different selection criteria , melanomas / naevi ratios , data set sizes , analysis methods .
instead of reporting selected results of competitors , we refer to table 1 of that thorough paper .
we just would like to comment on very high sensitivity scores ( over 95% ) . with the noticeable exception of seidenari   et al .
, such scores seem to have been attained either with very small data sets , or with high melanoma percentages , so in situations which appear to be rather far from real - world ones . even counting them ,
the result of our cross - validated test r1 is placed in the top third of the reported scores .
of course , the single - set test s places us at an even higher rank . it would be interesting to compare  as suggested by a referee  the asymmetry assessment given by our method with the one given by an expert dermatologist .
this is unfortunately not possible , since our evaluation does not consist of a single measure , but of 66 ( see section 5 ) , what compelled us to use support vector machines for classification . in   a comparison of the performance of our system and of human operators ( three dermatologists and three general practictioners )
the true novelty of the presented method consists in the use of a qualitative but objective mathematical tool , the size functions , to evaluate asymmetry ( of boundary , color , and mass distribution ) .
three experiments with 977 lesions , carried out under cross - validation , show very good performances .
but its good hit ratio , together with the complete independence from the competitors ' tools , make our method a tempting candidate for integration . in this line of thought ,
 size functions and support vector machines are used to implement a new automatic classifier of melanocytic lesions . 
 this is mainly based on a qualitative assessment of asymmetry , performed by halving images by several lines through the center of mass , and comparing the two halves in terms of color , mass distribution , and boundary . 
 the program is used , at clinical level , with two thresholds , so that comparison of the two outputs produces a report of low - middle - high risk . 
 experimental results on 977 images , with cross - validation , are reported . 
1. Introduction
2. Size Functions
3. Classification
4. Segmentation
5. Asymmetries
6. Experimental Results
7. Comparison
8. Conclusions