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Size Theory As a Topological Tool for Computer Vision Size Theory as a Top ological Tool for Computer Vision P Frosini and C Landi Department of Mathematics University of Bologna Pzza Pta S Donato I BOLOGNA ITALY frosinidmuniboit CIRAM BOLOGNA ITALY Department of Mathematics University of Pisa V Buonarroti I PISA ITALY landidmunipiit Abstract In this pap er we give an outline of Size Theory and its main results The usefulness of such a theory in comparing shap es is high lighted byshowing some examples The robustness of Size Theory with resp ect to noise and o cclusions is p ointed out In addition an algebraic approach to the theory is presented Keywords shap e Size Theory size function natural size dis tance Intro duction Comparing shap es of ob jects is a ma jor task in Computer Vision Size Theory is a new mathematical to ol for dealing with this task anditisnow the sub ject of exp erimentation cf eg and In this pap er we shall give the main denitions and results in this theory and show its main prop erties by displaying some examples For the sake of conciseness we shall omit the pro ofs This work was partially supp orted by MURST Italy ELSAG Bailey Italy and ASI Italy of our results they can be found in and An extension of the theory app ears in and The fundamental ideas in Size Theory are the concepts of nat ural size distance and size function Both of them provide a way of measuring the extent to which the shap es of two compact top olog ical spaces resemble each other They are mo dular concepts in the sense that they dep end on the arbitrary choice of particular func tions called measuring functions which can b e set in order to obtain invariance under those transformations that are required to preserve shap e in each sp ecic context Furthermore these concepts are in trinsically related We underline that Size Theory can b e applied for studying all data that can be seen as a compact top ological space not only images even though such a theory was originally conceived for use in Computer Vision After giving some mathematical results on natural size distances and size functions we shall deal with the problem of computing the latter Next as far as comparison of images is concerned we shall exhibit some examples showing the prop erties of invariance noise resistance and o cclusionresistance of size functions with resp ect to the choice of suitable measuring functions Finallywe shall givean algebraic representation of size functions in terms of formal series and discuss its usefulness Natural Size Distances Some Denitions and Results In this section we shall consider the set Size of all pairs M called size pairswhere M is a compact top ological space and is a continuous function from M to the set IR of real numb ers called measuring function In some cases in order to obtain particular results we shall assume that M is a suciently regular submanifold of some Euclidean space Our goal is to dene a distance that allows us to measure the extent to which the shap es of M and N are similar to each other We shall do so with resp ect to the continuous functions and which have b een chosen arbitrarily Denition Let M N betwo size pairs and let H M N b e the set of homeomorphisms from M onto N Let us consider the function that takes each homeomorphism f H M N to the real number f max jP f P jWe shall call the P M natural size measure in H M N with resp ect to the measuring functions and In plain words measures howmuch f changes the values taken by the measuring function Prop osition The function Size Size IR fgdened by setting M N inf f if H M N f H MN and otherwise is a pseudometric on Size Denition The metric induced by the pseudometric will be called the natural size distance in Size where denotes the equivalence relation dened by setting M N ifand only if M N The equivalence class of M will be denoted by the symbol M More details on the passage from a pseudometric to a metric can befoundinWehave used the term natural b ecause our manner of dening a pseudometric between compact top ological spaces is a particular case of a more general metho d cf Before pro ceeding we shall give a trivial example of natural size distance Example In IR consider the unit sphere S with equation x y z and the ellipsoid E with equation x y z On S and E consider resp ectively the measuring functions and that take every point of S and E to the Gaussian curvature of the given manifold at that point We have that S E In fact S fg while E fr IR r g and therefore for every f H S E we have f Denition We shall call optimal in H M N every homeomor phism f H M N such that M N f We p oint out that an optimal homeomorphism do es not generally exist even in cases when M and N are regular compact and without b oundary manifolds and are regular measuring functions cf However if we assume such hyp otheses in particular that M and N are manifolds of class C and are measuring functions of class C we have the following theorem Theorem Let us assume that an optimal homeomorphism exists in H M N Then the natural size distance between M and N is the Euclidean distance between a critical value of and a critical value of The ab ove theorem no longer holds if we drop the hyp othesis of existence of an optimal homeomorphism cf Such a theorem makes the computation of natural size distances less dicult A survey on natural size distances can be found in Size Functions Denitions and Prop erties In general natural size distances are dicult to compute as they in volve the study of all homeomorphisms b etween two compact top o logical spaces On the other hand they can compare compact top o logical spaces with resp ect to given measuring functions in a very powerful manner and quantify the dierence Thus we need a to ol to easily obtain information on natural size distances without com puting them directly the concept of size function is such a to ol In addition size functions are useful for comparison of shap es even in dep endently of natural size distances In the following denitions we shall assume a size pair Misgiven in M Denition For every y IR we dene a relation y Q P Q M if and only if either P Q or by setting P y there exists a continuous path M such that P Q and y for every In this second case we shall say that P and Q are y homotopic and call a y homotopy from P to Q Remark It is easy to show that is an equivalence relation y on M for every y IR Denition For every x IR we shall denote by Mh xi the set fP M P xg Denition Consider the function IR IR IN fg M dened by setting x y equal to the nite or innite number M of equivalence classes in which Mh xi is divided by the equiva Such a function will b e called the size function lence relation y associated with the size pair M Remark When x y size functions have a simple geometric in terpretation in such a case x y is equal to the number of M arcwise connected comp onents of Mh y i containing at least one p oint of Mh xi Now we shall give some simple examples of size functions Example In Fig weshow the size function of an ellipse E with re sp ect to the measuring function that takes each p oint to its distance from the barycentre of E In every highlighted region the constant value taken by the size function is given by the number displayed This size function has been computed by means of a computer and therefore some errors due to the necessary discretization app ear in the form of small triangles near the diagonal Remark We shall often display only the part of a size function inside a square x minxmax y miny max with x min y min and x max y max When this is the case the values x min y min x max y max will b e shown in each gure Fig Size function of an ellipse with resp ect to the distance from the barycentre Example Consider the size pair E where E is the abc abc ellipsoid with equation ax by cz in IR a b c and is the dimensional measuring function that takes each point of E to the Gaussian curvature of the ellipsoid at that point In abc Fig weshow the function IR IR IN fginevery E abc highlighted region the constant value taken by the size function is given for the case abc Fig Size function of the ellipsoid with equation ax by cz cb a with resp ect to the Gaussian curvature Example Consider the size pair M where M is the curve depicted in Fig left and is the dimensional measuring function that takes each p ointofM to its distance from the barycentre of M In Fig right weshow the function IR IR IN fgin M every highlighted region the constantvalue taken by the size function is given Main Prop erties of the Size Function M x y is nondecreasing in x and nonincreasing in y M x y is nite for x y under the hyp othesis that M is M not only compact but also lo cally arcwise connected Fig Size function of the curve depicted on the left with resp ect to the distance from the barycentre used as measuring function x y forevery xmin P M P M x y is equal to the numb er of the arcwise connected com M p onents of M for every x y max P P M x y for every x y such that there exists a non M isolated p oint Q M for which xQ and yQ We complete this summary of prop erties of size functions by giving a useful theorem which helps us in lo calizing the disconti nuity p oints of the size function in cases when M is a closed M ie compact and without b oundary submanifold of some Euclidean space Theorem Let us assume that M
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