Technical Report: Bessel Filter Analysis

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1 Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, sm3@ecs.soto.ac.uk I this techical epot, we pove two fudametal theoems fo a edge detectio algoithm based o a Bessel filte. Theoem 1: The gadiet magitude of the covolved image u + : R R calculated by u ( x, y = h * g is ubouded o the discotiuities of a give piecewise-costat image + + g : Ω R whee Ω ad R ae the image domai ad the set of positive eal umbes espectively. Poof: Befoe pesetig the poof fo this theoem, we biefly explai the stuctue of the poof. The covolutio (i Catesia coodiates betwee a modified Bessel fuctio of the secod kid ad zeo degee ad a iput image (a piece-wise costat fuctio depicted i figue (A (see the ed of sectio A of this documet at the cete of coodiate system (poit is evaluated i pola coodiate system. Next, we calculate the diectioal deivative of this covolutio (at poit alog a lie passig though poit with ay give slope betwee ta ( ϑ ad ta ( ϑ φ + (see figue (A i this documet. By exploitig the asymptotic popeties of the Bessel fuctio, we the pove that this diectioal deivate is ifiity (ubouded. We immediately coclude that the gadiet magitude of the covolved fuctio at poit is also ubouded. To explai the poof i details, let us ow coside the piecewise costat image show i figue (A descibed i pola coodiates as: ( g(, θ = ah ( H ( θ ϑ H ( θ ϑ ϕ (A-1 whee H(., a,, ad θ ae the Heaviside step fuctio, a costat epesetig lumiace, adial ad agula coodiates espectively ad ϑ φ [, π, ae costats

2 I figue (A, the oigi of the coodiate system, poit is placed o the image discotiuity fomig a shap coe. We aim to show that the gadiet magitude of u at (i.e. u is ifiite. If u is the covolved image with the Bessel filte, we evaluate u u = whee (, = cos( θ i + si( θ j, θ ( ϑ, ϑ + φ, ad i ad j ae the uit vectos alog x ad y axes espectively: u h ε ε h( x, y ε h( x, y = * g = Lim g( x x, y y dx dy + Lim g( x x, y y dx dy ε ε h( x, y h( x, y + Lim g( x x, y y dx dy Lim g( x x, y y dx dy ε + ε ε (A- whee h is a Bessel filte defied as: x ε K( x > h( x = ε ε K( x I the above equatios, K is the modified Bessel fuctio of the secod type ad zeo degee. At poit i figue (A i pola coodiates, we ca wite: u π h (, = Lim g(, θ ddθ ε (A-3 u (, = a Lim ϑ + ϕ + π ε ε ϑ + π h ddθ (A-4 the othe had, h ca be calculated as: h h h = h = cos( θ + si( θ x y (A-5 Equatio (A-4 ca the be witte as: K ( u a ϑ + ϕ + π (, = Lim cos( dd ε ε ϑ + π θ θ θ (A-6

3 Theefoe: K ( u a (, = ( si( ϑ θ si( ϑ + ϕ θ Lim d ε (A-7 Equatio (A-7 ca be itegated by usig itegatio by pat: u a (, = ( si( ϑ θ si( ϑ + ϕ θ Lim K ( K ( d ε ε (A-8 Usig asymptotic behavios of K (, as = ε ad ad l Hopital s ule fo idetemiate tems, we ca detemie the asymptotic behavio of tem K i equatio (A-8, whe = ε ad, i.e.: Lim K ( = Lim Log( ad Lim = ( π K = Lim exp( = Equatio (A-8 ca theefoe be ewitte as: ( u a (, = ( si( ϑ + ϕ θ si( ϑ θ Lim K( d ε (A-9 as ε, the itegatio tem i equatio (A-9 appoaches to ifiity. u Sice (, = u, the u = We otice that fo φ = π the wedge show i figue (A chages to a edge alog a staight lie o which poit ca be egaded as a abitay poit. I such a case, equatio (A-9 is educed to u a (, = ( si( ϑ θ Lim K( d ε (A-1

4 The absolute value of the ight had side of equatio (A-1 also appoaches ifiity as ε, hece u =. We also ote that by cosideig a egula cuve appoximated by a ifiite umbe of ifiitesimal staight lie segmets ad by assumig poit to be o oe of these ifiitesimal staight lie segmets, we exploit the above agumet obtaied fom (A- 1 to coclude that appoaches ifiity as ε fo this case as well. It is also oted u that fo boudaies fomig staight lies ad/o egula cuves the uit vecto is coveietly cosideed to be omal uit vecto to the bouday. y φ ϑ x Fig (A: A oigial image cotaiig discotiuities fomig a shap coe at the cete of coodiates + Theoem : The gadiet magitude of the covolved image u : R R calculated i u ( x, y = h * g has local maxima o discotiuities of a give piecewise-costat + image g : Ω R. Poof: At the begiig, let us biefly explai the stuctue of the poof. The diectioal deivative of the covolutio betwee the isotopic Bessel fuctio ad the iput image show i figue (A alog a staight lie passig though poit with a slope betwee ta ( ϑ ad ta ( ϑ φ + (see figue (A is cosideed. We iitially assume that the local maximum associated to poit is located at poit with pola coodiates (, θ. By usig the asymptotic popeties of the Bessel fuctio, we fially pove that poit is located at the cete of coodiates, i.e. poit is the same as poit. To explai the poof i details, let us ow coside agai g i figue

5 (A. We ow aim to pove that u has a maximum at poit, as ε. By u cosideig U (, = (,, we ca calculate U (, as (, π U h h = cos ( θ θ si ( θ θ gddθ whee h is the Bessel filte ad g is give i equatio (A-1. By itegatig the above equatio with espect to θ ad itegatig by pat with espect to fo the fist tem i the above equatio, we ca wite: (, dhε ( si ( ( si ( ( si ( ( si ( ( ( ( U = a Lim ϕ + ϑ θ ϑ θ d = a ϕ + ϑ θ ϑ θ Lim h ε d ε (B-1 Let us assume that U has a local maximum associated with poit i (, θ whee epesetig the distace betwee the local maximum ad the cetal poit, is a small displacemet, θ ϑ ad θ ϑ + φ. We otice that the local maxima with ad agle θ = ϑ o θ = ϑ + φ ae associated with poits o lowe ad uppe discotiuity edges (alog staight lies of the wedge ad theefoe ot associated with poit. A Mclaui s seies ca be witte i the eighbouhood of maximum poit, i.e.: = + + θ + (, θ (, (, (, U U U U ( whee = cos( θ θ e si( θ θ e, ( θ = si θ θ e + cos( θ θ e θ ( is pepedicula to, e ad eθ ae uit vectos alog adial ad agula coodiates espectively. Sice is a small displacemet, we igoe tems with highe degees. the othe had, U (, θ vaishes, because U has a local maximum at,, i.e.: ( θ U U U U = = + + θ (, θ (, (, (, We ca the calculate as:

6 U (, = U U + θ (, (, (B- The secod diectioal deivatives of U alog the uit vectos ad ca also be calculated as: U (, dh = a ( si ( ϑ θ si ( ϑ + ϕ θ Lim d d ε (B-3 U (, 1 dh = a( si ( ϑ + ϕ θ cos ( ϑ + ϕ θ si ( ϑ θ cos( ϑ θ Lim d d ε (B-4 By usig equatios (B-1, (B-3 ad (B-4, equatio (B- ca be witte as: = Lim ( si ( ( ϕ + ϑ θ si ( ( ϑ θ h( ε 3 3 ( si ( ϑ θ si ( ϑ ϕ θ θ ( ( ( si ( ϑ ϕ θ cos ϑ ϕ θ si ( ϑ θ cos ϑ θ 1 dh d d ε The above tem is idetemiate as ε. By usig l Hoptial s ule fo idetemiate tems, we ca theefoe wite as: dh ( si ( ( ϕ + ϑ θ si ( ( ϑ θ dε ( ( ( = Lim = dh ( si ( ϑ θ si ( ϑ + ϕ θ + θ si ( ϑ + ϕ θ cos ϑ + ϕ θ si ( ϑ θ cos ϑ θ ε dε Theefoe as ε, the local maximum poit fo U = u appoaches poit. Theefoe u at poit has a local maximum. Fo φ = π, the bouday passig though poit is a staight lie ad is theefoe calculated as: ( si ( ( si ( ( ϑ θ + θ ϑ θ ( ϑ θ ε π + ϑ θ ϑ θ = Lim = 3 ( si ( ( si ( cos (B-5 Boudaies fomig egula cuves passig though poit ae cosideed to cosist of ifiite umbes of ifiitesimal staight lie segmets. Fo ay poit o such ifiitesimal lie segmets,

7 the above agumet is applicable. The uit vecto is cosideed to be omal to boudaies fomig a staight lie ad/o a egula cuve. Ackowledgmets. This wok was suppoted i pat by the IST pogam of the Euopea Commuity, ude the PASCAL Netwok of Excellece, the IST ad PiView poject with gat umbe 1659

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