Do We Really Need Gaussian Filters for Feature Detection? (Supplementary Material)

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1 Do We Relly Need Gussin Filters for Feture Detection? (Supplementry Mteril) Lee-Kng Liu, Stnley H. Chn nd Truong Nguyen Februry 5, 0 This document is supplementry mteril to the pper submitted to EUSIPCO 0. Lemm. The derivtives of the Gussin functions re g (t;σ) ( t σ Proof. Direct clcultion. Normlized Derivtive. e g(t;σ) e σ σ ) nd g (t;σ) ( σ σ σ)e. Scle-spce xiom requires tht there is no enhncement to the locl mximum nd locl minimum [. Therefore, the derivtive must be normlized. Defining the mth order γ normlized derivtive opertor s σm,γ (σ ) mγ t m, we tke γ nd m, nd pply the opertor to g(t;σ) to yield σ,[g(t;σ) (σ ) t [g(t;σ) σ g (t;σ). Therefore, in the pper, insted of using g(t;σ) we use σ g(t;σ). In this cse, σ g (t;σ) ( π σ 3 t σ )e σ.

2 Definition. Let T > 0 be constnt. The signl I(t) is defined s {, T t T I(t) 0, otherwise. () Definition. The noise n(t) is i.i.d. Gussin with zero men nd vrince σn, denoted by n(t) N(0,σN ). Proposition. Letting v(t,σ) σ g (t;σ) I(t), nd w(t,σ) σ g (t;σ) n(t), it holds tht Hence, v(t,σ) [ t+t (t+t) π σ e σ + t T (t T) σ e σ, () E[w(t,σ) 0, (3) E[w(t,σ)w(t +τ,σ) σ σ N g ( τ,σ ) g (τ,σ ). () Vr[w(t,σ) E [ w(t,σ) 3σ N 8 πσ. (5) Proof. For simplicity we drop the vrible σ. First, note tht I(t) u(t+t) u(t T), where u(t) is the unit step function. Consequently, v(t;σ) σ g (t) I(t) σ g (τ)i(t τ)dτ ( τ σ σ [ τ π t+t t T π t+t t T ) σ τ e σ [u(t+t τ) u(t T τ)dτ σ 3 σ [ τ σ 3 [ t+t (t+t) π σ e σ Tht the men is zero is due to linerity of convolution: [ E[w(t) E σ g (τ)n(t τ)dτ e τ σ dτ e τ σ dτ π t+t t T + t T (t T) σ e σ. τ σ e σ dτ σ g (τ)e[n(t τ)dτ 0.

3 The utocorreltion of the output is [ E[w(t)w(t +γ) E σ g (τ )n(t τ )dτ σ g (τ )n(t+γ τ )dτ σ g (τ )g (τ )E[n(t τ )n(t+γ τ )dτ τ σ g (τ )g (τ )σn δ( τ γ +τ )dτ dτ σ σ N σ σng (τ ) g (τ )δ(τ (τ +γ))dτ dτ }{{} g (τ )g (τ +γ)dτ σ σ N g ( γ) g (γ). Since E[w(t) 0, we hve Vr[w(t) E[w(t) E[w(t)w(t + γ) γ0. Therefore, Vr[w(t) σ σ N g ( τ) g (τ) τ0. The convolution is g (τ) g ( τ) Putting τ 0 yields [g (τ) g ( τ) τ0 π (γ σ σ )e ( γ τ e σ γ σ σ +τ) σ)((γ σ σ )e ( γ γ γ σ8e γ γ πσ 5. ( (γ +γτ) σ 8 σ 8 γ σ 6 + σ )e σ 8e σ dγ π σ dγ π [ ( )!! σ 8 (+) ( σ ) ((γ +τ) σ σ )e γ +γ +γτ+τ σ dγ (γ+τ) σ dγ γ (γ +τ) σ6 σ 6 + γ +γτ σ )e σ dγ. γ σ dγ γ σ 6 e γ σ dγ + π γ γ σ 6e πσ σ 6 σ dγ + π πσ6 + πσ σ γ σ σe dγ γ σ e σ dγ Therefore, Vr[w(t) σ σ Ng ( τ) g (τ) τ0 3σ N 8 πσ. 3

4 Proposition. Given T, v(0,σ) is locl minimum of v(t,σ) if nd only if σ > T/ 3. Furthermore, the optiml scling prmeter σ for which v(0,σ)/ σ 0 is σ T. Proof. By KrushKuhnTucker optimlity condition, v(0, σ) is locl minimum of v(t, σ) if nd only if v (0) 0 nd v (0) > 0. The first order derivtive is v(t, σ) t [ t0 π σ e (t+t) σ [ π σ e (T) σ ) ( (t+t) σ 3 ) ( (T) σ 3 e (t+t) σ e (T) σ + e ( T) σ σ + + σ e (t T) σ + ) ( ( T) thus the first order criteri is stisfied. The second order derivtive is v(t,σ) t0 t [( t+t π σ 3 + (t+t) ) σ 3 (t+t)3 σ 5 e (t+t) σ [( t T π σ 3 + (t T) σ [(3T T3 [3T T3 σ σ ) e T σ + e T σ Therefore, v (0) > 0 if nd only if 3T T3 σ > 0, which is T σ < 3. ) (t T)3 σ 5 (3T T3 σ σ 3 ) ( (t T) t0 e (t T) σ ) e T σ σ 3 e ( T) σ 0, t0 e (t T) σ The second sttement cn be derived by considering the first order optimlity condition of v(0, σ) over σ, i.e., find σ such tht v(0,σ) σ 0. Since v(0,σ) [ T T e σ π σ, t0 we hve v(0, σ) σ T [ ( )σ e T σ +(σ )( ) T π T [ T π σ σ e T σ T T σ 3 e σ [ T σ e T σ. Setting v(0,σ) σ 0 yields σ T.

5 Proposition 3. Given σ, nd let C rgmin C [ σ g (t;σ) h (t;σ) dt, then C ασ where α 0.9 is constnt. Proof. We let f(t) def σ g (t) ( t π σ 3 ) e σ σ, nd ˆf(t) def h (t) 6C [u(t+3c) u(t+c) 6C [u(t+c) u(t C)+ 6C [u(t C) u(t 3C), where u(t) is the unit step function. Our gol is to find C such tht it minimizes the residue C rgmin C where f(t) ˆf(t) f(t) ˆf(t) L, (6) L [f(t) ˆf(t) dt. Observe tht f(t) ˆf(t) f(t) f(t), ˆf(t) + ˆf(t). L We now investigte ech term individully. ˆf(t) C 3C (6C) [U(t+3C) U(t 3C)dt C +3C 36C (6C) [U(t+C) U(t C)dt+ 36C dt+ + C 36C 3C. C C (C +C) 36C 36C dt+ + 3C C 36C 3C C 36C dt (6C) [U(t C) U(t 3C)dt Before we clculte f(t), ˆf(t), we clculte b f(t)dt in dvnce. Due to the unit step function, 5

6 the integrl will be in certin intervl [, b. b f(t)dt b ( t π σ 3 ) e σ σ dt [ b t t π σ 3e σ dt b e σ σ dt [ t [ b e σ π σ + b π t σ e σ dt π [ b [ t b e σ π σ [ b b e σ π σ + e σ σ t σ e σ dt Using previous result, we cn simplified the following eqution. f(t), ˆf(t) f(t)ˆf(t)dt C f(t)dt C f(t)dt+ 3C f(t)dt 6C 3C 6C C 6C C [ C 6C C π σ e σ 3C [ C σ e 9C σ 6C C π σ e σ C σ e 9C σ [ + 6C 3C π σ e 9C σ + C C e σ σ [ 6C 6C C π σ e σ 6C σ e 9C σ [ e C σ e 9C σ. Finlly, we hve f(t) f (t)dt ( t π σ 3 σ ( t σ [ t σ dt σe ) e σ dt ) σ + e σ dt ( t π σ 6 σ + ) σ e σ dt t e σ σ dt+ [ σ t e σ dt 0 σ t e σ dt+ 0 [ ( )!! πσ σ + 0 πσ6 σ + πσ [ 3 σ 8 σ5 π σ3 π + πσ σ 3 8 πσ e σ dt e σ dt Therefore, f(t) ˆf(t) 3C [ e C σ e 9C σ πσ. 6

7 Given vlue C, we cn find the locl minim by considering the first derivtive of of the cost function ε(σ) (f(t) ˆf(t) with respect to σ. ε(σ) σ [ σ 3C ( ) e C σ e 9C σ πσ 0 [ σ e C σ +σ ( C π + [ σ e 9C σ +σ ( 9C π )( )σ 3 e C σ )( )σ 3 e 9C σ +( σ 3 8 π ) [ (C σ σ ) e C σ ( 9C σ σ ) e 9C σ 3σ π 8 π. No nlyticl solution exists for solving ε(σ) σ 0. However, numericl results suggest tht there exists liner reltionship between C nd σ, shown in Fig.. 5 Optiml σ given C σc/0.98 Fixed vlue C nd optiml sigm 0 optiml σ Fixed Vlue C Figure : Liner reltionship between C nd σ is found with the rtio σ C

8 Proposition. Letting v(t) h (t;σ) I(t), nd w(t) h (t;σ) n(t), then 3, t 0, v(t) 3 t C, 0 < t C, nd Vr[w(t) 3 + t 6C, C < t C, σ N 3C. Proof. SinceE[w(t) 0,wehveVr[w(t) E[w(t) E[w(t)w(t+γ) γ0. Therefore,Vr[w(t) σ N h( τ) h(τ) τ0. The convolution is the summtion of squre of mplitude times the intervl Therefore, the Vr[w(t) σ N 3C. h( τ) h(τ) τ0 C (6C) + C (6C) + C (6C) (7) C 36C 3C. (8) References [ Tony Lindeberg. Feture detection with utomtic scle selection. Interntionl Journl of Computer Vision, 30:79 6,