Gaussian related distributions

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1 Gaussian related distributions Santiago Aja-Fernández June 19, Gaussian related distributions 1. Gaussian: ormal PDF: MGF: Main moments:. Rayleigh: PDF: MGF: Raw moments: Main moments: px = 1 σ π exp x µ σ R = M X t = exp µt + σ t Mean µ Median µ Mode µ Variance σ X 1 + X X i 0, σ px = x σ exp x σ M X t = 1 + σte σ t / π σt erf + 1 µ k = σ k k/ Γ1 + k/ Mean σ π Median σ log 4 Mode σ Variance 4 π σ 1 π µ 1 = σ µ = σ π µ 3 = 3 σ3 µ 4 = 8σ 4 1

2 3. Rician: R = X 1 + X X i A i, σ PDF: where A = A 1 + A. Raw moments: R = X X = A 1, σ + ja, σ p M M A, σ = M M +A σ e σ where L n x = M n, 1, x = 1 F 1 n; 1; x. Main moments: AM I 0 σ µ k = σ k k/ Γ1 + k/l k/ A σ Mean Median Mode Variance π µ 1 = L 1/ π µ 3 = 3 L 3/ σ π L 1/ A σ σ + A πσ L 1/ A σ A σ A σ σ µ = A + σ σ 3 µ 4 = A 4 + 8σ A + 8σ 4 um, 3 π µ 1 = L 1/ x σ µ = σ 1 + x π µ 3 = 3 L 3/ x σ 3 µ 4 = 4σ 4 + 4x + x Var = A 4 + 8σ A + 8σ 4 σ 1 1 4x 1 8x + Ox 3 with x = A σ Series expansion of Hypergeometric Functions: L 1/ x = x 1 + π π x πx + O x 5/ 3/ L 3/ x = 4x3/ 3 π + 3 x 3 + π 8 π x πx + O x 5/. 3/ 4. Central Chi: on-normalized version R = L Xi X i 0, σ

3 PDF: Raw moments: Main moments: R = L Y i Y i = 0, σ + j0, σ px σ, L = 1 L ΓL x L 1 σ L µ k = σ k k/ Γk/ + L ΓL µ 1 = ΓL + 1/ σ ΓL µ = Lσ µ 3 = 3 ΓL + 3/ σ 3 ΓL µ 4 = 4LL + 1σ 4 Γk/ + L Var = σ L ΓL x e σ 4 5. on central Chi: on-normalized version R = L Xi X i A i, σ PDF: with A L = i A i Main moments: µ 1 = µ 3 = 3 R = L Y i Y i = A 1,i, σ + ja,i, σ pm L A L, σ, L = A1 L L σ ML L e M L +A L AL M L σ I L 1 σ um L, 5 ΓL + 1/ 1F 1 1 ΓL, L, A L σ ΓL + 3/ 1F 1 3 ΓL, L, A L σ σ µ = A L + Lσ σ 3 µ 4 = A 4 L + 4L + 1A Lσ + 4LL + 1σ 4 µ 1 = ΓL + 1/ 1F 1 1 ΓL, L, x σ µ = σ L + x µ 3 = 3 ΓL + 3/ 1F 1 3 ΓL, L, x σ 3 µ 4 = 4σ 4 x + L + 1x + LL + 1 ΓL + 1/ Var = σ 1 + x ΓL σ L 3 8L + 4L + 4x 8x + Ox 3 1F 1 1, L, x with x = A σ 3

4 6. Gamma: PDF: MGF: Main moments: k 1 exp x/θ p x x = x Γkθ k ux 6 M X t = 1 θt k for t < 1/θ Mean kθ Median Mode k 1θ Variance kθ k = µ 1 Var θ = Var µ 1 mode = µ 1 Var µ 1 7. K distribution: PDF: x +1 x p x x = K ux 7 aγ + 1 a a In ultrasound we consider the following parameters: a = 1 b = 4α b E{A } = 4α σ = α 1 and then the PDF becomes: General form of the PDF: p x x = b Γα p x x = α xb K α 1 bx ux 8 with p R x/y a Rayleigh distribution Rσ = a y: and p γ y a Gamma distribution γ + 1, 1: Moments: Main moments for ultrasound: E{x k } = p R x/y = x a y e 0 p γ y = p R x/yp γ ydy x a y ux y Γ + 1 e y Γk/ + 1Γ k/ a k Γ + 1 µ 1 = Γ3/Γα + 1/ σ µ = σ αγα µ 3 = σ 3/ Γ5/Γα + 3/ α 3/ Γα [ Var = σ 1 1 α µ 4 = σ α ] Γ3/Γα + 1/ Γα 4

5 Sample Local Variance Gamma Approximation The biased SLV of an image Mx is defined as VarMx = 1 ηx p ηx M p 1 ηx p ηx Mp 9 with ηx a neighborhood centered in x. If = ηx, we define the random variable V = VarMx with moments E{V } = 1 1 µ µ 1 E{V } = 1 3 [ + 1µ µ µ µ µ µ 3 µ 1 ] VarV = 1 [ µ 3 µ µ µ µ + + 1µ µ 4 ] 1 For the main models: 1. Rayleigh E{V } = σn 1 π 4 σn 1 π 4 σ V = σ4 n π π + O 1. Rice E{V } = 1σ n σ4 n 4 + π π 1 1 4x 1 8x + O x 3 for x = A σ. n σv = 4σ4 n with KL = 1σ n σ V = 1σ4 n 1σ4 n 3. Central Chi E{V } = σ n 1 ΓL + 1/ L ΓL L + 8L Γ L + 1/ Γ 4 Γ4 L + 1/ L Γ 4 L L Γ L+1/ Γ L. 1 x x + O x 3 4 ΓL + 1/ΓL + 3/ 1 Γ + O L 5

6 4. on central Chi E{V } = σ n L 3 8L + 4L + x 8x + O x 3 σ n 1 σv = σ4 n L + O x x σ4 n 1 5. K distribution with E{V } = σ 1 1 K 1 α σ V = σ4 K α + O Γ3/Γα + 1/ K 1 α = 1 1 α Γα K = a + 1 a Γa 4 [ πγa + 1/ Γa a 3πΓa + 1/ Γa π Γa + 1/ 4] On the four cases, the PDF of the SLV may be approximated using a Gamma distribution. The mode of the distribution for each case is: 1. Rayleigh. Rice mode{v } = E{V } σ V E{V } = σn π 1 σn π mode{v } = σ n σ 5π 16π + O1/ π O σ/a 3. Central Chi mode{v } = σ n L Γ L + 1/ 1 Γ + O L KLσ n 4. on central Chi O σ n /A mode{v } = σ n σ n 6

7 5. K distribution mode{v } σ n 4 8K 1 K + 4K1 K 1 1 1/ 3 Combination of Gaussian Variables Let X i A, σ, i = 1,, be a set of Gaussian random variables. 1. Constant added and multiplied: aµ, σ + b aµ + b, a σ. Sum of zero mean Gaussian variables: IID 0, σ1 + 0, σ 0, σ1 + σ 3. Sum of Gaussian variables: IID L L i 0, σi 0, σi µ 1, σ1 + µ, σ µ 1 + µ, σ1 + σ L L L i µ i, σi µ i, 4. Difference of Gaussian variables: IID µ 1, σ1 µ, σ µ 1 µ, σ1 + σ 5. Product of Gaussian variables: IID µ 1, σ1 µ, σ σ i with PDF: px = 1 x K 0 πσ 1 σ σ 1 σ 6. Sum of square zero mean Gaussian variables: IID Xi σ γ/, σ χ σ / with X i σ = 0, σ and χ σ / a chi-square with degrees of freedom. 7. Square root of the sum of square Gaussian variables: IID X1 0, σ + X 0, σ Rσ with Rσ a Rayleigh distribution. X 1 A 1, σ + X A, σ R c A 1 + A ; σ 7

8 with R c A; σ a Rician distribution. L Xi 0, σ χ C L, σ i 1 with χ C L, σ a central Chi distirbution with parameters σ and L. L Xi A i, σ χ A L ; L, σ i 1 with χ A L, L, σ a non central Chi distirbution with parameters σ, L and A L = i A i. 8. Sample variance of Gaussian S = Combination of Rayleigh Variables Let R i σ, i = 1,, be a set of Rayleigh random variables. 1. Sum of square Gaussian variables: with X i σ = 0, σ. Sum of square Rayleigh variables: X i X 1 σ γ, 1 Xi σ γ/, σ Ri σ γ, σ with PDF: p x x = x 1 e x/σ Γσ ux 3. Sample mean of square Rayleigh variables: 1 Ri σ γ, σ χ σ / with PDF: p x x = x 1 Γσ e x/σ ux 4. Sum of Rayleigh variables: R i σ 8

9 with PDF approx: with 5. Sample mean of Rayleigh variables: e x /b p x x = x 1 1 b Γ ux b = σ [ 1!!]1/ ] σ e σ π 4 1 R i σ with PDF approx: p x x = x 1 1 b Γ e x /b ux 6. Square sample mean of Rayleigh variables: 1 R i σ χ b/ γ/, b with PDF approx: p x x = x 1 b Γ e x/b ux 7. Square root of sum of square Rayleigh variables: Ri σ χ, σ with χ, σ a central chi with PDF: p x x = x 1 Γσ e x /σ ux 8. Square root of sample mean of square Rayleigh variables: 1 Ri σ χ, σ / with χ, σ / a central chi with PDF: 9. Local sample variance of Rayleigh variables: p x x = x 1 Γσ e x /σ ux V = 1 Ri σ 1 R i σ 9

10 For x 0 and an even number of degrees of freedom the PDF is p V x = x 1 exp 1 4 α+ x 1 Γσ1 σ 1 ρ γ k=0 with σ1 = σ n, σ σ n π 4 γ = Γ + k Γk + 1Γ k [ σ σ 1 + 4σ 1σ 1 ρ ] 1/ σ 1 σ 1 ρ α + = γ + σ σ 1 σ 1 σ 1 ρ, k γ x and ρ the correlation coefficient, which for large may be approximated as ρ 5 Combination of Rician Variables π 4 16π 4π 0.95 Let R i σ, i = 1,, be a set of Rician random variables. 1. Sum of square Rician variables: Ri A i, σ χ x σ, A σ with χ a noncentral chi square with A = i A i, and PDF px = 1 1/ x σ e x+a xa σ I 1 σ A. Sample mean of square Rician variables: 1 L Ri A i, σ χ with χ a noncentral chi square with PDF px = 1 1/ σ x 1/ e x+a σ A 3. Sum of square Rician variables: approx. L R i A i, σ x σ, A σ with PDF L 1 pt = tl c1 c e t c b c tb 1 I L 1 c b c 1 c with c 1 and c constants, t = x/ LKΩ L and b = K+1, [?]. 4. Square of the sample mean of Rician variables: 1 L R i A i, σ χ xa I 1 σ x c, b c 1 10

11 6 Combination of K Variables Let K i σ, α, i = 1,, be a set of K-distributed random variables. 1. Transformation of K-variables: If then p S x = with T = fr i and R i Rayleigh variables.. Sum of square K variables: with PDF: 3. Sum of K variables: with PDF approximation p S x = p S x = fk i 0 p T x/yp γ ydy M Ki σ, α M+α x ΓMΓαa α+m x M+α 1 M K i σ, α x 1 K α M a M+α a K3M M+α ΓMΓα K α M ux x a K 3 M ux with K 3 M = M 1!! 1/M 7 Logarithmic transformation of RV 7.1 The lograyleigh distribution Let Rσ be a Rayleigh RV with σ parameter, the transformation, la transformacin lrσ = a logrσ + b is a log-rayleigh withpdf with σ a = σ e b a. Main moments p lr x = 1 x aσa e x a e e a σa E{x} = a [ logσ a γ ] where γ is the Euler s constant. E{x } = a 4 [ logσa + b ] a γ + π 6 11

12 MGF: Varx = π a 4 Φ lr ω = σ a jω a Γ j a ω Operation over log-rayleigh 1. Sum of lograyleigh lr i σ then the MGF is Φ P lrω = σa jω a Γ j a ω + 1 The mean is. Sum of two lograyleigh with PDF E{x} = a [ logσ a γ ] lr 1 σ + lr σ p t = 1 e aσa 4 e t/a t/a K 0 σa 7.3 The LogRician distribution If we define the variable Lrx = log Mx = log A + nr σn + n i σn this variable follows a LogRician distribution with PDF p L x = ex σn e e x +A σ Ae x n I 0 σn with x, +. The moments of this distribution are E{x} = 1 Γl 0, A σn + loga 1 E{x } = 1 4 log σn 1 + log σn [ ] Γ 0, l A A σn + log σn + 1 A 4Ñ1 σn 13 σx = 1 ] A Ñ 1 4 σn + [Γ 0, l A A σn + log σn 14 1 A A Ñ 1 log 15 4 σ n σ n with Γ l a, b the lower incomplete Gamma function and Ñ k x = e x i=0 x i [ ] ψi + k + ψ 1 i + k. Γi + 1 1

13 ψx is the polygamma function and ψ 1 x is its first derivative. It is easy to see that 1 σx A σn. The same solution for the variance may be obtained using a Series expansion on eq. 10 Lrx loga + n 1 A + n A n 1 A +... loga + n 1 A σ L r = σ A 7.4 The Log-on Central Chi distribution For parallel acquisitions, if we define the variable Lcx = log M L x = 1 L log Clr x + C li x 16 this variable follows a Log-on central Chi distribution with PDF p Lc x = A1 L L σn e x+1 e e x +A L AL e x σn I L 1 σn with x, +. The moments of this distribution are E{x} = 1 log σn 1 + E{x } = 1 [ log σn 4 ] A L +ÑL σ x = σ n [ Ñ L A L σ n ψl + l=1 A L Lσn F 1, 1 :, 1 + L; A L σn + log σ n ψl + A L Lσn Lσ n ψl 18 F 1, 1 :, 1 + ; A L σ n log σn A L Lσn F 1, 1 :, 1 + ; A L σn ] A L F 1, 1 :, 1 + ; A L [ ] A Ñ L A L σn log L σn 8 About the Coefficient of variation 8.1 CV of a Rayleigh distribution The square coeffiecient of variation CV of a random variable x is defined to be σ n C x = Varx E {x} = E{x } E {x} E = E{x } {x} E {x} 1 If x is a random variable that follows a Rayleigh distribution, then C x = 4 π σ n = 4 π π σ n π 13

14 So, the CV of a Rayleigh distribution does not depend on the value of σ n. In the paper, the value of the sample CV is used to tune the filter. 8. Sample CV of Rayleigh Let R i σn, i = {1,, } be a set of random variables with Rayleigh distribution. Then, if X = 1 Ri 1 Y = R i the sample CV may be defined as Z = X Y Y = X Y 1 3 Both X and Y are Chi-Square random variables. σ X χ n b Y χ The PDF of U = X Z assuming that X and Y are independent i i f U u = Γ σ Γ n b u 1 bu + σn 4 and making b σ n π 4 As Z = U 1 then with mode f U u = Γ π u 1 Γ 4 πu/ f Z z = Γ π z Γ 4 πz + 1/ f Z z Γ 4 z Γ z π/4 mode{f Z z} = 1 4 π 4 π + 1 π π 8.3 CV in a Rician distribution The square CV is Cx = E{x } E {x} 1 = = A [ σ π 4σ n e n 1 + A σn A [ π σ e n 1 + A σn σn + A I 0 + A/σ n I 0 A 4σ n A 4σ n + A σ n + A σ n I 1 A 4σn I 1 A 4σn ] 1 ] 1 14

15 If R = A σ n we can write C = 1 + R π e R [ 1 + R I 0 R + RI1 R ] 1 15