D.6 Diferença de cores através do HSB
D.6.2 Análise do ângulo de abertura θ
cos(θ) = ITl (t)Bl(t) ||Il(t)||||Bl(t)|| = (sI(t) + vI(t))T(sB(t) + vB(t)) ||sI(t) + vI(t)||||sB(t) + vB(t)|| = sTI(t)sB(t) + sTI(t)vB(t) + vTI(t)sB(t) + vTI(t)vB(t) q s2 I(t) + v2I(t) q s2 B(t) + v2B(t) . (D.8)
sTI(t)sB(t) = (sIcos(hI)ux+ sIsin(hI)uy)T(sBcos(hB)ux+ sBsin(hB)uy)
= sIcos(hI)uTx(sBcos(hB)ux+ sBsin(hB)uy) +
sIsin(hI)uTy(sBcos(hB)ux+ sBsin(hB)uy)
= sIsB
cos(hI) cos(hB)||ux||2+ sin(hI) sin(hB)||uy||2
+
sIsB(cos(hI) sin(hB) + sin(hI) cos(hB)) uTxuy
= sIsB(cos(hI) cos(hB) + sin(hI) sin(hB))
= sIsBcos(hI− hB), (D.9) sTI(t)vB(t) = (sIcos(hI)ux+ sIsin(hI)uy)TvBuv = sIvBcos(hI)uTxuv+ sIvBsin(hI)uTyuv = 0, (D.10) vTI(t)sB(t) = sTB(t)vI(t) = (sBcos(hB)ux+ sBsin(hB)uy)TvIuv = sBvIcos(hB)uTxuv + sBvIsin(hB)uTyuv = 0, (D.11) vTI(t)vB(t) = vIvBuTvuv = vIvB. (D.12) Substituindo (D.9)-(D.12) em (D.8) tem-se cos(θ) = sIsBcos(hI− hB) + vIvB q s2 I+ v2I q s2 B+ v2B = sI q s2 I + v2I sB q s2 B+ vB2 cos(hI− hB) + vI q s2 I + v2I vB q s2 B+ vB2
= sin(ϕI) sin(ϕB) cos(hI − hB) + cos(ϕI) cos(ϕB)
= 12(cos(ϕI − ϕB) − cos(ϕI + ϕB)) cos(hI− hB) +
1
2(cos(ϕI + ϕB) + cos(ϕI− ϕB))
= 12(cos(ϕI − ϕB) cos(hI− hB) − cos(ϕI+ ϕB) cos(hI − hB) + cos(ϕI+ ϕB) + cos(ϕI − ϕB))
= 12(cos(ϕI − ϕB) (1 + cos(hI − hB)) + cos(ϕI+ ϕB) (1 − cos(hI− hB)))
= cos(ϕI− ϕB) cos2 hI− hB 2 ! + cos(ϕI+ ϕB) sin2 hI− hB 2 ! . (D.13)
Definindo as diferenças angulares ϕI − ϕB e hI− hB como
∆ϕ = ϕI − ϕB (D.14)
e substituindo-as em (D.13), obtém-se
cos(θ) = cos(∆ϕ) cos2 ∆h
2 ! + cos(2ϕB+ ∆ϕ) sin2 ∆h 2 ! . (D.16)
A Equação (D.16) depende das variáveis ϕB, ∆ϕ e ∆h, onde
• o valor que ∆ϕ assume deve-se: (a) apenas à variação da saturação (ver Figura D.8.a), (b) apenas à variação do brilho (ver Figura D.8.b) ou (c) à variação conjunta tanto da saturação como do brilho (ver Figura D.8.c);
• por outro lado, o valor que ∆h assume deve-se à variação das matizes. Portanto, esta variação será maior se a relação das cores dos píxeis Il(t) e Bl(t) for mais de caráter
complementar (ver Figura D.9.a) que análogo (ver Figura D.9.b); tendo um valor de zero se a relação é monocromática (ver Figura D.9.c).
(a) (b)
(c)
Figura D.8: Possíveis casos que dão origem ao ∆ϕ: (a) devido unicamente a uma variação da saturação;(b) devido unicamente a uma variação do brilho;(c)a uma variação conjunta tanto da saturação como do brilho.
Portanto, em termos gerais pode-se dizer que o cos(θ) depende das componentes de brilho e cromaticidade dos píxeis Il(t) e Bl(t). Agora se é considerada, em forma restrita, a
problemática da detecção de variações, onde para o caso de um pixel Il(t) pertencente
ao fundo ele fica numa pequena vizinhança centrada em Bl(t), como é representado na
(a) (b) (c)
Figura D.9: Relações dos matizes: (a) complementarias;(b) análogas;(c) monocromáticas.
(uma representação gráfica é apresentada na Figura D.10.b), e, assim, a Equação (D.16) é aproximada por cos(θ) ≈ cos2 ∆h 2 ! + cos(2ϕB) sin2 ∆h 2 ! = 1 − (1 − cos(2ϕB)) sin2 ∆h 2 ! = 1 + 4 sin(ϕB) sin ∆h 2 !!2 . (D.17)
A Equação (D.17) indica que para o caso de detecção de variações, e considerando um pixel
Bl(t) constante, o cos(θ) dependerá somente do valor da diferença de matizes, ∆h. Assim,
as componentes de cromaticidade de Il(t) e Bl(t), especificamente a diferença de matizes,
∆h, define o valor de cos(θ).
(a)
(b)
Figura D.10: Píxeis Il(t) e Bl(t) considerado a problemática da detecção de variações (a)
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