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EVALUATION AND PRACTICAL ISSUES OF SUBPIXEL IMAGE REGISTRATION USING PHASE CORRELATION METHODS

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Fabrice Humblot1,2 , Bertrand Collin1 and Ali Mohammad-Djafari2 1

DGA/DCE/CTA/DT/GIP

2

Centre Technique d’Arcueil

LSS (CNRS-SUPELEC-UPS) ´ ´ Ecole Sup´erieure d’Electricit´ e

94114 Arcueil, FRANCE.

91192 Gif-sur-Yvette, FRANCE.

[email protected]

[email protected] [email protected]

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1st February, 2005 – PSIP 2005 – Toulouse 1

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Contents • Principle of the Phase Correlation Method • Extension to Subpixel Registration • Extension Using Image Gradients • Registration Equation • Numerical Evaluation • Evaluation with Real Unknown Data • Conclusions &

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Principle of the Phase Correlation Method [Kuglin & Hines, 1975] I2 (x, y) = I1 (x − x0 , y − y0 ) Iˆ2 (u, v) = Iˆ1 (u, v)e−j(u×x0

+ v×y0 )

Iˆ2 (u, v)Iˆ1∗ (u, v) −j(u×x0 = e |Iˆ2 (u, v)Iˆ∗ (u, v)|

+ v×y0 )

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δ(x − x0 , y − y0 ) &

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Extension to Subpixel Registration [Foroosh, Zerubia & Berthod, 2002] Iˆs1 (u, v) =

1 MN

M −1 N −1 X X

Iˆ

u+2πm v+2πn , N M

m=0 n=0

Iˆs2 (u, v) =

1 MN

M −1 N −1 X X

Iˆ

u+2πm v+2πn , N M

m=0 n=0

C(x, y) = IFT &



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e−j(

∗ (u,v) Iˆs2 (u,v)Iˆs1 ∗ (u,v)| |Iˆs2 (u,v)Iˆs1




u+2πm x0 , v+2πn y0 M N

)



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Extension to Subpixel Registration (2) C(x, y) ≈

∆x

=

x0 M

sin(π(M x−x0 )) sin(π(N y−y0 )) π(M x−x0 ) π(N y−y0 )

=

C(xs ,yp )xs ±C(xp ,yp )xp C(xs ,yp )±C(xp ,yp )

xs = x p ± 1 ys = yp ± 1

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Extension Using Image Gradients [Argyriou & Vlachos, 2003]

Gh (x, y) = I(x + 1, y) − I(x − 1, y)

Gv (x, y) = I(x, y + 1) − I(x, y − 1)

G = Gh + j × G v &

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Registration Equation Bilinear Approximation dx

x y

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dy

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  x = X + d (X, Y ) ∈ Z2 0 x  y0 = Y + dy (dx , dy ) ∈] − 1; 1[2 7

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Registration Equation Bilinear Approximation (2)

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I1 (x, y) = I0 (x − X, y − Y ) avec (X, Y ) ∈ Z2

I2 (x, y) =

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                            

(1 − |dx |)(1 − |dy |)I1 (x, y) + |dx |(1 − |dy |)I1 (x − sign(dx ), y) + (1 − |dx |)|dy |I1 (x, y − sign(dy )) + |dx ||dy |I1 (x − sign(dx ), y − sign(dy )) 8

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Numerical Evaluation Accurate Case of an Analytical Function

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  xˆ = 1.340 0  yˆ0 = 1.847

  x = 1.350 0  y0 = 1.850 9

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s[i, j] =

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Z

i+0.5 i−0.5

Z

j+0.5

h0 (u, v) du dv j−0.5

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Error on dx Estimations

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Neglicence of Integration Feature

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Simulated Movement Using a Real Image

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  x = 10.30 0  y0 = 12.60 13

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Simulated Movement Using a Real Image (2)

  x = 10.30 0  y0 = 12.60

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  xˆ = 10.16 0  yˆ0 = 12.69 14

  xˆ = 10.17 0  yˆ0 = 12.70

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Evaluation with Real Unknown Movement

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Evaluation with Real Unknown Movement (2)

  x =? 0  y0 = ?

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  xˆ = −0.07 0  yˆ0 = −0.12 16

  xˆ = 9.51 0  yˆ0 = 2.48

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Conclusions • Quick process thanks to FFT (1 sec. for 220 × 320 pixels2 image, with 500 MHz Intel Celeron processor) • Image’s gradients method: more efficient. Works well with images that have faint and badly drawn areas. • With a good evaluation model, maximum error committed is 2.4% (1st method), and 1.3% (2nd method). • The best to reconstruct the unknown area of a registered image is to take this area identically in the reference image. • Can be used for a local purpose, on sections of images, choosing preferably side lengths that are power of 2 (FFT). &

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References [Kuglin & Hines, 1975] C. D. Kuglin and D. C. Hines, “The Phase Correlation Image Alignment Method,” in Proc. IEEE 1975 Int. Conf. Cybernetics and Society, September 1975, pp. 163–165. [Foroosh, Zerubia & Berthod, 2002] H. Foroosh, J. Zerubia, and M. Berthod, “Extension of Phase Correlation to Subpixel Registration,” IEEE Trans. Image Processing, vol. 11, no. 3, pp. 188–200, 2002.

[Argyriou & Vlachos, 2003] V. Argyriou and T. Vlachos, “Sub-Pixel Motion Estimation Using Gradient Cross-Correlation,” in The 7th International Symposium on Signal Processing and its Applications (ISSPA), Paris, 1–4 July 2003.

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