geometric correction

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CHAPTER 4A. DERMANIS: REMOTE SENSING2is12 2= ∑ ( ik− mi) ≡ ( si) , (34)nks2j12 2= ∑ ( jk− m j ) ≡ ( s j ) , (35)nksxy1= ∑ ( xk− mx)(yk− my) ≡ sxsyρnkxy, (36)sxi1= ∑ ( xk− mx)(ik− mi) ≡ sxsiρxi, (37)nksyi1= ∑ ( yk− my)( ik− mi) ≡ sysiρyink(38)sxj1= ∑ ( xk− mx)(jk− m j ) ≡ sxsjρxj , (39)nksyj1= ∑ ( yk− my)( jk− m j ) ≡ sysjρyj. (40)nkFor the rotation-scaling-translation model⎡i⎤ ⎡ a⎢ ⎥ = ⎢⎣ j⎦⎣−bb⎤⎡x⎤⎡ti⎤⎥⎢⎥ + ⎢ ⎥ , (41)a⎦⎣y⎦⎣t j ⎦a = s cosθ , b = ssinθ(42)the least squares solution is given bys + saˆ= , (43)s + sxi2xyj2yˆs − sb = , (44)s + syi2xxj2ytˆi= m − aˆm − bˆm , (45)ixytˆ= m + bˆm − aˆm(46)jjxy8
Image registration (geometric correction)ˆvik= i − aˆx − bˆy − tˆ, (47)kkkiˆv j k= j + bˆx − aˆy − tˆ, (48)kkkjb syi− sxjtanθ ˆ = ˆ= , (49)a ˆ s + sxiyj2 2 122sˆ= aˆ+ bˆ= ( sxi+ syj) + ( syi− sxj) . (50)s + s2x2y4.2. ResamplingFigure 4-4The transformation from the new image or map coordinates ( x,y)to theoriginal image pixel coordinates ( i , j), established as described in theprevious section, can be implemented in order to produce a new“registered” or “geometrically corrected” digital image, whichcorresponds to either the “target image” in image-to-image registration,or to the particular cartographic projection in image-to-map registration.The center of every map pixel has integer valued coordinates ( X , Y )which the established transformation i = i( x,y), j = j( X , Y ) maps (fig.4.4) into values I = i( X , Y ) , J = j( X , Y ) , which are not integer but realnumbers in general. We need therefore to evaluate values =V i( X , Y ). j(X , Y ) = VX. YV I . J= for all the map pixels or equivalently for all theresulting positions ( I,J ) on the original image, from the available valuesV i , j at the pixel centers ( i , j). This can be achieved with the help of aninterpolation of the available grid values V i , j , which is calledresampling.Figure 4-5: Closest neighbor interpolationThe simplest interpolation-resampling method is the method of the near-9
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geometric correction

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