geometric correction

CHAPTER 4A. DERMANIS: REMOTE SENSINGFigure 4-2: The image to map transformationThe image-to-map registration (fig. 4.2) would be realized in this case bythe combined transformations( x,y)= χ−1−1ψ ( ( i,j)), ( i,j)= ( ψ ( x,y))χ . (4)If the projective transformations for two images a and b are( i a , j a ) = χ a ( λ,φ)and ( i b , j b ) = χ b ( λ,φ)respectively, the image-toimageregistration would be realized by the combined transformations( i , jbb−1−1χ ( ( i , j )), ( i , j ) = ( χ ( i , j ))) = χbaaaaaχ . (5)We shall restrict from know on the discussion to the image-to-map registrationproblem, bearing in mind that the treatment of the image-to-imageregistration is completely analogous.Between the two transformations in (4) the one we need is in fact thesecond, i.e. the map-to-image transformation (fig. 4.3). Indeed our purposeis not to find where the image pixels are located on the map, butrather where the pixels of a “digitized” map are located on the image. Inthis way values for the map pixels can be assigned on the basis of thevalues of the image pixels and the digital map can be realized in the formof a new digital file, the “registered to the map” image.−1In our search for the transformation ( i,j)= χ ( ψ ( x,y)) the cartographicprojection transformation ψ is always well defined, since it is a matterof choice from a set of available such mathematical transformations. Thisis not the case however for the image projective transformation χ ; it dependson the sensor position and orientation, it is affected by their temporalvariation during the time interval required for the observation ofthe image and furthermore involves the heights h of points, i.e., it hasthe form ( i,j)= χ(λ,φ,h). For this reason a simpler “empirical” approachis followed. The unknown transformationabbb−1χ ψ is replaced by2