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  • Lecture 7 - 1 - 11/5/2003 ____________________________________________________________________________________

    Concept Hell/Pfeiffer February 2003 ___________________________________________________________________________________

    7. Differential rectification 2 hours Aim: The need for differential rectification Theory: Indirect rectification ___________________________________________________________________________________

    7.1. Principles of differential rectification

    7.1.1. Mathematical principle The main idea of differential transformation is based on the transformation of slit of the source

    image into the slit into orthoimage. The parameters that define this transformation are relative

    positions of centers or two slits (small rectangular areas with very large ratio between length and

    height), the angle orientation and change in scale. All this parameters can be obtained by the

    coordinates of the center of the slit, and coordinates of middle points of opposite short sides. The

    calculation of these parameters is based on transformation of quadrilateral grid between two

    images [Kraus K., 1993]. The transformation of every cell of such grid (and arbitrary point inside

    it) is described by bilinear transformation

    01 11 21 31

    02 12 22 32

    a a x a y a xa a x a y a x


    = + + += + + +


    7.1.2. Instrumental realization The instrumental realization of this transformation could take several forms. More often used is

    optical transfer with digital image control. The plane of the image is separated into vertical

    strips. The width of the strips depends of the height diversity of the terrain. The criteria of such

    choice is the limit of height difference inside the slit. The set of profiles usually vertical is

    produced. The nearest points on the adjacent profiles are used to create rectangular grid. For this

    grid the x, h coordinates in input image are calculated. All this data are recorded.

    During the rectification a very narrow slit of length S, corresponding to the step between profiles

    is moved in y-direction over the xy-plane of the orthophhoto. During this continuous movement

    the corresponding line element from , plane is projected onto xy-plane.

    The line element is transformed by:

    1. Two translations of its center ( ' ; )C C

    2. Rotation through angle ; ____________________________________________________________________________________ FH-KA - Master course Photogrammetry 2003 B. Marinov

  • Lecture 7 - 2 - 11/5/2003 ____________________________________________________________________________________

    3. Change of scale ( ' )S S

    Original photograph Orthophoto (transformed image)

    Figure 7.1. Transformation of line element

    The following computation are made in real time during the process of scanning

    1. Determination the coordinate of center of the center of each line element c, c

    2. Rotation angle arctan


    3. Scale factor k - 2 2'S

    S Sk + = =

    After one raster grid quadrilateral is transformed, the next in the same vertical strip is transformed

    too. After finishing one strip the next neighbor strip is transformed in the same way.

    This principle is adopted in Wild Avioplan OR1 of Leica Heerbrugg and Orthocomp Z2 of Zeiss


    7.1.3. Technical realization of differential transformation principle The functioning of the instrument Avioplan OR1 is described. Its schematic diagram is shown


    ____________________________________________________________________________________ FH-KA - Master course Photogrammetry 2003 B. Marinov

  • Lecture 7 - 3 - 11/5/2003 ____________________________________________________________________________________

    Figure 7.2. Digital control of transformation

    The kernel of the system is control computer. Input data for real time processing are coordinates

    of vertical profiles in photo plane. These data are the ends x, h of line elements in the photograph.

    Computer calculates necessary elements for transformation of the strip (translation x, h, rotation

    angle a, and scale factor). These calculations are made in synchronisation of drum rotation that

    corresponds to the movement in y direction. The increment of motion in y-direction is 10m.

    The transformation data are calculated at every step of rotation. The numerical values control:

    1. shift of the photograph in two directions

    ____________________________________________________________________________________ FH-KA - Master course Photogrammetry 2003 B. Marinov

  • Lecture 7 - 4 - 11/5/2003 ____________________________________________________________________________________

    2. the angle of the Dove prism;

    3. The scale factor to operate the zoom lens.

    Only the light from small area of the photograph through slit mask passes to the film over the

    surface of drum and exposure of line elements one after another.

    7.1.4. Technical data of differential rectifiers Avioplan OR1.

    Slit parameters are: length from 3mm to 16mm in step of 1mm. The width of the slits is 0.1mm

    for monochromatic film and 0.3mm for colour.

    Angle of rotation =94 gon.

    Range of zoom lens 0.26 to 15.

    Maximum speed of orthophoto film 30mm/s.

    Maximum orthophoto format 90cm x 75cm.

    Figure 7.3. Wild Avioplan OR1

    The principle of operation of Zeiss Orthocomp Z2 is the same but it uses more powerful

    computer that allows more of the computation to be performed in the process computer not in the

    external computer like it is in Wild system.

    7.2. Differential rectification of different types of distorted images

    7.2.1. The deformation due to the terrain height

    ____________________________________________________________________________________ FH-KA - Master course Photogrammetry 2003 B. Marinov

    The main application of differential rectification is rectification with account of terrain relief.

  • Lecture 7 - 5 - 11/5/2003 ____________________________________________________________________________________

    Figure 7.4. Deformation due to the terrain heights

    The calculations over vertical profiles allow estimating the deformations introduced in digital

    image by height differences and by projective deformation of image.

    7.2.2. The rectification of tilted photographs Differential rectification has an advantage in respect to perspective transformation, when it is

    used for large angles of tilt more than 15gon.

    For such images the transformation procedure is as follow [Kraus K., 1993]. The interpolation

    grid in the plane XY of transformed image is defined. The perspective grid in xh-plane can be

    computed from colinearity equations by setting Z=0 and if parameters of orientation are known. ____________________________________________________________________________________ FH-KA - Master course Photogrammetry 2003 B. Marinov

  • Lecture 7 - 6 - 11/5/2003 ____________________________________________________________________________________

    The could be determined as a result of special resection with known control points, by stereopair

    processing or after block adjustment.

    The accuracy of differential transformation is the same as if perspective rectification is applied.

    Some interpolation error can be estimated because of the different method for determination of

    centers of grid sides by bilinear interpolation and perspective transformation.

    The example of such application is transformation of tilted at about 39gon plane image, that can

    not be transformed by perspective rectification [Kraus, K., 1993].

    Figure 7.5. Differential rectification of tilted plane

    Differential transformation suggests elegant solution in the case of combination between affine

    and perspective transformation. The task is to produce orthogonal projection of tilted object over

    the horizontal XY-plane from tilted photograph (central perspective projection).

    The steps of computations are:

    1. Definition of square grid in XY-plane

    2. Projected of coordinate grid over the tilted plane (calculation XYZ-coordinates of the grid


    3. Central projection of the object point with their 3D coordinates by colinearity equatio