Like analog plotting, automatic correlation cannot do everything. It may be impossible to correlate some details (presence of clouds, concealed slopes, shade, land occupancy too different on the two images, etc.).
The purposes of smoothing are:
· to correct incorrectly plotted portions, by continuity: this is elimination of defective matching;
· filling portions in which matching was impossible (several correlation peaks, similarity index too weak): this is plotting of missing parts.
Several procedures may be used, including the "elastic grid" method developed by G. de Masson d'Autume (1).
3.3.1 "Elastic grid" method
This method consists of adjusting z(i,j) values corresponding to the equilibrium position of a set of rigid rods elastically hinged to each other, onto known values of Z(i,j) some of which may be missing.
Its principle is to introduce a uniformity constraint onto the surface being searched, with the definition of a minimum curvature.
Break lines can also be introduced.
Mathematically, z is the solution of the minimization of energy problem.
qij >0, rij >0 in general, null on a break line,
pij >0 if Z(i,j) exists, otherwise null.
E(z) can be written in matrix form:
E(z) = zT A z - 2 ZT P z + ZT P Z
= zT A z - 2 bT z + bT
Z
where P = diagonal matrix pij,
b = PZ,
A = positive defined matrix (obtained from weighting matrices
P, Q, R).
A is of the form CTQC + DTRD + P.
C and D are matrices built with blocks starting from the 3 uniformity coefficients 1, -2,
1.
The identity: E(z) = (z-A-1b)T*A*(z-A-1b) - bTA-1b
+ bTZ
shows that E(z) reaches its minimum value -bTA-1b + bTZ
when Az = b, which can be solved by an iterative method.
Top of the page
3.3.2 Interactive corrections
Despite everything, the automatic smoothing process cannot eliminate all sources of
errors. This is why an interactive checking and correction tool is necessary to obtain a
good result (display of curves superposed on anaglyph images, stereoscopic vision of
epipolar images, etc.).
These corrections will consist of:
· putting water surfaces flat;
· eliminating obviously incorrect measurements under clouds, and replacing them by
continuity;
· smoothing areas which have low correlation coefficients (usually concealed parts of
some slopes or radiometric inversions) taking account of characteristic lines.
Top of the page
3.4 Calculating the DEM
3.4.1 Use of parallax
Orientation of the pair reconstitutes the exact positions of perspective bundles when the
view was taken. Therefore, the ground coordinates (x,y,z) of the intersection of
perspective rays can be calculated for each pair of corresponding points. In this way, an
irregular grid in (x,y) is obtained.
Top of the page
3.4.2 Calculating a regular grid
A regular grid (x,y), which is the required DEM, is obtained by inversion of this
irregular grid (interpolation between the nodes).
Top of the page
3.5 Conclusion
The ideal correlator must make all possible matches successfully, and must not create any
false correlations !
Problems:
· can we be sure that a match is correct, once it has been found?
These corrections will consist of:
· would the result have been better if some parameters had been set differently?
· are all non-matches obtained from points that were impossible to correlate?
It also has a better mapping generalization type of smoothing, capable of eliminating high frequency residuals.
