matrices multiplication in our new ARMLSM. A new multi- 
plication symbol is defined in our new ARMLSM: ©, Ny, © X 
means the multiplication of two matrices N, and X where the 
subscript j in array N, is changed according to the current 
corresponding column number in matrix X, and X © N, has 
the same meaning where subscript i in array Nz is changed 
according to the current corresponding row number in matrix 
X. And the new normals are: 
E«dX4-N,OdX--dXON, 2U-N,QX '-X QN, 
(22) 
where E and U, are the same as in equation 14. 
4.2 ARMLSM solution under the variable weight model 
Assuming: 
ESE+BR-S a c= e,;/(nmns) 
Ny=P +Ny—P. pr=3) ) puj/(man2) (23) 
N:=P,+N.—P: Pz=)) paij/(nin2) 
putting equation 23 into equation 22 and moving (e—e)*d Xo, 
((Ny — P1) Q dXo and dXo Q (Nz — P2) into the right hand 
side where dX is the last time corrections, we derived: 
EdX + p1S7 (21, + M1 )*S1d0X + dX SF (212 + A2)^ Sapz — 
U, — N, © X° — X° © N, — (E — 6) + dX° 
—(N, — Pr) © dX° — dX° © (N, — P) 
(24) 
The N, and N; are based on the elements values in equa- 
tions 20 or 21. The normals are diagonallized by premultipli- 
cation with X; and postmultiplication with ST , and consid- 
ering STS = I, we derived: 
D x (S1dX ST = S1UxST m 
S,U,ST — G * (81 X°ST) — S1[(E — ©) * dX0]ST (25) 
where 
gj n Ou)» O2)5)* — (ag) 
dij = € +49gi; 
Assuming hi; = 1/di;: 
dXxy41 — ST(H * SıU457 )S2 (27) 
Equation 27 the iterative formula of the variable weight model 
ARMSLM. Due to $1U&4157 — H*S1U&S1 , so HxS,U,57 
can be used as S1d X? S7 for the next iteration computation, 
need not be computed again. 
When the values of all elements in P, or P? are equal (say 
equal to pi or p», respectively), equation 24 becomes equa- 
tion 16. It means the uniform weight model ARMLSM is the 
special case of the variable weight model ARMLSM. 
4.3 Object space based ARMLSM 
If the ARMLSM is performed in object space, the matching 
grid will be based on the object space coordinate system in- 
stead of the parallaxes grid in image space. Assuming only 
one stereo image pair is used, object space based ARMLSM 
has all the similar formulas except the parallaxes array X? 
and dX need to be replaced by the elevation array Z° and 
dZ, and the E and U; are changed into: 
wi {ei} = les) 
u e.c Ag) 
UM a Hu s 8g y (28) 
1-48 + By 82 dz 8Z 8y 8Z 
Ag — gi(zi,y1) — g2 (22, yz) 
980 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
4.4 Weight models 
The weight model plays an important role in the new 
ARMLSM. Ideally the weight model should reflect the affec- 
tion both of the terrain feature and image intensity functions, 
but the two functions are often inconsistent, it results in the 
difficulties of choice of the weight model. The main factor 
which effects the single point LSM is the image texture in- 
formation [1], so currently our weight model is only relative 
with the image intensity. The variance, differential, gradient, 
entropy can be considered to determine the weights. The 
following is our method of selecting weights: 
1. computing the mean differentials (gz,gy) of differen- 
tials gzi; and gyi; in the matching window around the 
grid node 
2. finding ming;, ming,, maxgz and max gy 
3. determining the weights pz;; and pyi; for each grid 
node point: 
Pyij = Ca  (maxgz — hai) zz TT (29) 
Paij = C1 + (max gy — hyis) max Teen Te 
where pi, p2, C1, C2 are experiential constants, we rec- 
ommend p, = p2 = 100 and C, = C2 = 10, they are 
adjustable based on the image texture and terrain un- 
dulation 
45 New ARMLSM's computation efficiency 
Assuming the matching grid size is n x n, generally the tra- 
ditional MLSM needs about (n)° multiplication operations. 
According to our new ARMLSM, the multiplication opera- 
tions in each iteration are 8n? (actually are 8n® + 3n?, while 
S1dXoST need not be calculated since the second iteration, 
so we simplify to 8n? for convenience), again assuming the 
total iteration is k, the operations count ratio between our 
ARMLSM and the traditional MLSM is: 
Ratio = 8k/n° (30) 
our experiments show that k usually takes 3 or 4 iterations. 
5 PROCEDURE OF ARMLSM 
We mentioned previously the new ARMLSM can be applied 
in both image space and object space, and employs a hierar- 
chical strategy, here are the algorithm procedures: 
Image space based ARMLSM 
1. generating the image pyramids both for left and right 
images, assuming total levels is Æ, let k = K 
2. resampling the right image using the approximate par- 
allaxes (X° = 0 when k = K) 
3. performing the radiometric correction 
4. array relaxing | times based on equations 25 to 27 
5. correcting prime X°, if the corrections are less a given 
limit, goto step 2, otherwise stop at level k 
6. transferring the results from level k to level k — 1, let 
k=k—1 
7. if k = 0, finishing all levels’ matching, otherwise goto 
step 2 
        
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