Array Algebra Automation in Photogrammetryand Range Sensing

Urho RauhalaArray Algebra (URAA)

Consultant andR&D ContractorSan Diego, CA

urauhala@msn.com858-675-9505

Array Algebra rethinking in

• Expansion of matrix and tensor calculus• “Fast” applied math and general theory of

estimation (extended signal processing)• Expansion of Least Squares Matching for

automated co-registration with on-line edit• On-line edit with new bundle methods and

their range/Moire/projector expansion key to automation of terrain/feature extraction

Topics to be reviewed

1. Idea of multi-linear array algebra2. Global (finite elements) on-line edit of local

Least Squares Matching for co-registration3. Nonlinear expansion of loop inverses and

array (unified matrix and tensor) calculus4. Technology and systems prototypes in

photogrammetry and range/Moire sensing

Topic 1: Multi-linear array algebra

1.1 General math of fast transforms1.2 General (loop inverse) theory of linear

estimation and matrix inverses1.3 Array Relaxation of array filtering in

automation of DEM validation, multi-source DEM merge and Global Least Squares Matching

1.1: General Theory of Fast Math• Multi-linear tensor products of matrices• Parameters and observations arranged into 2-D,

3-D,… arrays (extended matrices and tensors) • Expansion of n tensor contractions into “fast” n-D

array multiplications with n small matrices• Fast inverse array multiplications (array solution)

with O(N) vs. O(N**3) for N parameters• Huge savings in RAM allowing N>millions to

formulate (otherwise unimaginable) problems

1.1 (cont.): 2-D Array Equations

1 2 1 2 1 21 1 2 2

1 2 1 21 1 2 2

1 2, ,1 ,1 ,1

1 2 1 2

2 11 1 2 12 1

1 2 2 21 1 2 22 1

1 2 1 2,,1 ,1

where , , ( ),( ) ( ) .

( ), ( ) ( ) .. . .

ˆˆ ( )

where

T

M N n n m mN M M m mm n n m

L L L L LT

N M n n m mN M n m m n

A x l v A X A L V

N n n M mm l vec La A a A

x vec X A A A a A a A

x A l A A l X A L A

= + ⇔ = +

= = =

⎡ ⎤⎢ ⎥= = ⊗ = ⎢ ⎥⎢ ⎥⎣ ⎦

= = ⊗ ⇔ =

1 11 1 1 1 2 2 2 2( ) , ( )L T T L T TA A A A A A A A− −= =

• Two small L-inverses n,m needed vs. one large N,M inverse with O(N) vs. O(N**3)

1.1 (cont.): 3-D array equations

3 3

1 2 3 1 2 3 1 2 31 1 2 2

31 2

1 2 3

3

1 2, ,1 ,1 ,1

1 1 1 2 2 2 3 3 3 1 2 3 1 2 3 1 2 31 1 1

1 2 3 1 2 3

( , ) ( , ) ( , ) ( , , ) ( , , ) ( , , )

where ,

T

n m

T

M N n n n m m mN M M m m mm n n m

nn n

j j j

A

A x l v A X A L V

a i j a i j a i j x j j j l i i i v i i i

N n n n M mm m= = =

= + ⇔ = +

= +

= =

∑∑∑

• Solution with three small inverses vs. one large• One small m,n matrix at a time in RAM vs. M,N• Solution of N=100**3=1,000,000 parameters with three 100x100 inverses vs. one large (million x million) matrix• RAM needed for nm + N + M vs. N**2+N+M elements

1.2: Loop inverse estimation• Parameter exchange to space domain• Full-rank adjustment of space domain parameters• Back-substitution to original parametric domain

0 0 0 0 0 0, ,1 ,1 ,,1 , ,1

1 12 1212

2 23 23 03,1 3,123

3 13 13

( ) , ( )

1 1 0 1 00 1 1 0 1 ,1 0 1 1 1

m m

m n n n m nm p n p

obs obs

obs obs

obs obs

A X L V AA L L V X A L A X L r A n

HH V V AH

= + ⇒ = + ⇐ = ⇐ = ≤

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− = + ⇒ = +⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

2,3

10 0 0 0 0 0 0 0

0 0 0 0 , ,1

0 0

1 1 00 1 1

ˆ ( ) , ( ) ( ) ( ) , ( )ˆ ˆ ( )

( )

m L m m m T T

m m m L Lm

n m m

T T L

L AA L r A r A r AA p n A A AA

X A L A AA L A L

A AA L

−

−⎡ ⎤= ⎢ ⎥−⎣ ⎦

= = = = ≤ =

= = =

=

Cont. of 1.2: Impact of loop inverses

• Expanded definition of unbiased estimators (u.e)• New foundations of matrix inverses• Starting point of nonlinear array algebra

0 0,1, ,1

0 0

Linear functions estim able under G auss-M arkov ( ) 0

even w hen ( ) and ( R ao's u.e)

. (Bjerham m ar-R ao -inverse)(M oore-Penrose pseudo-inverse)

np n p

Lm Lm

Lm Lm Lm Lm

Lm

A X L E V

iff A A A A p r A AA A A

A AA A vs AA A A GA A +

• = =

= < ≠ ∉

• = =

=1

,

0 0 0

0

in the special case ( )= ( ) in the full-rank special case ( )

' (1 / 2 " * *2 1 / 6 '" * *3 ... )

w here ' and ", '" are B laha's Q -surface tensor partials

L T T

m n

m

p r AA A A A r A n

K L K L K L L V

K AA K K

−

=

= =

• + + + = +

=

1.3: Array Relaxation (AR)• Array Filtering for “fast” solutions of large-scale problems• Automated DEM validation (fill-in and filtering), geomorphic

merge and global image matching at speeds of 10**6 nodes/sec/iteration using PC

• Finite element modification with the reduced dx normals of LSM acting as weighted “boundary conditions”. Array algebra models for properly weighted continuity equations

1 2 1 2 1 2 1 23 3

1 1 2 2 ,3 3

Fast 2-D (3-D...) singular value array solution in the inner loop using convolvers.... ( ) ( ) ( )

ˆ ˆ( * )

Deviations

k lT T

ij kl i k j lk l

X N XN M XM U I I N N M M vec X vec U

X S H S US S x h u=+ =+

+ +=− =−

•+ + + = ⇔ ⊗ + ⊗ + ⊗ =

= ⇔ =

•

∑ ∑ˆ of local LSM from their mean shifted to right hand side using latest

Shift invariant space domain convolution applied and iterated (PERS No4, 1989) X

•

Topic 2: Least Squares Matching (LSM)

2.1 Basic ideas of LSM (1976-1986 period)2.2 Global and direct local solution of hybrid

linear and nonlinear LSM model2.3 Global Minimum Residual Matching

(GMRM) expansion of LSM (1986-1996)

2.1: Least Squares Matching (LSM)

F(x+dx,y+dy)=G(x,y)+V, F’x dx+F’y dy = G(x,y)-F(x,y)+V• “Best” minimum variance estimator of shifts dx,dy with

typical 0.02-0.2 pixel x-shift accuracy• Global AR solution for automated pull-in and fill-in of

dense dx with on-line relative orientation using dy• Multi-pair Epipolar Bundle Adjustment (EBA) of dx,dy

at sparse tie point locations• Transform EBA adjusted dx,dy from epipolar space to

unrectified images to serve as tie points for bundle block adjustment, followed by terrain extraction over entire models with no or minor local dy adjustments

2.2: Global LSM• Parametric model of bias b and epipolar shifts dx,dy after

reshaping of g(x-dx,y-dy) using neighboring points• Linear bias b and one shift term eliminated. Reduced and

weighted shift equations feed the global adjustment models of dx or dy at high speed, resolution and quality

• Direct local estimate found for the shift term without Taylor truncation of the nonlinear parametric model

2

11 12 1 1 12 11

21 22 2

( , ) ( , ) ( , )1-D example of local LSM with n=2, ( ) 1 / 2 " '(0) (0) :

[ ( , , ) ( )] / ( ) ( ( )) /

( , ) ( ) reduced normal in is

x m

x m

b f x dx y dy g x y v x yf x b f x f dx f

f b dx x g x b m b m dx u b u m dx m

m b dx m dx u dx

=

=−

• + + + = +

= + + +

• ∂ − ∂ = + = ⇒ = −

+ = ⇒

∑

1 12 11

2n-3 (vs. 2n-1) degree polynom.ˆ " '(0) '(0) [ '(0) '(0)] / " [1992 ISPRS Comm II]

ˆ ˆ ˆ Back substitution ( ( )) / has a closed form nonlinear solution in

f dx g f dx g f f

b u m dx m dx

= − ⇒ = −

• = −

2.3: Minimum Residual Matching (MRM)

• Least squares object function minimizing the 2nd power of residuals v extended to an arbitrary power p of abs(v)

• Extended Newton-Raphson (N-R) using multiple initial values, high order partials and uncertainty basket ub>0

• The special case p=0 and ub=0 corresponds to un-normalized cross correlation (inferior to LSM and MRM)

• Laplace estimation with p=1, LSM with p=2, ub=0 equals augmented normalized correlation (Ruyten template matching used in PSP wind-tunnel method)

• Robust estimation 1.7<p<2.3 together with the new N-R cured LSM problems (canopy pull-in, blunder detection..)

Topic 3: Nonlinear expansion of estimation theory, matrix and tensor

calculus

3.1 Nonlinear least squares normals for LSM and normalized correlation

3.2 Nonlinear expansion of projective equations

3.3 Nonlinear expansion of loop inverses and estimation theory

3.1 Nonlinear LSM normals

• Avoid Newton-Gauss Taylor truncation (after linear term) • Derivation of nonlinear normal equations as product of

nonlinear partial derivative matrix and nonlinear functional model (here without the linear radiometric model)

2

20 0 0 0 0

2

00 0 0 0

( )min / [ ( ) ] 2 '( ) [ ( ) ] 0

Example: ( ) 1 / 2( ) " ( ) ' , '( ) ( ) ' ,

'( )[ ( ) ] 0, 0, 1, 2, ...

"(/ 2 1 / 3{ '/ " ( ) / '

T T

m

i i i ii m

F X G VV V dX F X G F X F X G

f x dx x dx f x dx f f f x dx x dx f f

f x dx f x dx g dx p dx q x m

fp f f f g f

=−

= +

= ⇒ ∂ − = − =

+ = + + + + + = + +

+ + − = + + = = ± ± ±

= + − +

∑2 2

0

2 22 3 0 0

0 00 0 0

1 2 ... )},'(2 1)

22( ) "(1 2 ... )" / 33 " 2 '(2 1) 3(2 1) '

i i

mf m

x gf g f mq f dxf f m m f

+ ++

− + += + + −

+ +∑

3.1 Normalized least squares correlation (Ruyten AT)= LSM

• F(X) consists of normalized cross correlation function

2

2 2

2

{ [ ( , ) ][ ( , ) ]}( , ) vs. LSM: [ ( , ) ] [ ( , ) ]/ ( , )

{ ( , ) } { ( , ) }( , ) max / / 0

( )( ) '( ) '( )

'( ) ' '

fg

fg fg fg

x x

f x dx y dy f g x y gJ dx dy f x dx y dy f g x y g c v x y

f x dx y dy f g x y gJ dx dy J dx J dy

f f g g f g g f g gx y

f g g f f fx

+ + − −= + + − − − =

+ + − −

= ⇔∂ ∂ =∂ ∂ = ⇒

− − − −∑ ∑ ∑

−∑ ∑

∑∑ ∑

2 2

2

1/ [ ( , ) ] ( )' '( ) 0 for symm.template

0'( )'( ) ' ' '

2' ' ' '[(LSM:

2' ' '

y

x y y

x x y x

x y y

c f dx dy f f fdx f f fxdy f f fyf g g f f fy

f f f f f fdxdyf f f

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ − − −⎡ ⎤ ∑ ∑⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=− − =−∑ ∑⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ∑⎢ ⎥−∑ ∑ ∑ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎡ ⎤∑ ∑ −⎡ ⎤⎢ ⎥⇒ =⎢ ⎥⎢ ⎥⎣ ⎦∑ ∑⎢ ⎥⎣ ⎦

00

0

) ( ) / ] ( )( ) for symm. template and ( )( )'[( ) ( ) / ]y

g g c f f g gcf f f ff f f g g c

⎡ ⎤− − − −∑=⎢ ⎥ − −− − − ∑⎢ ⎥⎣ ⎦

∑∑

3.1 Contrast corrected LSMContrast parameter c in observed values g

2

2

2

2,

[ ( , ) ] ( ) ( , )( , ) min. / ( , ) / ( , ) / ( , ) 0

/ ( , ) ( ) [ ( , ) ] 0

{ ( )( ) '( ) '( )}/ ( )

'

x y

m

f x dx y dy f c g g v dx dyv dx dy v c v dx dy v dx v dx dy v dy v dx dy

v c v dx dy c g g f x dx y dy f g

c f f g g dx f g g dy f g g g g

F

+ + − − − =

= ⇔ ∂ ∂ = ∂ ∂ = ∂ ∂ =

∂ ∂ =− − + + + − =

⇒ = − − + − + − −

∑ ∑ ∑ ∑∑ ∑ ∑

∑ ∑ ∑ ∑

,1 2,11

2

2

2

' " " ..( ) ( ) ' '

' " " ..

( ) '( ) '( )( )(

'( ) ' ' '

'( ) ' ' '

0m

x xx xyx y

m y yx yy

f f dx f dyT g g c f f f dx f dyf f dx f dy

g g f g g f g gx y c f f gf g g f f f dxx x x y

dyf g g f f fy x y y

V+ + +⎡ ⎤

⎡ ⎤= − − + − + + =⎢ ⎥⎣ ⎦+ + +⎣ ⎦⎡ ⎤− − −∑ ∑ ∑⎢ ⎥ − − −⎡ ⎤⎢ ⎥⎢ ⎥− =−∑ ∑ ∑⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥−∑ ∑ ∑⎢ ⎥⎣ ⎦

∑

for) ( )( )'( ) 0 symm.

0 templ.'( )

g f f g gf f fxf f fy

⎡ ⎤ ⎡ ⎤− −∑ ∑⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥− =−∑⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−∑⎣ ⎦ ⎣ ⎦

3.2 Nonlinear loop inverse normal equations of projective equations• Traditional ground-to-image transform assumes

perfect (infinite weights) flatness• New transform starts from

0

03,3

0

'' where is the 3x3 rotation matrix

by assigning random errors to all observed image values ', ', cincluding and treating as adjustable parameter for ea

x X Xy k Y Y Mc Z Z

x yc k

M−⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

31

ch image point.Then, form rigorous nonlinear normals (no Taylor truncation) for all threeobservations using their (closed form vs. Taylor) nonlinear partial derivatives.ONLY THEN eliminate / (k c m X X≠ −[ ]0 32 0 33 0) ( ) ( ) .m Y Y m Z Z+ − + −

3.2 Nonlinear normal equations of projective equations (cont.)

• Partial derivatives wrt three attitude angles replaced by tensor product of partial derivatives wrt sum of rows of the 3x3 rotation matrix

• Perform summation over i=1,2,3 of observed values/pt and over j=1,2,…m points/image

• Accumulate nonlinear normals along a strip of images• Apply AR double “strip tie” point idea by assigning new

strip variant point coordinates to “double strip ties” with properly weighted difference equations

• Future bundle methods ready for development after their privately funded URAA inventions with no support from the R&D institutions in USA (repeat story of D.C. Brown/DBA Systems and Uki Helava/Bendix)

3.3 Nonlinear Array AlgebraSIAM J. MATRIX ANAL. APPL. Vol. 24 No 2 pp. 490-528

• Nonlinear expansion of loop inverses and Blaha’s Q-surface (Kalman, Schnabel) tensor methods

• New matrix and short-hand tensor operator tools to derive high order Taylor expansions in space domain, their nonlinear normals and solutions

• Inverse Taylor expansion from the space domain least squares solution to the nonlinear parametric domain = closed form (direct nonlinear) solution of X from observables L

• Unlimited # of Taylor terms included in nonlinear Lm-inverse by evaluating the function and its 1st derivative in 4-6 steps, starting with N-G solution in the space domain

3.3 Nonlinear Array Algebra (cont.) Short-hand array algebra of Taylor terms

, , , , , ,,1,1,1

. , , , , ,,

( ) ' 1/ 2 " **2 1/ 6 '" **3 ...( )

where A**p contracts the p last indices of array A (having p or more indices).' " **1 1/ 2 '" **2 ...'( )

m n m n n m n n nnmm

m n m n n m n n nm n

F X F dX F dX F dXF X dX

F F dX F dXF X dX

N

= + + + ++

= + + ++

,1,1 , ,1

. . .,1 ,1,1 ,1 , ,

2. , , , , , , , ,1,1

2

( ) [ ] 0'( ) ( )

' [ ] { ' ' [ ]* " }( ) ( )

1/ 2{ '* "} **2 " **1 ' 1/ 2 '" **2 [ ]( )

....1/12 '

T

mn n m m

T TTn m n m m nm mm m m n n

T T

m n m n n m n n m n n n mm

X dX GF X dX F X dX

F G F F G F dXF X dX F X dX

F F dX F dX F dX F dX GF X dX

F

+ = − =+ +

= − + + − ++ +

+ + −+

+ 2" **2 '" **3 (5th order array polynomial)where operator * in * contracts the first index of arrays , .

TdX F dXA B A B

3.3 Nonlinear Lm-inverse expansion of general inverse and estimation theory

1 1 1,1

2 2 2,1

1,1 ,1

1 1 1, ,1 ,1

Nonlinear expansion of Array Filtering and sequential Kalman updating of( )

( ) .

Nonlinear transform of parameters to (unknown true values)

obsp

obsm p

n p

obsp p p p

F X L V

F X L V

X L

I L L V

K

−

= +

= +

= +

02 1 2 2 2 1 2 1 2 1 2 1

, , , , , ,,1

12 2 1 1 1 1 1

,

2 1 2 2 2 1 1 1 1,

, , ,1

( ) ( ) ' 1 / 2 " **2 1/ 6 '" **3 ..

where ' ' ' , ' ' ( ' ' ) ,

" **1 ' ( " ' ") ( ' ) **1 '

obsm p p m p p p m p p p pm p

m m T T

m p p

m m

m p pm p n n n

L L V K L K dL K dL K dL

K F F F F F F

K dL dK F K F F dL F

− − −−

−

−

−−

= + = + + + +

= =

= = − ,

2 1 2 2 2 1 1 1 1 2 1, , , ,

, , , ,1

and

'" **2 2 ' ( '" ' '") ( ' ) **2 ' ' ' .

n p

m m m

m p p n p m p n n pm p n n n n

K dL dK F K F F dL F F− −

−

= = − − ∆

3.3 Nonlinear Lm-inverse (cont.)2 1

02 1 2 1 2 1 2 1

, , , , , ,

10

1 1 2 1 2 1

Perform similar Taylor expansion of the nonlinear function '( )

'( ) ' ( ) " **1 1/ 2 '" **2 ..

and form the normals of parameters

( ) '( ) ( ) 0.

m p p m p p p m p p p p

T

K L

K L K L K dL K dL

dL

N L dL K L K L

− − −

•

= + + +

+ = =•

01 1 1 1

"Hyper iteration" solution of UNLIMITED Taylor terms in matrix notations expanding Blaha's inverse Taylor technique of tensors. Solution

ˆ ˆ ˆ is combined with the inverse transform of = tdL L L dL+

1 1 1 1,1, , , , , ,

3 1

ˆo from' 1/ 2 " **2 1/ 6 '" **3 ..

New mr-inverse for the inverse Taylor derivative is re-used in one "super iteration" of the reverse parameter transform ˆ ' [

np n p n n p n n n

mr

XdL F dX F dX F dX

X X F F

= + + +

•

= + 1 3 1 1 4 1 1 5 1

1 1 1 2 1 2 1 1 1 1 1 1 1 3

ˆ ˆ ˆ( ) ( ) ( ) ...]

' ' { '( ) [ '( ) '] ' [ '( ) ']} ' '( )mr m m m m

X L F X L F X L

F F F X F X F F F X F F F X

− + − + − +

= − − −

3.3 Impact of Nonlinear Lm-inverse1

1 1 1 1 1

ˆ Superior pull-in range over traditional Newton-Gauss as the solution is nearly linear even when ( ) is highly nonlinear. Step 1 consists of linear

ˆ ˆtransform ' where is tm

dLF X

dX F dL dX ddX dX

•

= = +

1 1

02

he linear Lm-solution ofˆ ˆthe first N-G iteration used to get = '

ˆ ˆ Step corrects the nonlinear terms of reverse transform The third step corrects for the remaining non-linearit

dL F dX

X X dX ddX• = + +•

3 2 1 1 2 1 1 2

1

y byˆ ˆ ˆ' [ ( ) '( )( )]

The remaining steps use the mr-inverse derivative to make the model domainˆ ˆ parameter estimate consistent with the space domain

Related nonlinear

mX X ddX F F X L F X ddX

X L

= + − − +•

• solution of Blaha's Q-surface technique had 10-100times better pull-in range than N-G to converge in 4-6 finite steps. Solves the general inverse problem of linear and nonlinear estimation.

Topic 4: Automated photogrammetryand range/Moire/PSP systems4.1 Early research work (1971-1996)4.2 Auto* System prototypes (1996-2003)4.3 New vistas in satellite, aerial and close

range mapping technologies

4.1 Early applications (1971-1984)• Film deformation and self-calibration model of Hasselblad

close range moon camera (1971)• Predecessor of retro targeting and PSP wind-tunnel attitude

measurements in medical (bone fracture) imaging (1971-72)• 1976-77 terrain edit of array filtering, progressive sampling

and LSM studies • 1977-84 GLSM ideas of DMA studies, industrial

photogrammetry (GSI) and inertial/laser surveying• Fast 3-D finite element reseau deformation model and

original formulation of wavelet theory (JPEG2000) and Replacement Sensor Models of satellite mapping

• Fast bundle adjustment of LSQCHOL (2.4 sec vs. 17h) in support of DMA modernization program proposals

4.1 Applied work (1984-1996)• Ideas of GLSM (1984 ISPRS poster paper) and

bundle adjustment prototyping helped to win new softcopy WS R&D programs with Helava for GDE

• GDE IRAD for on-line DTED validation and geomorphic merge with GLSM prototyping (reported at ASPRS/ISPRS 1986,87,88,89,90)

• GLSM systems integration attempts in Socet Set (1992 ASPRS/ISPRS/IGARRS and SPIE 1993)

• FELSM (Foerstner Interest Operator expansion to Global Hough Transform), SPIE 1995

• DEM Matcher and Global Image Rectifier (Vienna 1996 ISPRS)

4.2 Auto* Systems Prototypes• Auto_Tie prototype for tie points: 1) Extract terrain using

GMRM pull-in down to 1:8-1:16, 2) Select ‘sweet spots’near nominal grid locations to extract 1kx1k 1:1 patches, 3) Refined registrations validated by Epipolar Bundle Adjustment (EBA) until a sufficient # of good points found, 4) Transform adjusted EBA results to unrectifiedimages and refine initial Ground-to-Image RPC model

• Auto_DEM/Clue for DEM production/on-line targeting of multi-ray and multi-sensor system (with terrain feedback)

• Revolution in DEM production by removing 95-100 % of manual edit (1-3 pts/sec). The fully automated speed of ONE MILLION match points in few sec is the key for overall production rate, quality and reduced costs

4.3 New vistas

• Full automation feasible using DEM patches as “shear tie points” (ELSM DEM Matcher, Vienna ISPRS 1996)

• New block triangulation techniques for automated DEM shear tie points of satellite and digital frame camera or close-range dot/line/Moire projector systems

• Extremely fast (paper and pencil) 7-D array solution of large area block adjustment feasible with improved accuracy of milli-pixels (URAA research in Relative Independent Model and BA)

• Nonlinear integral calculus of latest array algebra with future potential in automated systems of range sensing and EO/SAR/IR multi-ray stereo mapping, photo geodesy and PSP/Moire/projector techniques