(X - App) 708-5 0; hag) X 
((x 7 x) 152; * €; I9) Y * 
(x - Xp) Myyy + Cy May) Z = 
((x - Xp) my; + cy my, 4) X; + 
( (x = Xpj) 10553 * Cj 2333) Y; * 
((x - Xpj) 7545 * C; m5) Zy 
(4) 
(Qr 7 yg) mi * 05m) X* 
((y-7X5)mj;*c;mj)yYs 
((y Ypj) Ifl333 * Cj 2543) Z = 
(r 7 ypj) mi; * C5 mij) Xy + 
((y 7 ypj) m3; * C; m;j) Y; * 
(y Yo} 1334 + Cy 15,3) Zj 
Equation 4 will form a pseudo least squares 
solution. 
3. Feature Space Intersection 
The photo image of a feature in space can be 
represented by discrete points on each photo. 
The discrete image points on the second photo 
can be represented by a single continuous 
function or a group of finite elements. The 
modified form of the collinearity equation for 
a point seen on two photos will be as follows: 
  
  
  
  
x 2X -c My, (X-X,) + m2 (Y-Y,) + M3(Z-Z,) 
; p tm (X-X,) + M2 (Y-Y,) + Ma (Z-Z,) 
a dg. m,(X-Xj) * mj(Y-Yi) + M,(Z-Z,) 
M Ei Tm (X-X,) + M2 (Y-Y,) + M3 (Z-Z;) 
= Ij (X-X,) + rn,(Y-Y,)) + 1,3 (Z-Z2) 
X) 7 Xp2 2 = = = 
Ir4,UC X) * r4(Y-Y,) * r4(Z-2,) 
- ras (XX) + 122 (Y Ya) + Tes (2-22) 
Nr 2 Vn ER Ti FY Ti) 
Ya = £(x;) 
(5) 
or 
X; = FX, (X 4 2) 
Yi = FY; (X ,Y ,2) 
X; zUFX, tX ,Y ,2) 
f(x) = FY, (X ,Y ,2) 
where: 
x; Yi photo coordinates of the point on 
photo 1 
X| Y, 73 object space coordinates of photo 
My; + + + -M33 orthogonal orientation matrix 
elements for photo 1 
X, ,Y, ,2; object space coordinates of photo 
612 
2 
Yi:«£3 orthogonal orientation matrix 
elements for photo 2 
X, Y, Z2 object space coordinates of the 
point. 
The unknowns in Eq. (5) are (X, Y, Z, and xj). 
This mathematical model will provide a unique 
solution for the problem. Each additional 
photograph will provide three equations and 
one additional unknown. 
The function f(x;) can be linear, polynomial or 
a group of finite elements. This function 
should be defined from the discrete points 
that defines the feature on the second 
photograph. The function f(x;) should exist. If 
it does not exist, we can use x, = h(y,). If 
the feature line slope changes between 0 and 
360 degrees, then we can use the following 
equations: 
  
  
  
  
  
X, - hj(z) 
y, * h,(z) 
where the unknowns will be (X, Y, Z, x, y; 
and z). 
The linearized form of equation 5 will be: 
BA = f 
OFX1 OFX1 OFX1 0 
ox oY oz 
OFY1 OFY1l OFY1 0 
ox ay oz 
P^larex2 arx2 arx 0 
ox ay oz 
OFY2 OFY2 ory2 Of 
ox oY oz ox, 
ó X 
^ ó Y 
Lt 6 Zz 
ó x, 
3. RESULTS 
To test the new mathematical model, ground 
features from two aerial photos, taken from a 
flying height of 1500.00 ft above the terrain 
with a camera whose focal length was 152.00 
mm, were mapped using the KERN  DSR-14 
analytical plotter. The same features were 
digitized using the KERN MK-2 mono-comparator 
using the same photographs. The average rms 
errors between the two methods were 0.06 ft in 
planimetric position and 0.181 ft in 
elevation. 
The ground features were a combination of 
straight and curved lines that defined the 
edges of a highway. The rms error in modeling 
the image coordinates on the second photograph 
was in the order of 0.006 mm. 
4. CONCLUSIONS 
Feature space intersection has been tested and 
can be used in digital photogrammetry, 3D 
robot vision, and single photo digitizing. In 
digital photogrammetry, the number of discrete 
points that define the features will allow 
more accurate mathematical representation of 
the feature in Eq. 5. 
intersection 
The new formulation of space