Li Weiqian 
THE DIFFERENTIAL FORMULAS FOR ANALYTICAL PROCESSING 
OF PANORAMIC STEREO PHOTOS TAKEN BY FISH-EYE LENS 
Li Guangwen 
Dept. of Surveying, Tongji University 
Siping Road,Shanghai China 
Abstract 
The rigid mathematical formulas for determining object-point coordinates of the panoramic photographs taken by a 
Fish-eye leus in stereo close range photogrammetry are deduced. That are general differential formulas and suitable 
to the modes of normal, parallel-averted, convergent and equal tilt photographs. The experiments are also presented. 
KEY WORDS: Fish-eye photograph, Analytical processing, Differential formula. 
1. Introduction 
The application for panoramic photographs by fish-eye 
lens in close range photogrammetry has been 
expounded (¢f.[2])). One group of the theoretic formula, 
in which, is the formula of FDLT(Fish-Direct Linear 
transformation) and FDDLT (Fish Distortion Direct 
Linear Transformation) bearing the correction for 
additional parameter derived on the basis of DLT 
formula. Using the formula as stated, the layout can’t 
he less than six control points in photographic object. 
It is difficulty. for some close range photogrammetry to 
do so before or after the event, such as the processing 
for all of a sudden accident etc.. The principal 
advantage for the panoramic photography by fish-eye 
lens is able to take huge object at very near 
distance(even at 1-2m).having advantages of large area 
coverd. high resolution,large negative content and 
etc..Its photography use a simple and easy stereo close 
range camera and has a fish-eye lens attached to it. 
The camera is proceed with some calibration, adjustment 
or refitting by general camera(e.g.see gull 135). In 
order to make fish-eye lens applicable to stereo 
photogrammetry in any state. The rigid mathematical 
formulas are derived in this paper and the tests are 
conducted. In this way, the control points can be 
greatly decreased and they only need to layout 
according to control requirement of general terrestrial 
photogrammetry. The experiment cited in this paper uses 
the stereo-camera made by Shanghai 832 works 
subordinate to Ministry of Public Security. Its 
components consists of camera 135, base-bar, linker up 
with cameras,tripod and so on. Four frame-marker are 
carved into register plate of camera. There are three 
specifications of base-bar 40cm, 100cm, 120cm. The 
focusing installation of camera has been refitted.The 
fish-eve lens is attached to the lens of camera 135 in 
the test.The principal distance value f is kept from 
change during photo process. Carrying on stereo 
photography, the photographic cases may select at will 
and the svnchronous photography may be performed. 
2. General formula of fish-eye panoramic photograph 
Procecd from the critical formula of determining ground 
point coordinale in terrestrial stereo 
photogrammetry, that is: 
The ground point coordinate for left photo of stereo 
pair is: 
Xe=Xs,+ NX] =X, + Ni[a,(x;— Xp) t 4 d 2,029] 
Ye=Y, TNY1=Ys,+ N Cb, (x, — x0) + baf1 + 6302, — 70) ] C1) 
Zg— Z,,"-NuZi m Zs, NEG 047 x) 9 6f d 005 729] 
The ground point coordinate for right photo of 
stereo pair is: 
Xem Xs, NAXEm Xs Nat O5 735) Tr aif 10,7200] 
Y; Y,,- NYImY ,Nb(G x) bif bin 7201402) 
ZemZntNQImeZu.tNO; 1:246; 20] 
where: ; 
N, and N, are the coefficients of projective 
magnification; a;b;¢: and a4bjc} are direction cosines on 
left and right photo respectively, it can be determined 
from the formula in paper[1]. Xoro and x424f, .are 
elements of interior orientation on Jeft and right 
photo respectively, 15: and %:% are measuring image 
coordinates on left and right photo respectively, X«Yc 
Ze is geodetic coordinate of object point, Xs.Ys.2s1 
and Xs.Ys.Zs. are coordinates of photographic station 
on left and right photo respectively. Eq. (1) and (2)are 
applied to photograph of central projection, whereas 
the photograph taken by fish-eye len is spherical 
central projection(2]: 
x^ z 
x= ; —————=———=+ 2=_= | ya EL 
f FELL f = fi_x/i—z/2 
Substitute into Eq.(1),(2), then get the coordinated 
formula for fish-eye panoramic photo pairs, the formula 
for left photo is: 
Xe = Xs. + NX 
x1 
xen [rpm + 
at 
Ta fa (^ JH mi -2)| 
Ya=Ysıt NY1 
  
  
7 
x, 
=Ys+WNi [6 trem) y (3) 
  
-— 
e 
tots (fy r=) 
Za=2s, +NZ1 
Xi 
=Zs+N1 lam —-2) + 
Haft elf m) J 
  
the coordinated formula on right photo of fish-eye 
panoramic photo pairs: 
  
Xe: =X 53+ NaX3 xut Liu) 
  
Hash+as (772