Very High Resolution Satellite Image Triangulation

Luong Chinh Ke Warsaw University of Technology

Institute of Photogrammetry and Cartography 1 Plac Politechniki, Warsaw, Poland

lchinhke@gazeta.pl

Wiesław Wolniewicz Warsaw University of Technology

Institute of Photogrammetry and Cartography 1 Plac Politechniki, Warsaw, Poland

w.wolniewicz@gik.pw.edu.pl

Abstract: Since beginning of 21st century Very High Resolution Satellite (VHRS) image has been available in practice for orthophoto and DEM generation with possibilities of lower cost and shortest time. For this purpose, at first, we have to calculate the parameters of very high resolution satellite image. In practice there are two group methods so-called rigorous and generic. Rigorous methods are related with the satellite orbit parameters and sometime they are named the physical methods. If the satellite orbit parameters are supplied by vendor the number of ground control points will be limited to minimum. In many cases the satellite orbit parameters are not given. In such cases the generic methods are proposed to use. The method RPC (Rational Polynomial Coefficients) in the generic group is frequently useful. Basing on the specification of very high resolution satellite image the image taking in the perspective projection can be transformed at first into parallel projection where block of transformed high resolution satellite image could be carried out. The paper presents actual problem of triangulation block of very high resolution satellite image, performed by different scientific centers. Keywords: Satellite Photogrammetry, Very High Resolution Satellite, Triangulation, Orthorectification, Linear Array CCD, Epipolar Plane, Perspective, Parallel Projection. 1. Introduction

On September, 1999 IKONOS satellite (SpaceImaging, USA) provided spatial resolution images of 1m, has

opened the new era of very high resolution satellite image. It is traditionally known that graphical position accuracy of topographic map is not bigger than 0.1mm in

map scale. For example, the map scale 1:5 000 requires position accuracy no larger than 0.5m in terrain. The optical images (photo images) that are used to mapping have ground sampling distance (GSD) no larger than 0.5m. Analogically, the map scales of 1:20 000 up to 1:10 000 require GSD no larger than 1m. GSD is defined as the distance of the center of neighbored pixels projected on the ground. The pixel size of IKONOS image (15µm) contains information corresponding to the aerial images in scale 1:80 000 and information content of QuickBird image with 11µm pixel size are compared with the aerial images in scale 1:50 000. Therefore, IKONOS and Quicbird images can theoretically be used to topographic mapping up to map scale 1: 10 000 and 1:6 000. For producing orthoimages, if we adopt orthoimages will have 8 pixels per 1mm, IKONOS and QuickBird can be used to do orthoimages in scale 1:8 000 and 1: 5 000 (Jacobsen, 2005).

Basing on the above information we consider that the triangulation network of high resolution satellite images VHRS can use to project the high route, train way as well as georeference network for mapping in scales litter than 1:20 000, or for processing orthoimages used in agriculture and forest (Grodecki, el. al.; 2003; Madani, 2005; Toutin, 2005).

The paper comprises five sections. In the second section there is overview of VHRS from film images to digital ones, from spatial resolution of hundred meters up to one meter and smaller. Some characteristics of VHRS related with specification of linear array CCD sensor will be presented in the third section. In the fourth section some models of block of VHRS with their experiments results will be described. Final conclusions have to be discussed in fifth section. 2. Overview of very high resolution satellite images

First time in history the satellite SPUTNIK-1 (Soviet Union) lunched on October 4, 1957 marked the beginning of new space era. 20 months later the US tests of CORONA system were started in 1959. The state of the art in that time allowed only the use of film. USA used film up to 1963 and declassified its images in 1995, but Russia made the last satellite photo flight in 2000.

It took until 1972 to lunch a satellite dedicated to civil spaceborne Earth surface mapping: the ERTS (Earth Resources Technology Satellite) spacecraft, later renamed to Landsat-1. First digital images acquired from

1

Lansat-1 were presented on ISPRS Congress in Ottawa, Canada (1972). The MMS (Multispectral Scanner System) instrument of Landsat-1 provided a spatial resolution of 80m and a swath width of 185km. Lansat-4 (1982) with TM (Thematic Mapper system give ground sampling distance (GSD) of 30m. Lansat-7 (1999) with improved system TM has GSD of 15m. On September, 1999 SpaceImaging Company (USA) put in motion IKONS-2 satellite with GSD of 1m. Later, two companies Earth Watch and OrbImage (USA) started Systems QuicBird-2 (2002) of 0.61m and 2.44m (for panchromatic image), and OrbView-3 (2002) with GSD of 1m and 4m (for panchromatic image), respectively. Table 1 presents some parameters of current and near future satellite images of USA in the world commercial market. OrbImage Company plans in 2005 with OrbView-5 image with GSD smaller than 0.5m.

Table1. Some parameters of USA VHRS in current market and in near future

Images

Com- pany

Data of lunching

Pixel

number on a line

[pixel]

Mode of image

GSD (nadir)

[m]

Swath (nadir)

[km]

View angle direction on along-

track

View angle direction on across-

track

Height of flight

[km]

Ikonos 2

Space Imaging

09/1999

13816 3454

Pan 4 MS

0,82 3,2

11,3

±45o

±45o

680

Quickbird 2 WorlView 1

Digital Globe

10/2001 *2006

27000 6700

Pan 4 MS Pan

8 MS

0,61 2,44 0.5 2

16.5 16

±30o

±30o

450 770

OrbView 3

OrbView 5

OrbImage

2002

*2006

8000 2000 …….

Pan (or ) 4 MS Pan MS

1 4

0.41 1.64

8

15

±50o

……….

±50o

……….

470 ……

French SPOT satellites, at first lunched in 1986, with the stereoscopic possibilities of 10m GSD were used

for generation DTM and updating map up to 1:50 000. Later, SPOT-5 lunched in 2002 with HRS and HRG systems provided GSD of 5m and 2.5m, respectively. Other countries as India, Israel, Germany, Japan, South Korea etc. have their satellites which are listed on the table 2. Six countries as USA, India, Israel, France, South Korea and Russia can support the images with GSD of at least 1m. Images with GSD up to 2.5m can be supplied by five countries as United Kingdom, Thailand, Brazil, China, and Malaysia [2]. In the Fig. 1 there are some satellites systems presented.

Fig. 1. Overview of several satellite systems

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Table 2. Some parameters of VRSI of other countries in current market and in near future

Country

System

Data of lunching

Pixel number

on a line

[pixel]

Mode of image

GSD

(nadir)

[m]

Swath (nadir)

[km]

View angle direction on along-

track

View angle direction on across-

track

Height of flight

[km]

France

SPOT 5

PLEAIDES-1

PLEAIDES-2

2002

…….. *2008 *2009

12000 Pan 6000 Ms

Pan/Ms Pan/Ms

Pan/Ms Pan/Ms

5(3) orbit 10 across

0.7/2.8 0.7/2.8

60

………. 20 20

± 20o

± 27o

822

........

IRS P6

India (ISRO)

Cartosat-1 Cartosat-2

2003

*2005 *2005

12288

Pan + 3 Ms

LISS-3 4 Ms Pan Pan

23

2,5 1

70 Pan 24 Ms

140

27/30

± 26o

+20o, -5o fixed

617

Japan ALOS

.

*2005

.

28000 14000

Pan-nadir Pan (±24o)

2,5 10

.

35 70

± 24o

692

South Korea

KOMPSAT-2

*2005

Pan/Ms

1/4

15

„free”

EROS A1 Cyprus-Israel (ImageSat)

EROS B EROS C

12/2000

2003/2004

*2005 *2009

7000

20 000 5 000

Pan

Pan Ms 4 bands

Pan Pan/Ms

1,8 (1)

0,82

0.7 0.7/2.8

12,6

16,4

14 11

±50o

±45o

±50o

±45o

480

600

Resurs DK1 Russia

*2005

28300

Pan/Ms

1/2.5-3.5 28,3

-

-

350

RazakSat Malaysia

*2005 Pan/Ms 2.5/5 20 - - -

THEOS Thailand

*2007 Pan/Ms 2/15

3. Geometry of VHRS

In the Institute of Photogrammetry and Cartography, Warsaw University, Poland, one elaborates rigorous

block triangulation from VHRS that is presented in third section, basing on the physical-geometrical model of single images acquired on the satellite orbit [4]. This model is related with parameters describing the spatial position of satellite moving with Kepler law and image attitude in space. Therefore, one also calls this model a parametric model. For the purpose of block triangulation elaboration, rigorous model of geometric relationship between image and Earth’s surface is presented abbreviationally in subsection 3.1.

3.1. Time-dependent collinearity equation

On the Fig. 2 the geometric relationship between image and Earth’s surface in geocentric reference system

OXYZ is presented. In figure 2 the marks mean: γ – vernal equinox, i – inclination angle, λ0 – Greenwich meridian, Ω – right ascension angle, K – ascending node, w – the argument of perigee, π – perigee point, τ – true anomaly or travel angle at time t, Λ – geocentric longitude, λ – geographic longitude, Φ – geocentric latitude, Ψ – geographic latitude,

OO – Earth’s radius, OS – orbital height, R=OO+OS – the distance from Earth’s geocentric origin to satellite at time t. There are four very important coordinate systems presented in figure 2: Oxyz – imagery coordinate system, SXSYS ZS – satellite coordinate system, OXLYL ZL – local geodetic system, OXYZ – geocentric system.

3

Fig. 2. Geometric relationship between image and Earth’s surface in geocentric reference system OXYZ

Three angles φ, ω, κ determine tilt angles of image (in space Oxyz) with respect to satellite coordinate

system SXSYS ZS. Four quantities R = OO +OS, u = w+τ, i, Ω determine the satellite position with respect to geocentric system.

On the basic of Fig. 2 we can write a relationship between image and Earth’s surface in geocentric reference system OXYZ as follows

Tα r t = k T U (t) R + Rc(t) (1) where R = [ X Y Z ]T

rt = [ 0 yt -f ]T U(t) = ait[φ(t), ω(t), κ(t), u(t), Ω (t), i(t) ] Rc(t) = [ Xc(t) Yc(t) Zc(t) ]T

with φ(t), ω(t), κ(t); Xc(t), Yc(t), Zc(t) – the image tilt angles and sensor perspective center (exposure station) coordinates as the second degree linear functions of time variable in time interval t, α – view angle direction on the along-track or across-track, T – transformation operator, R – vector of ground point coordinates, rt - image vector coordinates in time interval t, k – scale factor, f – focus length, yt – point ordinate of line in time interval. ait with i =1 to 9 – elements of rotational matrix with size 3x3 that are functions of image exterior orientation elements and satellite orbit parameters (see Eq. 2).

The Eq. (1) is so-called the time-dependent collinearity one. Vector Eq. (1) can be written in non-linear equation in the following form

xt’= - f. L / M ; yt’ = - f. N / M (2)

where: L = a1t (X – Xc(t)) – a2t (Y – Yc(t)) + a3t (Z – Zc(t)) N = a4t (X – Xc(t)) – a5t (Y – Yc(t)) + a6t (Z – Zc(t)) M = a7t (X – Xc(t)) – a8t (Y – Yc(t)) + a9t (Z – Zc(t))

and xt’, yt’ – image transformed coordinates with respect to angel α.

Basing on the Eq. (2) and Fig. 2 we can write a general form of geometric relationship between image and Earth’s surface in geocentric reference system OXYZ in time interval t. Fx’t (Rt, Ωt, it, ut, φt, ωt, χt, X, Y, Z) + xt’ = 0 (3) Fy’t (Rt, Ωt, it, ut, φt, ωt, χt, X, Y, Z) + yt’ = 0

The Eq. (3) determines time-dependent collinearity equation for VHSR. 3.2. Epipolar geometry of HRSI

Image resampling into epipolar geometry is an important task in digital fotogrammetry for different work

such as image matching, DEM and orthophoto generation, aerial triangulation, map compilation and stereoscopic viewing. The purpose of epipolar geometry is finding conjugate (corresponding) points in the left and right image. They should have no y-parallax in the resampled images, but have an x-parallax that is linearly proportional to the corresponding object height. The epipolar lines of photo images are the straight lines, parallel to image x-axis (Fig. 2.a). They can be determined basing on the collinearity or coplanarity condition. The

4

epipolar lines is defined as the intersection of the epipolar plane with an image or the locus of all possible conjugate points of p on the other image by changing the height of the corresponding object point. The epipolar plane for a given image point p is defined as the plane that passes through the point p and both image perspective centers.

a)

b)

Fig. 3. Epipolar geometry: a) – epipolar geometry of photo image, b) - epipolar geometry of HRSI

HRSI that is acquired with scanning technology by linear array CCD sensor has complicated epipolar

geometry. Kim [3] proved that the epipolar line is no longer a straight line, rather has hyperpola-like shape (Fig. 3.b). In addition, the author proved that epipolar lines do not exist in conjugate pairs. Morgan [7] proved the general form of epipolar line for VHRS as follows

y’i .E1 + y’i .i.E2 = i.E3 + E4 (4)

where i – the scan line number on the right image, y’ – point ordinate on the right image, E1, E2, E3, E4 – are the function of interiors orientation parameters IOP and exteriors orientation parameters EOP of the scanner.

The epipolar line described by Eq. (4) become straight line when E2 = 0. It is equalized to following equation

E2 / E1 = (v1 x v2 ).v3 / (v1 x B). v3 (5) where: v1 , v2 , v3 , B are the four vectors presented in the Fig. 3b. It is clear that when E2/E1=> 0 → epipolar line is straight.

The problem of epipolar line can be easily solved in parallel projection Morgan el. al. [7]. Well-founded motivations are:

- Imaging system has narrow angular field of view (AFOV) – e. g. for IKONOS, it is less than 1o; for SPOT it is about 4o. The perspective light rays become closer to being parallel.

- The scanning time for individual image is very short (about 1sek. for IKONOS). At a result, the planes containing image and there perspective centers, are parallel to each other.

- At a result of very short capturing time, the imaging system travels equal distances in equal time interval (constant velocity). Therefore, same object distances are mapped into equal image distances.

Basing on above considering, the epipolar line equation, adopting the parallel projection, can be written as x.G1 + y.G2 + x’.G3 + y’.G4 = 1 (6) where x, y; x’, y’ are the image coordinates on the left and right image, respectively; G1, G2, G3, G4 are the parameters determining the epipolar line equation. 3.3. Relief displacement on VHRS

The relief displacement of the terrain ∆r caused by the elevation difference ∆h on image with focus length f, taking in flight height H can be calculated as

∆r = ∆h(f/H)R/(H- ∆h) (7)

For IKONOS-2, QuicBird-2 and OrbView-3 the quantities of relief displacement are presented in the table 3

5

Table 3. Quantities of relief displacement for VHRS

Elevation difference ∆h [m]

IKONOS-2 f=10m; H=680km

R=6,5km ∆r [mm]

QuickBird-2 f=8m; H=450km

R=8,5km ∆r [mm]

OrbView-3 f=8m; H=470km;

R=4km ∆r [mm]

50 0.006 0.017 0.007 100 0.012 0.036 0.014 200 0.024 0.067 0.029 500 0.061 0.168 0.073 1000 0.122 0.145

It is obvious that relief displacement is proportional to the elevation difference ∆h. The quantities of

relief displacement for IKONOS-2 are not bigger than their pixel size of 15µm when terrain elevation difference ∆h is not lager than 100m. Terrain elevation difference ∆h is not lager than 100m; it causes relief displacement for QuickBird-2 and OrbView-3 no lager three times than their pixel sizes of 11µm and 8.5µm, respectively.

4. Block triangulation of VHRS

Defining imaging process on satellite orbit by its parameters determines a physical or rigorous model. The second model is represented by characteristic models, based on rather different transformations such as affine, polynomial or rational function method, often so-called a rational polynomial coefficients (RPC) etc.

In this section the rigorous and generic block triangulation are presented. Each method has both advantages and disadvantages. The advantages of rigorous model of block triangulation are the possibilities to correct sensor distortion as well as other ones caused by Earth motion and map projection, to use time-dependent equation, orbital constraints and exterior orientation registered by GPS/INS. To the disadvantages of rigorous model of block triangulation can be unknown sensor physical parameters, sensor model no published by vendor, requiring specialized software, changing real-time loop math for each sensor. The advantages of generic model of block triangulation are the sensor independent, sufficient speed for real-time loop, supporting any map projection system, easy inclusion of external sensor. To the disadvantages of generic block triangulation can be impossibilities to correct local distortion, no having physical meaning, accuracy decreases when image gets larger and over parameterization. 4.1. Rigorous block triangulation Kepler model 1

Rigorous block triangulation based on Kepler model, in first stage of theoretical investigation, has been

carried out in the Institute of Photogrammetry and Cartography, Warsaw University, Poland [4]. The fundamental principle of method is represented as follow

The images can be acquired on the same satellite orbit by more than one sensor looking at the Earth with different angles, by rotating sensor in the along track direction (IKONOS, QuickBird, EROS, etc.) or by taking from two adjacent orbits (across track, SPOT). The advantages of along track images compared with across track images are that they are acquired in almost the same ground and same atmospheric condition. It is better for matching and later for conjugate point measurement. The fundamental benefit of an along track images is to orient them simultaneously in strip triangulation. In the Fig. 4 two models of across track stereo images and along track stereo images are represented.

a)

b)

Fig. 4. Orientation elements of stereomodel: a) across track stereomodel, b) along track stereomodel

6

There are different versions of rigorous method, depending on various orbit determination-propagation assumptions. The initial, fundamental and common assumptions of rigorous method are:

• The satellite motion on an orbit is a Keplerian motion. • The coordinates of perspective centers are in same scale. It means that the satellite velocity is constant. It is

balanced to constant time interval. • The rotation angles of satellite remain constant during the acquisition time of each image. • A geocentric coordinate system is used to orient all the images.

Rigorous block triangulation presented in this section becomes built from independent images of Kepler model. It is known that the satellite movement is subjected to Keplerian law. Its orbit is an ellipse-like shape. The

distance R in the Eq. (3) is a function of the semi-major axis “ r ” and the eccentricity “ e ”of an ellipse orbit. Six parameters (r, e, w, Ω, i, τ) fix the position of the satellite on space orbit. Basing on the above assumption, with given orbit four parameters (r, e, w, i,) remain constant (see Fig. 2). In general, for image “ j ” of orbit (strip) “ k ” in time interval “ t ” the Eq. (3) have the following form

Fx’jkt (rk, ek, wk, ik, Ωjkt, τjkt, j φjkt, ωjkt, χjkt, X, Y, Z) + xt’ = 0 (8) Fy’jkt (rk, ek, wk, ik, Ωjkt, τjkt, j φjkt, ωjkt, χjkt, X, Y, Z) + yt’ = 0

From Eq. (8) it is obvious that individual (single) image has nine orientation parameters, in which there are six parameters related with the orbit and three – with same image. The stereo pair (two images) has now twelve orientation parameters and three images have now fifteen etc. (table 4).

The major components of dynamic motion are the Earth’s rotation and the satellite movement along defined orbit. The parameters Ωjkt, τjkt, j φjkt, ωjkt, χjkt are time-independent. For given “ j ” and “ k ” these parameters can be modeled with second order linear functions of time variable as

Ω(t) = Ω0 + Ω1t + Ω2t2, Ω(t) = Ω0 + Ω1n + Ω2n2, τ (t) = τ 0 + u1t + u2t2, τ(t) = τ 0 + u1n + u2n2, φ(t) = φ0 + φ1t + φ2t2, or φ(t) = φ0 + φ1n + φ2n2, (9) ω(t) = ω0 + ω1t + ω2t2, ω(t) = ω0 + ω1n + ω2n2, χ(t) = χ0 + χ1t + χ2t2, χ(t) = χ0 + χ1n + χ2n2, where: n – the number of scan lines; Ω1, Ω2, τ1, τ2, φ1, φ2, ω1, ω2, χ1, χ2 – linear coefficients and Ω0, τ0 φ0, ω0, χ0 are the parameters of base line (center line) of j-image.

On the basing of Eq. (8) and (9) it is considered that for individual image the number of unknown parameters and linear coefficients will be 19, for two along track stereo images – 34. The total number of unknowns in strip having n images is (15xn) + 4. The number of unknown parameters in a block with N strips will be [(15xn) +4] N. There are M points in a block that need to be determined. Therefore, the total number of unknowns to be determined in block will be [(15xn) + 4] N + 3M. If the Eq. (9) is modeled with first order linear functions of t, the number of unknowns in block will be decreased about 1/3 times.

After differentiating functions Fx’jkt, Fy’jkt (Eq. 8) with inspect of unknowns (parameters and ground point coordinates), basing on M image point observations in k strip, we can create correction equation system written in the matrix form Av + Bd + L = 0 with weight matrix P (10) where: v = - residual matrix to image point coordinate observation, L = - free term matrix; A, B – the matrix of partial derivatives of observations and unknowns, d – unknown matrix of parametric increments and ground point coordinates, P – diagonal matrix of weights.

The Eq. (10) is written for following point groups: ground control and check points, conjugate points in the strip and in the block, the points that must be determined for determined purpose, other interesting points,

From equation (10) we form the normal equation system. After solving it the unknown matrix d becomes computed. Having determined parameters the transformation of any image points into geocentric reference system will be performed and next, into geodetic reference system.

It is necessary to remember that all Ground Control Points (GCP’s) have to be, at first step, transformed into geocentric reference system, in which all the operations will be done. At last step, interesting image points transformed in geocentric reference system will be carried out into geodetic system.

• If we like to give consideration to errors of sensor internal orientation, we write to Eq. (8) next eight parameters related with the errors of internal orientation elements: dx0, dy0, df – the errors of internal orientation; t1, t2, t3 – coefficients charactering error of symmetrical distortion; p1, p2 – coefficients charactering error of asymmetrical distortion.

• From subsection 3.3 it is obvious that in order to reduce an influence of ground height differences ∆h on the displacement of image points we must correct their coordinates.

7

• Second version of Keplerian method is based on the satellite velocity against satellite orbit parameters. Two along track stereo images have in total 12 unknown parameters. Other models so-called Gauss-Lambert and combined one are presented in the work of [6]. The number of orientation parameters of different models is presented in table 4.

Table 4. Number of unknown parameters for single and along track models

The number of parameters Models

Parameters Two images Three images Four images Five images

Individual

Xo, Yo , Zo ux, uy, uz φ, ω, κ

18

27

36

45

1

r, e, i, w Ω, τ φ, ω, κ

12

15

18

21

Kepler

2 Xo, Yo , Zo ux, uy, uz φ, ω, κ

12

15

18

21

Gauss-Lambert Xo, Yo , Zo φ, ω, κ

12 - - -

Combined method

Xo, Yo , Zo φ, ω, κ

- 18 24 30

4.2. Block triangulation by generic model - Rational Polynomial Coefficient (RPC)

The fundamental advantages of generic models of HRSI as writing on the beginning of section three are not required to known the physical-geometric sensor. In practice ground control points X, Y, Z needed usually to describe this model so-called Rational Polynomial Coefficient (RPC) as follows:

∑∑∑

∑∑∑

∑∑∑

∑∑∑

= = =

= = =

= = =

= = = == 1

0

2

0

3

0

1

0

2

0

3

01

0

2

0

3

0

1

0

2

0

3

0 ; n

i

n

j

n

k

kjiijk

m

i

m

j

m

k

kjiijk

n

i

n

j

n

k

kjiijk

m

i

m

j

m

k

kjiijk

ZYXd

ZYXcy

ZYXb

ZYXax (11)

In many publications there are other models used to present linear array CCD sensor as direct linear transformation, self calibration direct linear transformation, 3D affine transformation, dynamic affine transformation and self calibration dynamic affine transformation.

Grodecki el, al. [1] performed original algorithm of model RPC. Last experiment results of block triangulation with RPC method for HRSI as IKONOS, QuickBird, SPOT elaborated by Intergraph Company, presented by Madani [5] are quoted in table 5. Display of QuickBird block with its calculated results is presented in Fig. 5.

Table 5. Accuracies of block triangulation with RPC method

Ground coordinates Accuracies of images X(m) Y(m) Z(m)

Attention

IKONOS

RMS control point RMS check point RMS limit Mean std dev Max residual Residual limits RMS image

0.024 0.000 1.000 0.108 0.042 3.000

0.7

0.031 0.000 1.000 0.108 0.061 3.000

0.5

0.008 0.000 1.000 0.109 0.012 3.000

• two stereo pair, US Control points: 6 Check point: 0 Observation: 12 Interactions: 2 Degree Of Freed.: 24

QuickBird

RMS control point RMS check point RMS limit Mean std dev Max residual Residual limits RMS image

0.286 0.000 0.500 0.275 0.972 1.000

0.3

0.285 0.000 0.500 0.289 0.889 1.000

0.4

0.174 0.000 0.700 0.931 0.933 1.000

• 11 basic images, 10m DEM, US. Control points: 72 Check points: 0 Observation: 137 Interactions: 2 Degree of freed.: 274

SPOT

RMS control point RMS check point RMS limit Mean std dev Max residual Residual limits RMS image

6.829 0.000 10.000 8.069 11.782 10.000

0.3

4.105 0.000 10.000 7.810 8.398 10.000

0.4

2.955 0.000

10.000 10.869 5.604

10.000

• 4 level-1A images, France Control points: 22 Check point: 0 Observation: 37 Interactions: 3 Degree Of Freed.: 74

8

Fig.5. QuickBird block triangulation on Workstation of Z/I Imaging Block triangulation by affine transformation of VHRS converted into parallel projection Motivation of using parallel projection against perspective one of VHRS has been quoted in the subsection 3.2. Assuming the scanner roll angle ψ is known. Point coordinate transformation of VHRS acquired with perspective projection (x, y) into parallel one (x’, y’) is written in the following form:

)tan()/(1

1';'ψfy

yyxx−

== (12)

Parallel projection parameters of HRSI are components of unit projection vector (L, M), image orientation

angles (φ, ω, κ), shift values (∆x, ∆y) and scale factor s. Mathematical relationship between an object space point (X, Y, Z) and its corresponding image point (x, y’) can be expressed in vector form (non-linear) as:

(13)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡∆∆

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

00' y

x

ZYX

sRNML

Rsyx

TTλ

with R - rotation matrix between the object and image coordinate systems, 221 NMN +−= , λ – the distance between the object and image points.

A non-linear form (13) of parallel projection model can be reduced to the linear form described by 2D affine transformation. Then, the relationship between an image point of perspective projection (x, y) and its corresponding image point in parallel projection, together with the scanner roll angel ψ can be created as:

( )8765

8765

4321

)tan()/1(1 AZAYAXAfAZAYAXAy

AZAYAXAx

+++++++

=

+++=

ψ (14)

Basing on the model expressed with Eq. (14) Zhang el. al. [9] have built test block of 15 IKONOS images

taken from three orbits with area about (35x57) km2 and terrain height difference ∆h≈1000m. The accuracies for test block is represented in table 6

Table 6. The accuracies of IKONOS test block built with Eq. (14)

Ground Control Point GCP (m) Check Point CP (m) σo

(Pixel) Nr GCP ±mX ±mY ±mZ Nr CP ±mX ±mY ±mZ 0.32 86 0.37 0.44 0.74 0.31 20 0.21 0.33 0.62 66 0.59 0.58 0.87 0.32 8 0.19 0.20 0.56 78 0.56 0.72 1.15 0.35 5 0.39 0.32 0.40 81 0.87 0.85 0.93

On the basic of Tab. 6 it is obvious that with 5 GCP the accuracies of test block reach to the level of image’s

ground resolution. Information contents presented in Tab. 5 and Tab.6 give us important evaluations of RPC and model-based Eq. (14) block.

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5. Conclusion

Very high resolution satellite image triangulation (VHRSIT) has been became one of the important research directions at last time. VHRSIT networks built with rigorous (parametric or physical) model have more advantages related with possibilities to correct different systematic errors. At the result, the accuracy of VHRSIT networks will be high. For more and more developed accuracy of VHRSIT networks our further studies are focused on:

- image coordinate correction caused of Earth’s motion influence, - founding the orbital constraints related with a satellite in space orbit, - founding the rational adjustment method to allow for influence of correlated parameters of exterior

orientation elements, - influence of Ground Control Point configuration on VHRSIT network accuracy.

Acknowledgements

We would like to thank the Institute of Photogrammetry and Cartography, Warsaw University for available

materials, as well as prof. S.Białousz for valuable attentions. References [1] Grodecki J., Dial G., 2003. Block adjustment of high resolution satellite images described rational function.

Photogrammetric Engineering and Remote Sensing, Vol. 69, Nr 1. [2] Jacobsen K., 2005, High resolution satellite imaging systems – overview. Proceedings of the ISPRS Workshop,

Commission I, WG I/5, Hanover, (on CCROM). [3] Kim T., 2000. A study on the epipolarity of linear pushbroom image. Photogrammetric Engineering and Remote Sensing,

Vol. 66, Nr 8. [4] Luong C. K., Wolniewicz W., 2005. Geometrical models for very high resolution satellite sensors, 2005. (Paper to be

printed on Geodesy and Cartography of Polish Scientific Academy). [5] Madani M., 2005. Satellite image triangulation. Proceedings of the ISPRS Workshop, Commission I, WG I/5, Hanover,

(on CCROM). [6] Michalis P., Dowman I., 2005. A model for along track stereo sensors using rigorous orbit mechanics. Proceedings of the

ISPRS Workshop, Commission I, WG I/5, Hanover, (on CCROM). [7] Morgan M., Kim K., Jeong S., Habib A., 2004, Parallel projection modeling for linear array scanner scenes. 20-th

Congress of ISPRS, Istanbul, (on CCROM). [8] Toutin T., 2005. Generation of DTM from stereo high resolution sensors. Proceedings of the ISPRS Workshop,

Commission I, WG I/5, Hanover, (on CCROM). [9] Zhang J., Zhang Y., Cheng Y., 2004. Block adjustment based on new strict geometric model of satellite images with high

resolution. 20-th Congress of ISPRS, Istanbul, (on CCROM).

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