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A RIGOROUS MODEL FOR HIGH RESOLUTION SATELLITE IMAGERYORIENTATION

F. Giannone

DITS - Area di Geodesia e Geomatica, Sapienza Universita di Roma

via Eudossiana 18, 00184 Rome, Italy

[email protected]

Dottorato in Infrastrutture e Trasporti - Curriculum Infrastrutture e Topografia - XIX ciclo

Tutor: M. Crespi

KEY WORDS: High Resolution Satellite Imagery, Orientation, Modelling, Accuracy, Software

ABSTRACT:

The correct georeferencing of remote sensing imagery is a fundamental task in the photogrammetric processing for orthoimages,

DEM/DSM generation and 3D features extraction. The work developed in the PhD thesis and summarized in this paper is focused on

the georeferencing of pushbroom sensors imagery with an original rigorous model, based on the collinearity equations. In particular,

we developed, implemented and tested a model for georeferencing single images collected by EROS A and Quickbird satellites, which

is at present the unique rigorous orientation model developed in Italy. The model, implemented in the software SISAR (Software

per Immagini Satellitari ad Alta Risoluzione), reconstructs the orbital segment during the image acquisition through the Keplerian

orbital parameters, the sensor attitude, the internal orientation, allowing for self-calibration parameters too. As regards the estimation

procedure, in order to avoid instability due to high correlations among some parameters leading to design matrix pseudo-singularity,

Singular Value Decomposition (SVD) and QR decomposition were employed to evaluate the actual rank of the design matrix, to select

the estimable parameters and finally to solve the linearized collinearity equations system in the least squares sense.

To test the effectiveness of the new model, the results were compared with other well known orientation models: the rigorous model

implemented in the software OrthoEngine 10.0 (PCI Geomatica), developed by Thierry Toutin; the RPC model with affine correction

implemented in Sat PP (software implemented at ETH Zurich by the research group of Prof. Gruen). In this paper four images are

concerned, selected within the nine processed, showing that SISAR performances are at the level of the OrthoEngine ones, better than

Sat PP.

Moreover, we investigated in details the problem of the evaluation of the orientation accuracy, crucial to state the actual potentialities

for the geomatic applications of the High Resolution Satellite Imagery; in particular we studied and implemented in the SISAR software

two methodologies for the accuracy evaluation: the Hold-Out Validation (HOV) and the Leave-One-Out Cross-Validation (LOOCV)

methods.

Future prospects of this work mainly concern the stereopair orientation and the production of Rational Polynomial Coefficients (RPC)

according to an original algorithm (both already implemented and presently under evaluation), and the model extension to Ikonos,

Cartosat-1, EROS B and Prism satellites.

1 INTRODUCTION

The High Resolution Satellite Imagery (HRSI) have relevant im-

pact for cartographic applications, as possible alternative to aerial

photogrammetric imagery to produce orthophotos at 1:5000 scale

or lower, for the generation of Digital Elevation Models (DEM)

and Digital Surface Models (DSM) and also for 3D feature ex-traction (e.g. for city modeling), especially in areas where the or-

ganization of photogrammetric surveys may result critical. How-

ever, the real possibility of using HRSI for cartography depends

on several factors: sensor characteristics (geometric and radio-

metric resolution and quality), types of products made available

by the companies managing the satellites, cost of the software for

the final processing to realize the cartographic products.

The first and fundamental task to be addressed is the imagery dis-

torsions correction, that is the so called orientation and orthorecti-

fication. The distortions sources can be related to two general cat-

egories: the acquisition system, which includes the platform ori-

entation and movement and the imaging sensor optical-geometric

characteristics, and the observed object, which takes into account

atmosphere refraction and terrain morphology. At present, HRSIorientation methods can be classified in three categories: black

models (like Rational Polynomial Function - RPF), consisting in

purely analytic functions linking image to terrain coordinates, in-

dependently of specific platform or sensor characteristics and ac-

quisition geometry; physically based models (so called rigorous

models), which take into account several aspects influencing the

acquisition procedure and are often specialized to each specific

platform and sensor; the gray models (like Rational Polynomial

Coefficients - RPCmodels), in which the mentioned RPF are used

with known coefficients supplied in the imagery metadata and

”blind” produced by companies managing sensors by their own

secret rigorous models.Since 2003, the research group at the Area di Geodesia e Geomat-

ica - Sapienza Universita di Roma has developed a specific and

rigorous model designed for the orientation of imagery acquired

by pushbroom sensors carried on satellite platforms, like EROS-

A, QuickBird. This model has been implemented in the software

SISAR (Software per Immagini Satellitari ad Alta Risoluzione).

The first version of the model was uniquely focused on EROS-

A imagery, since no commercial software including a rigorous

model for this platform were available at that time. Later, the

model was refined and extended to process QuickBird Basic im-

agery too.

The model structure is outlined in §2. A particular care was de-

voted into the estimable and significant parameters selection. In

fact not all the parameters included in a rigorous model may bewell estimated from the available data; therefore, estimable pa-

rameters have to be automatically identified, whereas the non es-

timable ones have to be properly constrained. This results was



achieved by a selection strategy based on a Singular Value De-

composition (SVD) followed by a QR decomposition.

This ‘tricky’ but crucial topic is resumed in §3.

Another topic studied in this research, and presented in §4, is the

evaluation of the images accuracy, strictly conditioning the ac-

tual imagery potentiality; for this reason two methodologies for

the accuracy evaluation were studied and implemented. In the

most common method (HOV) the known ground points (GPs)

are partitioned in two sets, the first used in the orientation model

(GCPs - Ground Control Points) and the second to validate the

model itself (CPs - Check Points); then the accuracy is computed

as the RMSE of residuals between imagery derived coordinates

and CPs coordinates. An alternative is the use of the Leave-One-

Out Cross-Validation method; it mainly consists in the iterative

application of the orientation model using all the known GPs (or

a subset of them) as GCPs except one, different in each itera-

tion, used as CP. In every iteration the residual between imagery

derived coordinates and CP coordinates (prediction error of the

model on CP coordinates) is calculated; the overall spatial accu-

racy may be estimated by calculating the usual RMSE or, better,

a robust accuracy index like the mAD (median Absolute Devia-tion) of the prediction errors on all the iterations.

In §5 some features of the SISAR software are resumed, whereas,

for the sake of brevity, the results of the comparisons with Ortho-

Engine and Sat PP regarding only 4 images with different fea-

tures are summarized in §6.

2 MODEL DEFINITION

The orientation model implemented in SISAR is a rigorous one,

based on a standard photogrammetric approach describing the

physical-geometrical imagery acquisition. Of course, in this case,

it has to be considered that an image stemming from a pushbroom

sensor is formed by many (from thousands to tens of thousands)lines, each acquired with a proper position (projection center) and

attitude. All the acquisition positions are related by the orbital

dynamics. Therefore, rigorous model implemented in SISAR is

based on the collinearity equations, with the reconstruction of the

orbital segment during the image acquisition through the knowl-

edge of the acquisition mode, the sensor parameters, the satel-

lite position and attitude parameters. The approximate values

of these parameters can be computed thanks to the information

contained in the metadata file delivered with each image; these

approximate values must be corrected by a least squares (LS)

estimation process based on a suitable number of Ground Con-

trol Points (GCPs). Moreover, in order to relate the image to

the ground coordinates, expressed in an Earth Centered - Earth

Fixed (ECEF) reference frame, a set of rotation matrices have tobe used. These matrices include those needed to shift between

sensor, platform, orbital and Earth Centered Inertial (ECI) coor-

dinate systems, while the transformation between ECI and ECEF

coordinate systems must take into account precession, nutation,

polar motion and Earth rotation matrices (Kaula, 1966).

2.1 Coordinate systems

In order to describe the collinearity equations (§2.5), the defini-

tions of some coordinate systems are needed:

Image system (I): is a 2-dimensional system describing a point

position in an image. The origin is in the upper left corner, the

pixel position is defined by its row (J ) and column (I ). The

column numbers increases toward the right and row numbers in-creases in the downward direction

Sensor system (S): the origin is in the perspective center (which

in this case may be considered coincident with the satellite center

of mass), the x-axis is tangent to the orbit directed as the satellite

motion, the z-axis is directed from the perspective center to pixel

array and y-axis is parallel to pixel array

Body system (B): it is aligned to the Flight system (see below)

when the angle Roll (ϕ), Pitch (ϑ) and Yaw (ψ) are zero

Flight system (F): the origin is in the satellite center of mass (per-

spective center), the X-axis is tangent to the orbit along the satel-

lite motion, the Z-axis is in the orbital plane directed toward the

Earth center of mass and the Y-axis completes the right-handed

coordinate system

Earth Centered Inertial system - ECI (I): the origin is in the

Earth center of mass, the X-axis points to vernal equinox (epoch

J2000 - 1 January 2000, ore 12 UT), the Z-axis points to celes-

tial north pole (epoch J2000) and the Y-axis completes the right-

handed coordinate system

Earth-Centered Earth-Fixed system ECEF (E): the origin is

in the Earth center of mass, the X-axis is the intersection of equa-

torial plane and the plane of reference meridian (epoch 1984.0),

the Z-axis is the mean rotational axis (epoch 1984.0) and the Y-

axis completes the right-handed coordinate system

The transformation matrix from the sensor systems to the ECI

one can be expressed through three rotations (Westin, 1990):

RSI = RFI · RBF · RSB (1)

Inertial-Flight matrix (RFI ): it allows the passage from the in-

ertial geocentric system (ECI) to the orbital one; it is a function of

keplerian orbital parameters and varies with the time inside each

scene (for each image row J)

RFI = [Rx(−π/2) · Rz(π/2)] · [Rz(U ) · Rx(i) · Rz(Ω)] (2)

where i is the inclination, Ω the right ascension of the ascending

node, U = ω + v with ω argument of the perigee and v true

anomaly

Flight-Body matrix (RBF ): it allows the passage from the or-

bital system to the body one through the attitude angles (ϕ, ϑ, ψ)which depend on time (for each pixel row)

RBF = Rz(ψ) · Ry(ϑ) · Rx(ϕ) (3)

Body-Sensor matrix (RSB ): it allows the passage from the body

to the sensor system. This matrix considers defect of parallelism

between axes (X ,Y ,Z )S and (X ,Y ,Z )B and it is considered

constant during a scene for each sensor; the elements of the ma-

trix are usually provided in the metadata files.

The product of REI and RT SI matrices allows the passage from

sensor S to ECEF system, the final rotation matrix being:

RES = REI · RT SI = Rz(K ) · Ry(P ) · Rx(W ) (4)

the angles (K , P , W ) define the satellite attitude at the moment

of the acquisition of image row J with respect to the ECEF sys-

tem.

The rotation matrix for the transformation from ECI system to

ECEF system (REI ) can be subdivided into four different steps,

considering the motions of the Earth in the space: precession, the

secular change in the orientation of the Earth’s rotation axis and

the vernal equinox (described by the matrix RP ); nutation, the

periodic and short-term variation of the equator and the vernal

equinox (described by the matrix RN ); polar motion, the coor-

dinates of the rotation axis relative to the IERS Reference Pole

(described by the matrix RP ) and Earth’s rotation about its axis(described by the Sideral Time through the matrix RS) (Mon-

tenbruck and Gill, 2001).

REI = RM · RS · RN · RP (5)



2.2 Keplerian parameters

The satellite orbit can be described using the well-known six Ke-

plerian elements. According to Keplerian laws, a satellite (con-

sidered as material point) under the effect of a gravitational field

generated by a mass concentrated in a point moves in a planedescribing an elliptic orbit; the satellite position at each generic

epoch T can be represented by seven parameters (Fig. 1):

semi-major axis (a): it is semi-major axis of the orbital ellipse

orbit inclination (i): it is the angle (positive if counter clock

wise) between the orbital plane and the equatorial plane; by con-

vention the inclination is a number between 0 and πright ascension of the ascending node (Ω): it is the angle (pos-

itive if counter clock wise observed from the North pole) at the

center of the Earth from the vernal equinox to the ascending node

eccentricity (e): it is the eccentricity of the orbital ellipse

true anomaly (v): it is the angle measured in the center of the

ellipse between the perigee and the position of the satellite at

generic epoch T - defined to be 0 at perigee -

argument of the perigee (ω): it is the angle between the nodalline (intersection between the orbital plane and the equatorial

plane) and the semi-major axis, measured in the orbital plane

from the ascending node to the perigee

time of the perigee passage (T p): it is the time referring to the

epoch when the satellite is nearest to the Earth

Figure 1: Keplerian parameters

2.3 Attitude angles

To define the sensor’s position during the acquisition it is nec-

essary to know its attitude with respect to the satellite center of

mass; the attitude is describe by the roll (ϕ), pitch (ϑ) and yaw(ψ) angles referred to the axes of orbital system (Fig. 2). The ap-

Figure 2: The attitude angles

proximate values of this angles are calculated with the metadata

file information; the corrections to these approximate values are

supposed to be modelled by a second order polynomials. Note

that there is not any physical meaning in doing this, but good

results seem to support this choice8>>>><>>>>:

ϕ = ϕ + a0 + a1τ + a2τ 2

ϑ = ϑ + b0 + b1τ + b2τ 2

ψ = ψ + c0 + c1τ + c2τ 2

(6)

τ is the time, in seconds, such as τ = J · integration time,

where the integration time is the time needed to physically ac-

quire a row and J is the row of the pixel. The nine coefficients

(ai, bi, ci) are unknown and need too be estimated within the LS

adjustment.

2.4 Internal orientation and self-calibration parameters

The internal orientation parameters describe the intrinsic geomet-

ric features of the sensor. Usually, for the pushbroom sensor

the modelled parameter is the focal distance f . Moreover self-

calibration parameters are used to correct the systematic errors

due to geometric distortion of CCD linear array sensor and thelens distortions. The main errors that may occur in the linear ar-

ray sensors are the shift or/and rotation of the CCD segment in

the focal plane with respect to their nominal position (Fig. 3):

• I 0 is the sensor shift in column-direction with respect to the

first pixel of array; its approximate value is equal to half of

the columns number (I 0approx = ncol/2)

• J 0 is the sensor shift in row-direction with respect to the

central pixel; its approximate value is zero

• k in the horizontal rotation in the CCD plane; its approxi-

mate value is zero

Figure 3: Self-calibration parameters: I 0 (a), J 0 (b) and k (c)

The lens distortion may be described by two coefficients ( d1, d2)

and modelled with a second order polynomial (d1(I − I 0) +

d2(I − I 0)

2

).

2.5 The rigorous model

As mentioned before, a rigorous model is based on the collinear-

ity equations and describe the imagery acquisition both from ge-

ometrical and physical point of view. It is now possible to write

the collinearity equations relating to the position of a point in

the image space to the corresponding point in the object space,

according to a central projection. In our case, the collinearity

equations may be conveniently expressed in the Earth Centered

Inertial (ECI) system in vector form (Fig. 4):

˛X tY tZ t

˛ = ˛X sY sZ s

˛+d ˛ux

uy

uz

˛ ⇒ ˛ux

uy

uz

˛ = 1

d˛

X t − X sY t − Y sZ t − Z s

˛ (7)

where: (X t, Y t, Z t) are the ECI coordinates of the ground point;

(X s, Y s, Z s) are the ECI coordinates of the satellite; (ux, uy, uz)



Figure 4: Central projection model

are the component in the ECI system of the unit vector u directed

from satellite to GP and d is the satellite to GP distance.

The collinearity equations in vector form written using the sensor

system is:

˛˛ xs

ys

f

˛˛ = RSI

˛˛˛

ux

uy

uz

˛˛˛ ⇒

˛˛˛

xs

ys

f

˛˛˛ =

1

dRSI

˛˛ X t − X s

Y t − Y sZ t − Z s

˛˛ (8)

where f is the focal distance and RSI is the rotation matrix from

ECI to sensor system.

The two collinearity equations for each GP are obtained dividing

the first two equations of (8) by third one:8><

>:

xsf

= RSI,11|Xt−Xs|+RSI,12|Y t−Y s|+RSI,13|Zt−Zs|

RSI,31|Xt−Xs|+RSI,32|Y t−Y s|+RSI,33|Zt−Zs|

ysf

= RSI,21|Xt−Xs|+RSI,22|Y t−Y s|+RSI,23|Zt−Zs|

RSI,31|Xt−Xs|+RSI,32|Y t−Y s|+RSI,33|Zt−Zs|

(9)

With simple geometric consideration (Fig. 5) we can write thecollinearity equations as a function of image coordinates, internalorientation and self-calibration parameters, Keplerian orbital andattitude parameters:8>>><>>>:

xsf

= tanβ = dpix

f [J − int (J ) − 0.5 − J 0 − k (I − I 0)]

ysf

= − tanα = −dpixf

nd1 (I − I 0) + d2 (I − I 0)2 +

+k [J − int (J ) − 0.5 − J 0](10)

Figure 5: The image coordinates in the sensor system

The collinearity equations are a function of the parameters de-

scribed in § 2.2, § 2.3 and § 2.4. The approximate values for allparameters may be derived from the information contained into

the metadata files released together with the imagery or they are

simply fixed to zero. In theory, these approximate values must be

corrected by an estimation process based on a suitable number of

Ground Control Points (GCPs), for which the collinearity equa-

tions are written; nevertheless, since the orbital arc related to each

image acquisition is extremely short (few hundreds of kilometers)

if compared to the whole orbit length (tens of thousandths), some

Keplerian parameters are not estimable at all (a, e, ω) and others

(i, Ω, T P ) are extremely correlated both among them and with

sensor attitude and viewing geometry parameters.

In order to avoid instability due to high correlations among some

parameters leading to design matrix pseudo-singularity, Singu-

lar Value Decomposition (SVD) and QR decomposition are em-

ployed to evaluate the actual rank of the design matrix, to se-

lect the estimable parameters and finally to solve the linearized

collinearity equations system in the LS sense.

3 MODEL COMPUTATION

3.1 Functional model

The functional model (Sanso, 1989) in synthetic form reads:

f (y, x) = 0. In this model the two collinearity equations F 1and F 2 (10) are written for each available GCP; if n the GCPsnumber, the matrices can be sketched as in the follows:design matrix

A =

˛˛˛˛˛˛˛˛˛

A1

A2

.

.

.An

˛˛˛˛˛˛˛˛˛

→ 1st GCP

→ 2nd GCP

→ nth GCP

(11)

AT i =

˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛

˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛

∂ a0F 1 ∂ a0F 2 ∅ ∅ ∅

∂ a1F 1 ∂ a1F 2 ∅ ∅ ∅

· · · · · · ∅ ∅ ∅

∂ c2F 1 ∂ c2F 2 ∅ ∅ ∅

∂ T P F 1 ∂ T P F 2 ∅ ∅ ∅

∂ f F 1 ∂ f F 2 ∅ ∅ ∅

∂ I0F 1 ∂ I0F 2 ∅ ∅ ∅

∂ J 0F 1 ∂ J 0F 2 ∅ ∅ ∅

∂ kF 1 ∂ kF 2 ∅ ∅ ∅

∂ d1F 1 ∂ d1F 2 ∅ ∅ ∅

∂ d2F 1 ∂ d2F 2 ∅ ∅ ∅

∂ XF 1 ∂ XF 2 1 ∅ ∅

∂ Y F 1 ∂ Y F 2 ∅ 1 ∅

∂ ZF 1 ∂ ZF 2 ∅ ∅ 1

˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛

˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛

(12)

unknown vector is X T

= |a0, a1, a2, b0, b1, b2, c0, c1, c2,T p, f , I 0, J 0, k , d1, d2, δX 1, δY 1, δZ 1, · · · , δZ n|observation vector

Y =

˛˛˛˛˛˛˛˛˛˛˛˛˛

Y 1...Y i

.

.

.Y n

˛˛˛˛˛˛˛˛˛˛˛˛˛

→ for ith

GCP → Y i =

˛˛˛˛˛˛˛˛˛

I

J XY Z

˛˛˛˛˛˛˛˛˛

→ imagecoordinates

→ groundcoordinates

(13)known vector

d =

˛˛˛˛˛˛˛

˛˛˛˛˛˛˛˛˛˛

d1d2

.

.

.

dnX1

X2

.

.

.Xn

˛˛˛˛˛˛˛

˛˛˛˛˛˛˛˛˛˛

→ 1st GCP

→ 2nd GCP

→ nth GCP→ coordinates of 1st GCP

→ coordinates of 2nd GCP

→ coordinates of nth GCP

(14)



matrix of observation coefficients

D =

˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛

D1 ∅

D2

. . .

∅ Dn

∅ · · · ∅

.

.

.∅ · · · ∅

˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛

→ 1st GCP

→ 2nd GCP

→ nth GCP

→ pseudo-observation

Di =

˛˛˛˛˛

∂ IF 1 ∂ J F 1

∂ IF 2 ∂ J F 2

˛˛˛˛˛

(15)

3.2 Stochastic model

The stochastic model is: C Y Y = σ20Q, where (a priori choice)

σ0 = 1 and the cofactor matrix is chosen as a simple diagonalmatrix for observation (I, J ; X t, Y t, Z t):

C Y Y = Q =

˛˛˛˛˛˛

C IJ ∅∅ C IJ

. . .

C XY Z ∅∅ C XY Z

˛˛˛˛˛˛

(16)

where 8>>>>>><>>>>>>:

C IJ =

˛˛ σ2

I ∅

∅ σ2j

˛˛

C XY Z =

˛˛˛σ2X

∅ ∅∅ σ2Y ∅∅ ∅ σ2Z

˛˛˛

(17)

The standard deviations of the image observations are set equal,

considering that manual collimation tests carried out indepen-dently by different operators showed that an accuracy ranging

from 1/3 to 1/2 pixel in image coordinates may be routinely

achieved; for the GCP coordinates standard deviations are usu-

ally set equal to mean values obtained during their direct surveys

or cartographic selection.

3.3 Singular Value Decomposition

The Singular Value Decomposition (SVD) is a very powerful

techniques to dealing with sets of equations or matrices that are

either singular or numerically very close to be singular.

The SVD of a matrix A ∈ ℜm×n

(with m ≥ n) is any factoriza-tion of the form:

A = U W V T (18)

where W ∈ ℜn×n is a diagonal matrix with positive or zero

elements (wij) that are the singular values of A; U ∈ ℜm×n and

V ∈ ℜn×n are orthogonal matrices, whose columns (uj , vj) are

called the left and right singular vectors.

For a system of linear equations (Ax = b), using the SVD we can

write ((Golub and Van Loan, 1993), §5.5.3):

U W V T x = b (19)

and the LS solution x minimizes Ax − b2.

Since the orthogonal matrix preserves the norm, for any x ∈ ℜn

we have:

Ax − b22 =‚‚(U T AV )(V T x) − U T b

‚‚22

=

=Pr

i=1(wiz i − uT i b)2 +

Pm

i=r+1(uT i b)2

(20)

where z = V T x and r is the rank of A.

Of course Ax − b22 = min if

rXi=1

(wiz i − uT i b)2 = 0 (21)

then using the SVD, the LS problem is now in terms of a diagonal

matrix, and finally8><>:

xi = uT i b

wivi if wi = 0

xi = anything if wi = 0

(22)

The advantage of using the SVD is that it can reliably handle the

rank deficient case as well as the full rank case.

3.4 QR decomposition

The QR decomposition of a matrix A ∈ ℜm×n (m ≥ n) is given

by:

A = QR (23)

where Q ∈ ℜm×m is an orthogonal matrix and R ∈ ℜm×n is an

upper triangular matrix. If the rank of A is equal to n, the first

n columns of Q form an orthonormal basis for the Range(A).

Thus, the calculation of the the QR factorization is a way to com-

pute an orthonormal basis for a set of vectors.

The standard algorithm for the QR decomposition involves se-

quential evaluation of Householder transformations. An appro-

priate Householder matrix, applied to a given matrix, can zero all

the elements, situated below of a given element, in a column of

the matrix. For the first column of matrix A an appropriate ma-

trix H 1 is evaluated, which put on zero all the elements below

the first element in the first column of A. Similarly H 2 zeroes all

elements in the second column below the second element and soon up to H n−1

R = H n−1 · · · H 1A (24)

where QT = H n−1 · · · H 1, i.e., Q = H 1 · · · H n−1.

The generic matrix H i zeroes all elements in the first column be-

low the first element for a sub-matrix of A (Ai ∈ ℜ(m−i)×(n−i)).

If A is rank deficient, the QR factorization do not give a basis for

the Range(A). In this case to calculate an orthonormal basis for

Range(A) it is necessary to compute the QR decomposition of a

column-permuted version of A, i.e., AP = QR ((Golub and Van

Loan, 1993), §5.4)

QT AP =

» R11 R12

0 0

– →

→r

m − r

↓ ↓r n − r

(25)

where P is a permutation, r is the rank of A, R11 is an upper

triangular and non singular matrix and Q and P are products of

Householder matrices Q = H 1 · · · H r , P = P 1 · · · P r.

For understanding the permutation matrix role it is necessary to

define the vector N ∈ ℜm for a generic matrix A ∈ ℜm×n:

Amn =

˛˛˛˛

a11 a12 · · · a1na21 a22 · · · a2n

..

. · · ·am1 · · · amn

˛˛˛˛

(26)

N m =˛

mPk=1

a2k1mPk=1

a2k2 ... mPk=1

a2kn˛

(27)

The element of the N are the square value of norm calculated

for each column of A. The permutation matrix P applied at the



generic matrix A makes a matrix AP = AP such that the el-

ements of the corresponding vector N are placed in decreasing

order. As for the generic matrix H i, the generic matrix P i per-

mute the column of a sub-matrix of A (Ai ∈ ℜ(m−i)×(n−i))(Fig.

6(a)); if k is the column with the maximum value of norm, the

permutation matrix P i exchange the columns i and k (Fig. 6(b)).

In a system of linear equations (Ax = b), if A ∈ ℜm×n and

(a) (b)

Figure 6: Ai before (a) and after (b) the permutation

has a rank r , the QR decomposition produces the factorization

AP = QR where R is described in the equation (25). As for the

LS problem we have:

Ax − b22 =‚‚(QT AP )(P T x) − QT b

‚‚22

=

= R11y − (c − R12z )22 + d22

(28)

where

P T x =

» yz

– →

→r

n − r

QT b =»

cd

– →

→r

m − r

(29)

If x is a LS minimizer we have

x = P

» R−1

11 (c − R12z )z

– (30)

If z is a set of zeroes in this expression, we obtain the basic solu-

tion:

xB = P

» R−1

11 c0

– (31)

xB has at most r non-zero components and so AxB involves a

subset of A columns.

3.5 Subset selection using the SVD and QR

For a system of linear equations (Ax = b), with A ∈ ℜm×n

(m ≥ n) it is necessary to select the estimable parameters. We

describe an SVD-based subset selection procedure, due to Golub,

Klema and Stewart ((Golub and Van Loan, 1993)§12.2), that pro-

ceeds as follows:

• we compute the SVD A = U W V T and use it to determine

a rank estimate r

• with the QR decomposition QR = AP we select an in-

dependent subset of A columns; if R11xB = QT b with

xB ∈ ℜr and we set:

y = P

» xB

0

– (32)

then Ay is an approximate LS predictor of b that involves

the first r columns of AP . The permutation matrix P is

calculated so that the columns of the matrix B1 ∈ ℜm×r in

AP = [B1 B2] are “sufficiently independent”

• we predict b with the vector Ay where y is described in the

equation (32), and z minimizes B1xB − b2

4 ACCURACY EVALUATION

One of the main objectives of our studies about orientation was

the definition of an effective methodology to assess the spatial ac-

curacy achievable from the oriented imagery, strictly condition-

ing their actual potentiality for geomatic applications. For this

reason considered two methods: the standard, commonly applied

method, named Hold-Out Validation (HOV) and the Leave One

Out Cross Validation (LOOCV), a new strategy, already applied

in spatial accuracy assessment but here applied for the first time

to HRSI orientation accuracy assessment.

4.1 Hold Out validation - HOV

Currently, the most used (essentially, the unique) method to as-

sess spatial accuracy of oriented HRSI corresponds to the hold-

out validation (HOV), also known as test sample estimation. Ac-

cording to it the data set (known GPs) is partitioned in two sub-

sets, selecting a training set (GCPs) and a test set (CPs). The only

restrictions on such selection is to have both sets of similar size

sufficiently well-distributed on the whole image; apart from this

consideration, the selection should be random. Once the model

is trained, accuracy is usually evaluated as Root Mean Squared

Error (RMSE) of residuals between the oriented image derived

coordinates with respect to CPs coordinates.

This method has the advantage of being simple and easy to com-pute, but it also has some drawbacks, as it is generally not reliable

and it is not applicable when a low number of GPs is available.

First of all, once the two sets are selected, accuracy estimate is

not reliable since it is strictly dependent on the points used as

CPs; if outliers or poor quality points are unfortunately included

in the CPs set, accuracy estimate is biased. In addition, when a

low number of GPs is available, almost all of them are used as

GCPs and very few CPs remain, so that RMSE may be computed

on a poor (not significant) sample. In these cases accuracy as-

sessment with the usual procedure is essentially lost. In addition,

this method makes a poor use of the available information, as a

large part of it is just used for validation purposes.

4.1.1 Modelling accuracy vs. GCPs number A first inves-tigation step was devoted to assess the accuracy and the mini-

mum number of GCPs to achieve it for each image, according

to the classical HOV. In fact, it was proved in many situations

that, when GCPs number increases, the image accuracy increases

reaching a maximum; no further enhancement may be got even if

other GCPs are added.

In order to study these trends a simple model it has been imple-

mented in order to represent it. The function suited for the fit has

been recognized to be a decreasing exponential:

y = a · ebx−t(33)

where y is the CPs RMSE, x is the number of GCPs used to build

the model, a and b are estimated with standard deviation (σa, σb)by the LS adjustment. The value of t is calculated iteratively

starting with t = 1 and estimating the two coefficients a and b;

if the difference from the asymptotic value at x = nGCP /2 (that

is y(nGCP /2) − a), is significant (e.g. > 1cm) the value of t



is increased of a unit and the estimates of a and b has to be re-

computed. This choice to constrain the slope of the function to

the asymptotic value has been done in order to avoid that possible

false oscillations of estimated accuracy on few CPs can affect the

estimation of a and b parameters.

The asymptotic value a ± σa enables the determination of the

number of GCPs (nGCP ) sufficient for achieve the maximum ac-

curacy, that is the integer value large then the x value so that

y = a + 1.96σa (95% upper confidence interval) under the hy-

pothesis of normal distribution of RMSE value.

nGCP = intsup t

s b

ln(a + 1.96σa) − ln a (34)

Figure 7: Example of RMSE fit

4.2 Leave One Out Cross Validation - LOOCV

Leave-one-out cross-validation (LOOCV) is the proposed alter-native to the HOV to perform accuracy assessments of oriented

HRSI. This method is a special case of the general k-fold cross-

validation method, which involves the partitioning of the original

data set in k subsets of equal size (approximately). The model

is trained k times, using each subset in turn as the test set, with

the remaining subsets being the training set. The overall accuracy

can be obtained averaging the accuracy values computed on each

subset (Brovelli et al., 2006).

LOOCV is k-fold cross-validation computed with k = n, where

n is the size of the original data set. Each test set is therefore

of size 1, which implies that the model is trained n times. This

method applied to HRSI orientation involves the iterative applica-

tion of the orientation model, using all the known GPs as GCPs

except one, different in each iteration, used as CP. In every it-eration the residual between imagery derived coordinates with

respect to CP coordinates (prediction error of the model on CP

coordinates) is calculated; the overall spatial accuracy achiev-

able from the oriented image may be estimated by calculating

the usual RMSE or, better, a robust accuracy index like the mAD

(median Absolute Deviation) of the errors on all the iterations.

In this way we overcome the mentioned drawbacks of the classi-

cal HOV procedure: it is a reliable and robust method, indepen-

dent from a particular set of CPs and on outliers, and it allows us

to use each known GP both as a GCP and as a CP, capitalising all

the available ground information.

The CP residuals are analyzed to point out the critical points, that

is the CP having the absolute value of residual greater than three

times the mAD (|resCP | > 3 · mAD). The position of thecritical points with respect to the perimeter of the area covered

by GCPs has been analyzed: this analysis evaluates if the critical

points are along the perimeter and if it is necessary to filter out the

effect of the anomalous residuals with a robust accuracy index.

5 SOFTWARE IMPLEMENTATION

The SISAR software was developed in C++ language; it consists

of one main program and 52 subroutines for a total amount of

about 7000 lines.

The software flow chart is sketched in the figure 8:

Figure 8: SISAR flow chart

6 ANALYSIS OF RESULTS

The SISAR rigorous model has been tested over nine images. In

the frame of this paper we discuss the results obtained in the pro-

cessing of two EROS A and two QuickBird images, with different

geometric characteristics (Tab. 1).

Area GSD

Off-nadir Scene

GPangle () coverage

[m] start end (Km×Km)

EROS A

ITA1-e1038452 (Rome) 1.80 9.1 9.4 13×10 49

ITA1-e1090724 (Rome) 2.60 31.0 40.1 17×12 49

QuickBirdAugusta (*P002) 0.75 2 8.2 (mean value) 20×19 39

Salerno 0.67 20.0 (mean value) 48×18 57

Table 1: Main features of EROS A and QuickBird images

The two images of Rome cover areas of different extensions but

the GPs were set only in the area shared by all the scenes. The

GPs were surveyed with static or fast static procedures by a geode-

tic quality GPS receivers and their coordinates were estimated by

Trimble Geomatic Office software with respect to available GPS

permanent stations (M0SE at Rome Faculty of Engineering). The

mean horizontal and vertical accuracies of the coordinates are be-

tween 10 cm and 30 cm.

The Salerno image was created stitching three different Quick-Bird images coming from the same orbital segment; the particu-

larity of this image is the wide latitude extension around 48 Km.

Also the GPs for the Augusta and Salerno images were surveyed

by geodetic quality GPS in RTK mode; the mean horizontal and



vertical coordinate accuracies in this case are between 5 cm and

10 cm.

The ground coordinates for all GPs were expressed in the WGS84-

ETRF89 reference frame (Italian realization by IGM95 network)

while the orthometric heights were obtained applying geoid un-

dulations according to the ITALGEO95 model (public model).

To test the effectiveness of the new model, SISAR results were

compared with other well known orientation models: the rigor-

ous model implemented in the software OrthoEngine 10.0 (PCI

Geomatica) and developed by Thierry Toutin (Toutin, 2004); the

RPC model with affine correction implemented in the software

Sat PP and developed at ETH Zurich in the research group of

Prof. Gruen (Zhang, 2005).

satellite software model

EROS A OrthoEngine rigorous

SISAR rigo rous

QuickBird

OrthoEngine rigorous

Sat PP RPC

SISAR rigo rous

Table 2: Software and models comparison protocol

Note that EROS A was not processed with Sat PP since this soft-

ware does not implement a rigorous model for asynchronous sen-

sors and EROS A imagery are not supplied with RPC.

For each image, the first investigation concerned the accuracy

analysis according to the classical HOV; afterward the LOOCV

method was used to evaluate the accuracy of two EROS A im-

ages in an alternative way by using the same GCP sets suited to

exploit the maximum accuracy according to the HOV. It is to be

remembered that the two strategies are generic and can be ap-

plied to the results of any software; the HOV strategy has been

largely applied to various images and to three different software

- SISAR, Sat PP and OrthoEngine - while the LOOCV has beenapplied only to SISAR and OrthoEngine, since when the strategy

was defined and implemented Sat PP was not available anymore.

6.1 EROS A results - HOV method

The EROS A images were oriented 11 times with different num-

ber of GCP (n GCP: 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49).

The RMSE results of the CPs have been fit, separately in North

and East components, in order to evaluate the number of GCPs

needed to stabilize the accuracy.

6.1.1 Rome ITA1-e1038452 The analysis of the image accu-

racy shows that the results of OrthoEngine are better than SISAR

ones (Fig. 9 and Tab. 3).

Figure 9: Image accuracy vs. GCP number for ITA1-e1038452

GCP

Ortho Engine SISAR

RMSE CP RMSE CP

East North East North

[m] [m] [m] [m]

9 2.05 3.75 2.28 3.23

13 1.91 2.28 2.25 3.16

17 2.06 1.91 2.34 2.8521 2.17 1.90 2.38 2.57

25 2.24 1.88 2.44 2.61

29 1.90 1.66 2.38 2.70

33 1.61 1.32 2.31 2.36

37 1.43 1.41 2.00 2.53

41 1.75 1.46 2.08 2.34

45 0.98 1.63 1.41 2.54

FIT Results

n GCP 9 16 4 15

Table 3: Results for ITA1-e1038452 (γ = 9.1 → 9.4)

6.1.2 Rome ITA1-e1090724 The SISAR results for this sec-

ond image are better than OrthoEngine ones, especially in North

component (Fig. 10 and Tab. 4).

Figure 10: Image accuracy vs. GCP number for ITA1-e1090724

GCP

Ortho Engine SISAR

RMSE CP RMSE CP

East North East North

[m] [m] [m] [m]

9 4.43 13.81 3.97 6.38

13 4.37 8.64 4.05 6.58

17 4.75 7.54 4.11 5.45

21 4.90 7.66 4.33 5.76

25 3.18 6.72 3.19 7.11

29 3.08 6.90 2.67 4.48

33 3.27 7.27 2.78 4.5537 3.35 7.66 2.82 4.98

41 3.19 5.60 2.43 3.20

45 3.42 4.34 3.04 3.35

FIT Results

n GCP 11 16 11 11

Table 4: Results for ITA1-e1090724 (γ = 31.0 → 40.1)

Comparing the results of ITA1-e1038452 and ITA1-e1090724

images, SISAR produce much competitive results when the off-

nadir angle is larger, thus allowing a wider range of acquisition

conditions.

6.2 QuickBird results - HOV method

6.2.1 Augusta (*P002) The image was oriented 8 times and

for each test we used a different number of GCP (n GCP:9, 13,

17, 21, 25, 29, 34, 39).



Figure 11: Image accuracy vs. GCP number for Augusta *P002,

rigorous model for SISAR and OrthoEngine

Figure 12: Image accuracy vs. GCP number for Augusta *P002,rigorous model for SISAR and RPC with affine correction for

Sat PP

For the image accuracy the three software show similar behav-

iors, except for the East component of Sat PP (Fig. 11, 12 and

Tab. 5).

GCP

OrthoEngine SISAR SAT PP

RMSE CP RMSE CP RMSE CP

East North East North East North

[m] [m] [m] [m] [m] [m]

9 0.80 1.11 1.09 0.89 1.51 0.94

13 0.93 1.06 0.96 0.94 1.26 1.02

17 0.80 1.17 0.93 1.00 1.27 0.96

21 0.79 0.99 1.02 1.08 1.38 0.98

25 0.83 1.02 0.97 0.99 1.41 0.77

29 0.90 1.00 0.97 0.88 1.44 0.81

34 0.99 1.05 0.87 0.40 1.25 0.33

FIT Results

n GCP 4 10 13 9 11 8

Table 5: Results for Augusta *P002 (mean γ = 28.3)

6.2.2 Salerno The image was oriented 13 times and for each

test we used a different number of GCP (n GCP: 9, 13, 17, 21,

25, 29, 33, 37, 41, 45, 49, 53, 56). This image was not orientedwhit Sat PP because the vendors RPC are not available.

The SISAR results are in good agreement with OrthoEngine one

except for the North component with few GCPs where the SISAR

results are better (Fig. 13 and Tab. 6).

Figure 13: Image accuracy vs. GCP number for Salerno, rigorous

model for SISAR and OrthoEngine

GCP

Ortho Engine SISAR

RMSE CP RMSE CP

East North East North[m] [m] [m] [m]

9 0.98 3.07 0.59 1.43

13 0.89 0.69 0.53 0.96

17 0.73 0.84 0.52 0.84

21 0.72 0.65 0.56 0.80

25 0.69 0.63 0.54 0.79

29 0.59 0.61 0.42 0.85

33 0.57 0.60 0.44 0.85

37 0.55 0.64 0.43 0.83

41 0.61 0.49 0.46 0.76

45 0.68 0.49 0.32 0.81

49 0.72 0.54 0.39 0.48

53 0.86 0.41 0.47 0.71

FIT Results

n GCP 17 19 13 17

Table 6: Results for Salerno (mean γ = 20.0)

6.3 EROS-A1 images - LOOCV method

The results analysis (Tabs. 8 and 9) highlighted that the same GP

(point 5) is critical for both software packages, but in different

images (Tab. 7); in these cases the contribution of a robust ac-

curacy index is remarkable, since LOOCV RMSEs are biased by

the extremely high residuals on point 5 but in different manners.

In any case, it has to be noted that when point 5 acts as a GCP,

no particular problems in the model precision and orientation ac-

curacy is evidenced. A possible explanation may be suggested

by the position of point 5, the northernmost along the perimeter

of the area covered by GPs (Fig. 14(a) and 14(b)): the estimated

orientation models are significantly different if point 5 acts (in

HOV) or not (in LOOCV) as a GCP, so that they cannot be di-

rectly compared. This is why it seems more correct to filter out

the effect of the anomalous residual by LOOCV mAD instead of

considering it as in LOOCV RMSE.

Finally, the number of LOOCV critical CPs, with North or East

residuals larger than 3mAD, is significantly lower for SISAR (1

in ITA1-e1038452 image and 1 in ITA1-e1090724 image both on

the perimeter) than for OrthoEngine (5 in ITA1-e1038452 image

on the perimeter, 2 in ITA1-e1090724 image).

Image OrthoEngine S ISAR

ITA1-e1038452 6.30 10.88ITA1-e1090724 41.48 10.50

Table 7: Point 5 residual modules (meter)



(a) ITA1-e1038452

(b) ITA1-e1090724

Figure 14: LOOCV critical CPs with North or East residuals

larger than 3mAD are evidenced; areas covered by GCPs are de-

limited (black lines) (GCPs:red dots and HOV CPs: black stars)

Accuracy index OrthoEngine SISAR

LOOCV East [m] North [m] East [m] North [m]

RMSE 2.20 2.42 1.55 3.58

mAD 1.64 0.90 1.32 2.05

Abs max 5.51 5.94 3.07 10.75

HOV East [m] North [m] East [m] North [m]

RMSE 2.06 1.91 2.34 2.85

Table 8: Comparison between models and accuracy indexes for

ITA1-e1038452 image.

Accuracy index OrthoEngine SISAR

LOOCV East [m] North [m] East [m] North [m]

RMSE 5.15 12.13 3.75 5.06

mAD 3.09 6.16 3.13 3.22

Abs max 14.43 41.37 8.16 13.69

HOV East [m] North [m] East [m] North [m]

RMSE 4.75 7.54 4.11 5.45

Table 9: Comparison between models and accuracy indexes for

ITA1-e1090724 image.

7 CONCLUSIONS AND FUTURE WORK

The development of an original rigorous model for the orientation

of basic imagery collected by QuickBird and EROS A and its im-

plementation into the software SISAR were discussed. To point

out the effectiveness of the new model, SISAR results were com-

pared with the corresponding ones obtained by the software Or-

thoEngine 10.0 (PCI Geomatica), where Thierry Toutin’s rigor-

ous model is implemented, and by Sat PP software, implemented

at ETH Zurich in the research group of Prof. Gruen.

The new rigorous model were tested on several images but here

the results attaining to four of them with different features were

discussed only, for the sake of brevity; for each image the ori-entation accuracy (RMSE of CP residuals) were analyzed with

two different methods (HOV and LOOCV), the second one be-

ing applied for the first time to High Resolution Satellite Imagery

orientation accuracy assessment.

The comparison among the three software underlines that SISAR

results are at the level of those derived from the other software.

Nevertheless, the analysis of EROS A images show that SISAR

produces competitive results when the imagery exhibit remark-

able off-nadir angles, thus allowing a wider range of acquisition

conditions.

As regards the accuracy assessment strategy, the LOOCV method

with accuracy evaluated by the robust index mAD seems to sup-

ply interesting results, especially to single out outliers or critical

points. Anyway, some critical cases were evidenced, mainly in

connection with GPs placed on the perimeter of the areas cov-

ered by them, when they act as CPs. It has to be supposed that

bad results stemming from these cases are not representative of

the sought mean orientation accuracy, so that they have to be fil-

tered out by a robust accuracy index an mAD; nevertheless, fur-

ther investigations will be performed on these cases.

Future prospects of this work mainly concern the stereopairorien-

tation, the production of Rational Polynomial Coefficients (RPC)

according to an original algorithm and the model extension to

Ikonos, Cartosat-1, EROS B and Prism satellites.

Some of these new research topics, as the Ikonos Carterra Geomodel, the RPC production and the orientation of stereopairs are

already under implementation and first results are encouraging.

ACKNOWLEDGEMENTS

This research was partially supported by grants of the Italian Min-

istry for School, University and Scientific Research (MIUR) in

the frame of the project MIUR-COFIN 2005 - ”Analisi, compara-

zione e integrazione di immagini digitali acquisite da piattaforma

aerea e satellitare” - National Principal Investigator: S. Dequal.

The Author thank very much:

• Valerio Caroselli (Informatica per il Territorio S.r.l.,Rome,Italy)

and Fabio Volpe (Eurimage S.p.A., Rome, Italy), who kindly sup-

plied the EROS-A1 and Quickbird imagery respectively• Laura De Vendictis, Francesca Lorenzon, Lucia Luzietti, Au-

gusto Mazzoni, Roberta Onori, who carried out the GP GPS sur-

veys

• Maria Antonia Brovelli and Eugenio Realini for the investiga-

tion about LOOCV method, jointly developed

I would like to thank Prof. Mattia Crespi for his continuous in-

terest and support.

REFERENCES

Brovelli, M., Crespi, M., Fratarcangeli, F., Giannone, F. and Re-alini, E., 2006. Accuracy assessment of high resolution satelliteimagery orientation by leave-one-out method. Submitted to IS-

PRS Journal of Photogrammetry & Remote Sensing.

Golub, G. and VanLoan, C. F., 1993. Matrix computation. TheJohns Hopkins University Press, Baltimore and London.

Kaula, W. M., 1966. Theory of Satellite Geodesy. BlaisedellPublishing Company.

Montenbruck, O. and Gill, E., 2001. Satellite orbits. Springer,Berlin.

Sanso, F., 1989. Il trattamento statistico dei dati. Editore CLUP,Milano.

Toutin, T., 2004. Geometric processing of remote sensing im-ages: models, algorithms and methods (review paper). Interna-tional Journal of Remote Sensing 10, pp. 1893–1924.

Westin, T., 1990. Precision rectification of spot imagery. Pho-

togrammetric Engineering and Remote Sensing 56, pp. 247–253.

Zhang, L., 2005. Automatic Digital Surface Model (DSM)Generation from Linear Array Images. Diss., Technische Wis-senschaften ETH Zurich, Nr. 16078,IGP Mitteilung.

rigorous model

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