back-projection of the

DSM

into image spaces, the hypotheti-

cal matching points, here referred to as corresponding points,

are detected (Figure 2b). For this projection Rational Function

Model (Equations 1) is used (Grodecki 2001):

=

=

(

)

=

= =

∑

1

2

3

4

0

( , , )

( , , )

( , , )

( , , )

, ,

0 0

∑∑

=

, ,

(1)

where

~ and

~ are normalized image coordinates, and

,

,

and

are normalized ground coordinates;

is generally set

to 3. In this study, since bi-temporal images are used, the

above equations are re-written for the image

(

= 1

2) and

ground point

as:

=

(

)

(2)

where

(

)

is the transformation based on the

RFM

equations.

However,

RPCs

provided by imaging vendors have inherent

uncertainties due to small attitude or ephemeris errors, which

manifest themselves as biases in the image coordinate space

(Fraser and Hanley, 2005). Fraser (2003) demonstrated that

the bias vectors, which are the result of a direct comparison

between the back-projected

GCPs

in the image space, using

RPCs

, and the corresponding image points, are fairly invariant

within an image and can be modeled using an affine transfor-

mation, disregarding the type of terrain. In their experiments,

the standard error of the biases is around half a pixel.

Applying the affine bias compensation to the image points,

the adjusted

RFM

equation can be written as:

ˆ

ˆ

(

)

=

(3)

where

ˆ and

ˆ are bias compensated image coordinates, and

is a 2D affine transformation given in:

ˆ

ˆ

=

=

11

12

13

21

22

23

0 0 1

1

(4)

where

,

∈

1:3,

∈

1:3 are the unknown coefficients of the

affine transformation for which at least three control points

are required. Therefore, using the adjusted

RFM

equations,

a back-projection from ground space to image space can be

generated with an accuracy better than one pixel. Figure 3

depicts how uncorrected

RPCs

and an affine transformation in

each image are used to relate the corresponding image points.

The process of finding the corresponding image points is

done indirectly using the

DSM

as an indicator. By finding the

corresponding points, a look-up-table (

LUT

) is generated. In

each row of the

LUT

, the

DSM

pixel ground coordinates and the

corresponding image points are given. Therefore, for each

DSM

pixel: [

1

,

1

,

2

,

2

,

,

,

,

]

, j

∈

{1,2,…,

}, is generated; where

(

1

,

1

) are base image coordinates, (

2

,

2

) are target image coor-

dinates, (

,

,

) are ground coordinates from the

DSM

.

is the

patch

ID

from the base image segmentation, and

is the total

number of pixels in the

DSM

.

In the second step, false change detection results, caused by

the effect of relief displacement and accordingly occlusion,

must be prevented. This effect is schematically depicted in

Figure 4. In this figure, AE is the hypothetical curve map-

ping the point A in the object space to the point E in the base

image space (AE is represented by a curve since in satellite

imagery the collinearity equations are replaced by

RFM

; there-

fore, AE is not a straight line). Although all of the points A, B,

Figure 3. Indirect matching of corresponding points using back-projection of DSM pixels into image spaces: (a) A schematic representation

of back-projection of a ground pixel, A, to the image spaces using RFM (G1 and G2 operators). Using the uncorrected RPCs, the ground point

A is projected to a~ ′ and a~ ′′ in the bi-temporal images. Afterwards, by performing an affine transformation (T1 and T2), a~ ′ and a~ ′′ are trans-

ferred to their correct places a′ and a′′ respectively, resulting in the indirect matching of points a′ and a′′; and (b) The same fact is shown with

a real example. Some sample pixels (the corner pixels of a building) in the DSM are projected to the bi-temporal images using uncorrected

RPCs. With T1 and T2 affine transformations in the image spaces, the associated image points are transferred to their correct positions.

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