International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B3, 2012
XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia
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(b) Projection based on incorrect model parameters
Figure 4: Precise image orientation and model parameters are
both required for correct model projection
In this paper, the image orientation parameters of the aerial
photographs are known and fixed, while floating models are
used for reconstructing geometric 3D building models. These
reconstructed geometric models are later used for refining the
image orientation parameters of the close-range photographs
taken by personal mobile computing devices. So the
reconstructed model parameters remain fixed while fitting
floating model to the close-range photographs.
3. LEAST-SQUARES MODEL-IMAGE FITTING
The principle of model-image fitting algorithm is to adjust
either model parameters or the image orientation parameters, so
the model projection fit the building images. Since the floating
model can be taken as a wire-frame model, the edge pixels are
selected as fitting targets. The optimal fit is achieved by
minimizing the sum of the perpendicular distances from the
edge pixels to the corresponding projected line of the wire-
frame model. Figure 5 depicts the optimal fitting procedure.
The selected primitive model is projected onto the image and fit
the extracted edge pixels
Edge :
JE 20
Model Base
Projection
Figure 5: Optimal model-image fitting
Optimal Fitting
Either for geometric modeling or image orientation, an
approximate fitting is required before applying the LSMIF
algorithm. An interactive program is developed for model
selection, approximate fitting, and visualization. To obtain as
close as to the right fitting, this program provides a user
interface that allows the operator to resize, rotate, and move a
model to fit the corresponding building images approximately.
Benefited from the approximate fitting, the LSMIF iteratively
pulls the model to the optimal fit instead of blindly searching
for the solution. To avoid the disturbance of irrelevant edge
pixels, only those edge pixels distributed within the specified
buffer zones will be used in the calculation of the fitting
algorithm. Figure 6 depicts the extracted edge pixels Tj; and
the buffer determined by a projected edge v;;v;; of the model.
The suffix i represents the index of edge line, j represents the
index of overlapped image, and K represents the index of the
edge pixel. Filtering edge pixels with buffer is reasonable,
because the discrepancies between the projected edges and the
corresponding edge pixels should be small, as either the model
parameters or the image orientation parameters are
approximately known.
*. Extracted Pixels
» Yi)
5 : > Tx jk» V ji)
> ijk? * oe Se
© .
e e e
$. Dr 3
"e, .
4 Vi (xo Va)
— x
Figure 6: Extracted edge pixels and buffer
The optimal fitting condition we are looking for is the projected
model edge line exactly falls on the building edges in the
images. In Eq.(1), the distance dj, represents a discrepancy
between an edge pixel Tj; and its corresponding edge line v;,,,
which is expected to be zero. Therefore, the objective of the
fitting function is to minimize the squares sum of dj. Suppose
a projected edge line is composed of the projected vertices
vi(Xij, Yır) and v;2(X;2, iz), and there is an edge pixel Ty (x,
Vi) located inside the buffer. The distance d;, from the point
Tj to the edge viv; can be formulated as the following
equation:
5 IO —Vn)X x + (Ay TX a * OX - Jas) (I)
J Xn) +On —yp)
where i = the index of the edge line
J = the index of the overlapped image
k = the index of the edge pixel
d,
ik
The photo coordinates v;;(x;;, V;1) and v;2(X;2, V;2) are functions
of the unknown model parameters, comparatively the exterior-
orientation parameters of photos are known. Therefore, dj will
be a function of the model parameters. Taking a box model for
instance, dj; Will be a function of w, /, h, a, dX, dY, and dZ,
with the hypothesis that a normal building rarely has a tilt angle
(f) or swing angle (s). The least-squares solution for the
unknown parameters can be expressed as:
Sdyl = X[Fg(w,l,h, a, dX, dY,dZ)] — min. 0)
Eq.(2) is a nonlinear function with regard to the unknowns, $0
that the Newton's method is applied to solve for the unknowns.
The nonlinear function is differentiated with respect to the
unknowns and becomes a linear function with regard to the
increments of the unknowns as follows:
oF, oF, oF, oF,
din —Fino 1 d x J [s i Act (3)
ow J, a ), oh ), a jJ,
oF, oF, oF,
B | AdX+| — | MY+| == | AZ
odX ), adY J, àdZ ),
in which, Fj, is the approximation of the function 5
calculated with given approximations of the unknow
Intern
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The author
the Nation:
2221-E-00
Braun, C.,
Cremers, /
for Photog
Graphics,