Submitted for review to 3DIM2005

Fast Alignment of 3D Geometrical Models and 2D Color Images
using 2D Distance Maps

Abstract

Kurazume[7] proposed the simultaneous registration algorithm using the reflectance image and the 2D texture images. Range sensors often provide reflectance images asside products of range images. This 2D reflectance image is aligned with the 3D range image because both images are obtained through the same receiving optical device.
Therefore, we can utilize the 2D reflectance image for solving the 2D-3D registration problem instead of the original
3D model. By using the 2D reflectance image, the 2D-3D
problem is modified to the simple 2D-2D problem. In [7],a number of photometric edges extracted from reflectanceimage and texture image were registered and the relativepose is determined using a robust M-estimator. Epipolar
constraints are also introduced to estimate relative posesof multiple texture images simultaneously. Elstrom[4] extracted feature points from the reflectance image and the
texture image by a corner detector, and determined correspondence between these feature points. Umeda[21] alsoutilized the reflectance image, but they introduced the optical flow constraint between the reflectance image and the
texture image. Intrinsic and extrinsic parameters are determined using the least squares method.

This paper presents a fast pose estimation algorithm of a
3D free form object in 2D images using 2D distance maps.
One of the popular techniques of the pose estimation of 3D
object in 2D image is the point-based method such as the
ICP algorithm. However, the calculation cost for determining point correspondences is expensive. To overcome thisproblem, the proposed method utilizes a distance map on
the 2D image plane, which is constructed quite rapidly by
the Fast Marching Method. For pose estimation of the object, contour lines of the 2D image and the projection of
the 3D object are aligned using the distance map iteratively
by the robust M-estimator. Some experimental results withsimulated models and actual images of the endoscopic operation are successfully carried out.

1. Introduction
Pose estimation problem of a 3D free form object in a 2D
image is one of the fundamental problems in the fields ofcomputer vision, such as object recognition, virtual reality,
and texture mapping. A lot of algorithms for 2D-3D registration have been proposed so far [1]-[4],[6]-[15],[18]-[23].
Brunie[2] and Lavallee[8] utilized a pre-computed 3D distance map of a free form object for 3D pose estimation. The
error metric is defined as the minimum distance between thesurface of the 3D model and projection rays, and the sumof the error is minimized using the Levenberg-Marquardt
method. To make the registration process efficiently, 3D
distance from the surface is precomputed and stored withthe octree structure. Zuffi[23] applied this algorithm for thepose estimation of a knee joint in a single X-ray image fortotal knee replacement surgery.
For the alignment of a 2D color image taken by a color
sensor and a 3D geometric model measured by a range
sensor, Viola proposed a technique based on a statisticalmethod [22]. This method evaluated the mutual information between the 2D image and the image of the 3D model
based on the distribution of intensity. Allen[19] also proposed a method using 3D linear edges and the 2D edges.
These methods work well on the surface with little albedo
variance. But these are easily trapped in a local minimumon surfaces with rich textures.

On the other hand, some registration techniques using
silhouette images or contour lines in 2D image plane havebeen proposed. Lensch[9],[10],[11] proposed a silhouettebased approach. The size of XOR regions of silhouette images of a 2D image and a 3D model is defined as the similarity measure, and the optimum pose which minimizes thesize of the XOR region is determined using the Downhill
Simplex method. Neugebauer[15] proposed simultaneous
registration algorithm for multiple texture images and a freeform object. To determine pose parameters of multiple images (6n dof: n is the number of images) and camera focal
length (2 dof), three objective functions are utilized, which
are i) distance between the feature point in the image and its
corresponding 3D point projected in the image, ii) distance
between the outline in the image and the outline of the projected 3D model, and, iii) similarity of the image attributeat corresponding points in different images. In the 3rd criteria, the distance map is introduced as an artificial image
attribute. This distance map is defined as the Euclidean distance from edges extracted by the sobel operator, and established using a 3x3 mask operation iteratively. A blend
1Fast Alignment of 3D Geometrical Models and 2D Color Images using 2D Distance Maps Submitted for review to 3DIM2005

of objective functions is minimized by the nonlinear leastsquares method, and optimum poses of multiple images are
determined simultaneously.
In contour-based approach, the error is computed as thesum of distances between points on a contour line in a 2D
image and points on a projected contour line of a 3D model,
and the optimum pose is determined based on the ICP algorithm [3], [14]. Though these algorithms are simple andquite robust against noise, they are computationally expensive since a number of point correspondences between two
contours have to be determined for calculating the registration error.
This paper presents a new registration algorithm of a 2D
color image and a 3D geometric model utilizing a 2D distance map. In this method, firstly, the boundary of an object on the 2D color image is detected and the 2D distancemap from the detected boundary is constructed by the Fast
Marching Method. Next, an optimum pose of the 3D modelis estimated to decrease the sum of the distance value on
projected contour points of the 3D model iteratively using arobust M-estimator. Our 2D-3D registration algorithm belongs to the contour-based approach. However, we utilize
a distance map for evaluating a registration error instead of
the point correspondence. Since there is no need to determine the point correspondence, our algorithm works fasterthan the conventional point-based approach, after the distance map is created. In addition, since the distance map
can be established quite rapidly using the Level Set Method
named the Fast Marching Method, our method applies tothe realtime tracking of a moving object. In this paper, weshow some fundamental experiments including tracking ofinternal organs in endoscopic images for a navigation system of the endoscopic operation.

−x+x
T, −DijT, 0)2 +(max(Dij
1−y+yT, −DijT, 0)2 ) 2 = 1/Fij
max(Dij(1)

Next, since the arrival time is propagated in one direction from the old to the new, a point which holds the oldest
arrival time in a whole region is chosen, and the arrival timeof the boundary at this point is determined from Eq.(1).
The concrete procedure is as follows:
Step.1 (Initialization) The whole region is divided into a
number of grid points with a proper grid width. Before
starting the calculation, all the grid points are categorized into three categories (known, trial, far) according
to the following procedure.
1. The grid points belonging to the initial front (denote the boundary hereafter) are added to the category of known and the arrival time of these grid
points is set to 0 (T = 0).
2. Among the 4 neighboring grid points of a grid
point belonging to the category of known, the one
which doesn’t belong to the category of known iscategorized into to the category of trial, and thearrival time of this grid point is calculated temporarily from the equation, Tij = 1/Fij . In addition, the arrival time of these grid points is stored
in heap data structure using the heap sort algorithm in ascending order of T .
3. The grid points except for the above grid points
are all categorized into the category of far, andthe arrival time is set to infinity (T = ∞).

2. Fast Marching Method for rapid construction of distance map

Step.2 Choose the grid point (imin , jmin ) which is placedat the top of the heap structure. This grid point has
the smallest arrival time in the category of trial. Remove this grid point from the category of trial andthe heap structure, and categorize it as the category ofknown. Run “downheap” algorithm to reconstruct theheap structure.

Before describing our registration algorithm, the Fast
Marching Method proposed by Sethian [16],[17] is briefly
introduced, which is utilized for the fast construction of a
distance map in our method.
The Fast Marching Method was initially proposed
as a fast numerical solution of the Eikonal equations(|∇T (p)| F = 1), where T (p) is arrival time of a front
at a point p and F is a speed function. It generally takes along time to obtain a proper solution since this equation issolved using a convergent calculation. However, by adding
restriction such that a sign of the speed function F is invariant, the Fast Marching Method solves the Eikonal equation
straightforwardly, and thus, quite rapidly. In this method,
the arrival time of a front at each point is determined in theorder from the old to the new.
At first, the Fast Marching Method modifies the Eikonal
equation into the next difference equation.

Step.3 Among the 4 neighboring grid points((imin − 1, jmin ),(imin + 1, jmin ), (imin , jmin −1),(imin , jmin + 1)) of the selected grid point(imin , jmin ), the one which belongs to the category of
far is changed to the category of trial.
Step.4 Among the 4 neighboring grid points of the selected
grid point (imin , jmin ), the arrival time of the one
which belongs to the category of trial is calculated using Eq.(1). Then, run “upheap” algorithm to reconstruct the heap area.
2

Fast Alignment of 3D Geometrical Models and 2D Color Images using 2D Distance Maps Submitted for review to 3DIM2005

Step.5 If there is a grid point which belongs to the categoryof trial, then go to Step 2. Otherwise, the process is
terminated.

6. Total force and moment around the COG (center ofgravity) of the 3D model is determined using the robust
M-estimator.

This Fast Marching Method constructs a distance map of
a whole region quite rapidly, which indicates the distancefrom a boundary (an initial position of the front) to a certain
point. To construct the distance map, the speed function Fij
in Eq.(1) is set to 1 at first. Next, the arrival time of the front
T is determined using the above procedure. Since the speedfunction is 1, T suggests the distance from the boundary
to the point. Figure 1 shows an example of the calculated
distance map.
boundary

7. The pose of the 3D model is changed according to thetotal force and moment.
8. Repeat from step 1 to 7 until the projected image of the
3D model and the 2D image coincide each other.
The above procedure is explained in more details withsome examples in the following sections.

contour lines of
distance map

3.1. Construction of distance map
Figure 2 shows the calculated distance map. At first,
a boundary is extracted from the color image using theLevel Set Method[17],[5]. Next, the distance map from thisboundary is calculated using the Fast Marching Method explained in Section 2.

+20 pixels+100 pixels

boudary+20 pixels

Figure 1. An example of the distance map calculation using the Fast Marching Method.
(a) Object

3. A New 2D-3D registration algorithm basedon the distance map(b) Obtained distance map

Figure 2. Detected boundary and distance map.

This section describes our fast 2D-3D registration algorithm using the distance map in detail. We assume that a 3D
geometric model of the object has been constructed beforehand, and is represented by a number of triangular patches.
Camera internal parameters are also calibrated and obtained
precisely.
The proposed algorithm is summarized as follows:

3.2. Fast detection of triangular patches of contour
points in 3D model
Figure 3 shows an example of the contour detection of
the projected 3D geometric model. The contour detection
and identifying triangular patches on the 3D model corresponding to points on the contour line are computationallyexpensive and time consuming. In our implementation, we
utilize the high-speed rendering function of the OpenGL
hardware accelerator and thus these procedures are executed quite rapidly.
The detailed algorithm is as follows: Initially, we assign different colors for all the triangular patches in the 3Dmodel and draw the projected image of the 3D model on theimage buffer using the OpenGL hardware accelerator. The
contour points of the 3D model are detected by raster scanning of the image buffer. By reading colors of the detected
contour points, we can identify the corresponding triangular patches on the 3D geometric model. Figure 4 shows thecolor image of the projected 3D model with the patches ofdifferent colors.

1. At first, a boundary of an object on the 2D color image
is detected using an active contour model (ex. Snakesor the Level Set Method[5]).
2. Next, the distance map from the detected boundaryon the 2D image plane is constructed using the Fast
Marching Method.
3. The 3D geometric model of the object is placed at anarbitrary position and projected on the 2D image plane.
4. Contour points of the projected image and their corresponding patches on the 3D model are identified.
5. Force is applied to the selected patch of the 3D modelin 3D space according to the distance value obtained
from the distance map at each contour point.
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Fast Alignment of 3D Geometrical Models and 2D Color Images using 2D Distance Maps Submitted for review to 3DIM2005

effectiveness of this term is shown in Figure 5. This termworks when the direction of the normal vector of the contour line does not coincide with the direction of the steepest
descent of the distance map. Therefore, this is effective especially for a long, slender and symmetrical object, such asarms or legs of human body, since a part of normal vectors
is almost perpendicular to the direction of the steepest descent during the alignment procedure as shown in Figure 5.

detected contour

Figure 3. Contour detection of 3D geometricmodel.

Color image

3D object

Boundary(a) 3D model(b) Object image with different
color patches

Figure 4. Detection of triangular patches of thecontour points.

3.3. Force and moment calculation using the Mestimator and determination of optimum pose
After obtaining the distance map on the 2D image andthe list of the triangular patches of the 3D model corresponding to the contour points, the force fi is applied to
all the triangular patches of the contour points (Figure 6).
The force fi is the vector perpendicular to the line of sight,
and the projection of the fi onto the 2D image plane coincides with the fDM , which is the vector obtained by thefollowing equation (Figure 7(a)).
fDM = DMi,jk · sign(n ·(1) First + second terms(2) First term only

Figure 5. Effect of Eq.(eq.fdm)

Then, total force and moment around the COG are calculated using the following equations as shown in Figure7(b).
ρ(fi )(3)F =∇DMi,j+| ∇DMi,j |i∇DMi,j∇DMi,j)(1− | n ·|)n (2)| ∇DMi,j || ∇DMi,j |

M

=



ρ(ri × fi )(4)iwhere DMij is the value of the distance map at (i, j), n
is a normal vector of the projected contour point of the 3Dmodel, k is an arbitrary constant, and sign(·) is a sign function which returns the sign of a number. The first term
stands for the force toward the direction of the steepest descent of the distance map. We assume that the magnitudeof this term is proportional to the value on the distance map. The second term stands for the force regarding the consistency of the normal vector of the contour line and the
direction of the steepest descent of the distance map. The

where r is a vector from the COG to the triangular patch andρ(z) is a particular estimate function. In practical scenario,
a part of the object is occasionally occluded by the otherparts, or the 2D image is corrupted by noise. In these cases,
the obtained boundary does not coincide with the projected
contour of the 3D model and the correct distance value cannot be obtained. Therefore, we introduces the robust Mestimator to ignore contour points with a large amount oferrors.
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Fast Alignment of 3D Geometrical Models and 2D Color Images using 2D Distance Maps Submitted for review to 3DIM2005

Let’s consider the force fi and the moment ri × fi as anerror i . Then, we modify Eqs.(3) and (4) as FE(P ) ==
ρ(i )(5)M

fi
f DM ifv

fi

Objectr
M
Fi

Steepest descent
Image plane
direction

where P is the pose of the 3D geometric model. The pose
P which minimizes the E(P ) is obtained as the following
equation.
 ∂ρ(zi ) ∂zi
∂E
=
=0
(6)
∂P
∂zi ∂Pinormal vector

Boundary(b)

Figure 7. Force and moment around COG.

band is a narrow band along the image boundary and theprecise distance map is costructed in this region using theFast Marching Method. In other region except the distanceband, the distance value is roughly estimated as the distancefrom the center of the image boundary. In the beginning of
the registration process, we use a coarse image and a wide
distance band in order to align the 2D image and the 3Dmodel roughly. After several iterations, we change the image to the fine one and compute the distance band with narrower width. In experiments, we used the distance bandwith 10 pixels and three images with difference resolution,that is, 160x120, 320x240, 640x480. Computation time ofthe distance band in each image is shown in Table 1.

In our implementation, we adopt the Lorentzian function
for the estimation function ρ(z), and gradually minimize theerror E in Eq.(5) by the steepest descent method.

w(z) =

v
Focul point

(a)

1 ∂ρ
w(z) =(7)z ∂zFrom the above equation, we obtain the following weighted
least squares method.

∂E
∂zi
=
=0
(8)
w(zi )zi
∂P
∂P
i

σ2
log(1 + (z/σ)2 )2
1
1 + (z/σ)2

Projected
contour point

f DM

Here, we define the weight function w(z) as the following
equation in order to evaluate the error term.

ρ(z) =

Projected
contour(9)(10)

Table 1. Computation time of the distance band by
the Fast Marching Method.

The pose P which minimizes the error in Eq.(5) is the estimated relative pose between the 2D image and the 3D geometric model.

Image size
160x120320x240
640x480fi

Whole region
[ms]
6.9
32.8
189.1

Distance band
(10 pixels) [ms]
0.48
1.1
2.7

3.5. Characteristics of our method
Characteristics of our method are summarized as follows:
1. The computational cost is expensive in the conventional contour based approach, since a number of point
correspondences have to be determined for calculating
the registration error. However, after the distance map
is created, our algorithm works faster than the conventional point-based approach, since there is no need todetermine the point correspondence.

Figure 6. Applying the force f to all the triangular
patches of the contour points.

3.4. Coarse-to-fine strategy and the “distanceband”
Though the 2D distance map can be constructed efficiently by the Fast Marching Method, computing the distance value in whole region of the color image is wasteful. Therefore we adopt coarse-to-fine strategy and a bandshaped distance map named a distance band. The distance2. The Fast Marching Method is utilized to construct a
distance map of a whole region quite rapidly.
3. The larger the number of points on the boundary becomes, the longer processing time is required for the5Fast Alignment of 3D Geometrical Models and 2D Color Images using 2D Distance Maps Submitted for review to 3DIM2005

conventional point-based approach. On the other hand,
total processing time of the proposed method is almost
constant even if the number of points on the boundary
increases.

models. The example of mapping the camera image insteadof the original texture is shown in Figure 10.
2D image 3D model

4. Experiments
In this section, we show some fundamental results of theregistration and tracking experiments using simulated images of a doll as shown in Figure 3 and actual images ofinternal organs for the endoscopic operation.

(a) Initial position(b) 1 iteration (0.01s)(c) 2 iterations (0.02s)(d) 3 iterations (0.03s)(e) 30 iterations (0.3s)(f) 60 iterations (0.6s)

4.1. 2D-3D registration of simulated images
Firstly, Figure 8 shows how the proposed 2D-3D registration algorithm works with a model of a small doll (aheight of 5cm).
The computation time of the force fi for one patch on
the 3D model is 0.30 [µs] for Pentium IV processor, 3.06
GHz. The total processing time is 9.6 [ms] for one update
period including projected contour detection, calculation offorce and moment, and the steepest descent method for Pentium IV processor, 3.06 GHz. The average and the varianceof the registration error between the boundary and the projected contour are 1.19[pixels] and 0.90[pixels2 ], respectively. Here, the number of points on the boundary is 652,
image size is 640 × 480 pixels, and the 3D model is composed of 73,192 meshes with 219,576 vertices. On the other
hand, the computation time of the force fi for the conventional point-based method [7] is 1.15 [µs], which uses the
k-D tree structure to search the nearest points between thecontour points of the 2D image and the projected 3D model.
Table 2 show the comparison of the processing time forseveral cases in which different numbers of points on theboundary were used. From these results, it is clear that theprocessing time of the proposed method is almost constanteven if the number of points on the boundary increases.
On the other hand, the point-based approach requires 3 to
7 times longer processing time than the proposed method.
Moreover, the larger the number of points becomes, thelonger processing time is required for the point-based approach. Therefore, it is verified that the proposed 2D-3D
registration algorithm has an advantage comparing with theconventional point based approach in views of execution
time, especially when there are a large number of points
on the boundary.
Next example is the tracking experiment of the moving
object. Figure 9 shows the tracking results of the doll whichmoves on the image plane. As seen in this example, the 2D3D registration can be performed even if the object moves
up to 20.5 pixels/second.
Though the 3D model shown in Figure 4 has a colored
texture, our method can be applied to any 3D models without texture since it does not use any color information on 3D

Figure 8. 2D-3D registration of simulation images.

Table 2. Comparison of processing time for one
patch

Number of points
on boundary
628
1265
1868
2490

Proposedmethod [µs]
0.30
0.30
0.30
0.30

Point-based
method [7] [µs]
1.15
1.50
1.70
2.22

4.2. 2D-3D registration using actual images of theendoscopic operation
We carried out fundamental experiments for a navigation
system of the endoscopic operation. In this system, a 3D organ model, which is constructed by CT or MRI images, issuperimposing endoscopic images to guide operation procedure.
Firstly, we tried the 2D-3D registration using actual images of the cirrhosis of the liver. Figure 11 shows the initial
results of the experiments. Figure 11(a) shows the actual
image of the cirrhosis of the liver taken by the endoscope.
Figure 11(b) and Figure 11(c) show the 3D model of theliver obtained by the CT images and experimental results
for the 2D-3D registration, respectively. The registration
is successfully carried out if the initial position of the 3D6

Fast Alignment of 3D Geometrical Models and 2D Color Images using 2D Distance Maps Submitted for review to 3DIM2005

120 pixels(a) t = 0 s
color image

3D model(b) t = 7.7 s
Figure 10. An example of mapping the camera image on the 3D model.

the proposed algorithm was verified through the fundamental experiments using simulation images and actual imagesof internal organs for a navigation system of the endoscopic
operation.

(c) t = 12.8 s

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(d) t = 18.1 s
Figure 9. An example of moving object tracking.

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5. Conclusion
This paper described a new registration algorithm of 2D
color images and 3D geometric models. This method utilizes the 2D images and their distance map created by theFast Marching Method, and determines a precise relativepose between 2D images and 3D models using the robust
M-estimator.
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Fast Alignment of 3D Geometrical Models and 2D Color Images using 2D Distance Maps Submitted for review to 3DIM2005

Initial position(a) Endoscpoic image of (b) 3D model of the liver
the cirrhosis of the liver(c-1) Initail position(c-3) after 36 iterations(c-2) after 20 iterations

Figure 12. 2D-3D registration in endoscopic video
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