3D shape recovery of complex objects from multiple silhouette images

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3D shape recovery of complex objects from multiple

silhouette images

Yen-Hsiang Fang, Hong-Long Chou, Zen Chen

*

Department of Computer Science and Information Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan

Received 7 July 2001; received in revised form 1 November 2002

Abstract

A reconstruction method is proposed which represents the object with a line-based geometric model. The method does not need the point correspondence information in recovering the 3D object geometry. It is based on the concept of volume intersection, but it is substantially diﬀerent from the existing octree-based reconstruction methods in the aspects of data structure, reconstruction process and representation uniqueness under a 2D rigid motion. For visualizing the 3D reconstructed object geometry a conversion from the line-based geometric model to a bounded triangular mesh model is developed. The experimental results show that the method is capable of capturing the diﬀerent details of the object. And it works fast and requires a relatively low memory space.

Keywords: 3D shape recovery; Volume intersection; Line-based model; Dynamic line resolution; Triangular mesh model

1. Introduction

Reconstruction of 3D objects from images is an important task in computer vision. The 3D geometric model of the object ﬁnds applications in part manufacturing and design, target recog-nition, and virtual reality, etc. We are concerned with computer vision techniques for recovering the geometry of a complex object from multiple silhouette images. Here complex objects refer to

objects containing parts of varying size, for in-stance, teapot with ﬂat handle and riﬂe with tiny trigger.

There are various vision-based reconstruction methods including:

(1) The stereo vision methods (Fua, 1997; Huang and Netravali, 1994; Okutomi and Kanade, 1993).

(2) The structured light methods (Proesmans et al., 1998; Chia et al., 1996; Hu and Stockman, 1989).

(3) The volume intersection methods (Slabaugh et al., 2001; Niem, 1999; Snow et al., 2000; Mo-ezzi et al., 1996; Fromherz and Bichsel, 1994;

*

Corresponding author. Tel.: 35-712-121; fax: +886-35-5723-148.

E-mail address:zchen@csie.nctu.edu.tw(Z. Chen).

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Garcia and Brunet, 1998; Szeliski, 1993; Sri-vastava and Ahuja, 1990; Potmesil, 1987; Chien and Aggarwal, 1986).

(4) The voxel coloring methods (Kutulakos and Seitz, 2000; Szeliski and Golland, 1999; De Bonet and Viola, 1999; Seitz and Dyer, 1999; Prock and Dyer, 1998).

The characteristics of these methods are given in Table 1 and each method has its own merits and shortcomings. For a practical object reconstruc-tion we are looking for a widely applicable method that is capable of producing a complete object geometric model that is reasonably accurate. The method we are going to propose has the following distinct features:

(1) It does not need the point correspondence in-formation in recovering the 3D object geome-try, as required by the stereo vision method. (2) It builds a complete and dense geometric model

of the object. Generally, the geometric model built by the stereo vision method is a sparse reconstruction due to the limited number of available feature point unless an active struc-tured light is used in the vision method. (3) It is based on the concept of volume

intersec-tion, but it is substantially diﬀerent from the related octree-based reconstruction methods (Garcia and Brunet, 1998; Szeliski, 1993; Sri-vastava and Ahuja, 1990; Potmesil, 1987; Chien and Aggarwal, 1986) in the aspects of data structure, reconstruction process, and representation uniqueness under a 2D rigid motion. We elaborate on these diﬀerences below:

(a) Diﬀerence in the data model/structure: In the octree-based method the basic unit of the geometric model is the octree cell and the octree cells are organized in a tree

data structure with a branch factor of 8. Our method uses line segment as the basic unit and the line segments are organized in a 2D array of linked lists.

(b) Diﬀerence in the reconstruction process: The reconstruction process of the octree-based method is recursive, while our method constructs the line-based model sequentially.

(c) Representation uniqueness under a 2D rigid motion:

In the octree-based method, a 2D trans-lation or rotation of a root cell will gen-erally end up with a dramatic change in the octree representation form. Thus, the octree-based representation is sensitive to the 2D rigid motion. However, in our line-based modeling the 2D object motion only leads to a similar motion of the 2D array representation of the object with-out changing its intrinsic data structure form. The insensitivity to the 2D object motion is useful to the applications such as object matching and spatial relation deduction.

The paper is organized as follows. Section 2 describes the line-based data model used to ﬁt the 3D object geometry and the phase-one recon-struction process using the intersection operation. Section 3 presents the dynamic line resolution ad-justment for the phase-two reconstruction based on the shape smoothness. For visualizing the 3D reconstructed object geometry a technique is given in Section 4 that converts the line-based geometric model to a bounded triangular mesh model. Sec-tion 5 gives the experimental results to illustrate the quality of the reconstruction results and the required processing time and memory space. Sec-tion 6 is the conclusions.

Table 1

Characteristics of the vision-based reconstruction methods

Method Sensing mode Data model Reconstruction scheme

Stereo vision Passive Point set Triangulation

Structured light Active Point network Triangulation

Volume intersection Passive Voxel set Volume intersection

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2. Reconstruction of the object line-based geometric model

The original (or starting) line-based data model used to fit the 3D object geometry is defined as a 2D array of line segments that have the same length and are perpendicular to a base plane at the regular grid points, as shown in Fig. 1. The uni-form spacing between the grid points determines the spatial resolution of the line segments. Each line segment may contain more than one line sec-tion intersecting with the object. The reconstruc-tion process consists of two phases. In the first phase the used line-based data model has a fixed line resolution. In the second phase the object line-based model obtained in the first phase is refined with a dynamic line resolution. The reconstruction of phase-one is described here and the phase-two reconstruction will be given in the next section.

The phase-one reconstruction algorithm: 1. Input: A set of silhouette images of the object

obtained with a calibrated camera and a rotat-ing turntable.

2. Output: The phase-one object line-based geo-metric model.

3. Method:

(1) Generate a 2D array of line segments with a ﬁxed line resolution and a uniform line length that based on the physical dimen-sions of the object to be reconstructed.

(2) For each view

(2.1) Project the 3D line segments to 2D image plane based on the camera cal-ibration parameters.

(2.2) For each projected line segment, cal-culate the 2D line sections that inter-sect with the object silhouette. (2.3) Back-project each found 2D line

sec-tion to ﬁnd the corresponding 3D line section onto the chosen line segment. (3) Find the intersection of the 3D line sections

obtained from all views.

The calculations involved in the phase-one re-construction are given below. For each view, the 3 4 projection matrix H is known. Using H , we can calculate the projected 2D line segment of each 3D line segment on the image plane. For a 3D line segment Liwith start point P0and end point P1, the

projection points p0 and p1 are given as:

p0¼ H P0

p1¼ H P1

By connecting p0 and p1, we can obtain the

corresponding 2D line segment li.

We use the Bresenham algorithm (Bresenham, 1965) to represent li by the discrete integer

coor-dinates stored in a linked list PL: PL¼ fðu0; v0Þ ! ðu1; v1Þ ! ðu2; v2Þ !

! ðun; vnÞg:

By tracing the coordinates of the line PL, we can obtain the 2D line sections that intersect with the binary object silhouette image, as shown in Fig. 2.

To obtain the corresponding 3D line sections intersecting the object, we need to back-project the 2D line sections to its original 3D line segment from which they are obtained. The back-projec-tion diagram is illustrated in Fig. 3.

In Fig. 3 the 3D line segment Li with the start

point P0 and the end point P1 is projected to 2D

line segment li from point p0 to point p1. Also, a

general 3D point P on Li is projected to the 2D

point q on li. The lengths of Liand liare D and d.

The length from q to p0 is l.

Fig. 1. The original line-based data model used to ﬁt the 3D object geometry.

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Given the projection matrix H and the coordi-nates of P0, P1, p0, p1, and q, we want to calculate

the coordinate of the unknown 3D point P and the length L between P and P0:

Let the line equations of Li and li be given by

x¼ x0þ tðx1 x0Þ y¼ y0þ tðy1 y0Þ z¼ z0þ tðz1 z0Þ 8 < : 0 6 t¼ L D61 ð1Þ and u¼ u0þ sðu1 u0Þ v¼ v0þ sðv1 v0Þ 0 6 s¼l d 61 ð2Þ where ðx; y; zÞT is the coordinates of the point P , and t and L are unknown parameters.

From projection geometry, we have

wj uj vj 1 2 4 3 5 ¼ H xj yj zj 1 2 6 6 4 3 7 7 5; j ¼ 0 and 1 ð3Þ and w " u v 1 # ¼ H "_x y z 1 # : ð4Þ

After some manipulations, we can obtain the fol-lowing linear equations:

u w þ ðw0u0 w1u1Þ t ¼ w0u0 v w þ ðw0v0 w1v1Þ t ¼ w0v0 1 w þ ðw0 w1Þ t ¼ w0 8 < : ð5Þ

Fig. 2. Trace the line segment to ﬁnd its line sections that in-tersect with the binary object silhouette image.

Fig. 3. Back-projection of a 2D point q on the image line to a 3D point P on the 3D line segment.

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Then, we solve for t to obtain t¼ w0s w1þ sðw0 w1Þ ¼ w0 l w1 d þ ðw0 w1Þ l :

Plugging the value of t into Eq. (1), we can calculate the coordinate of P and length L.

Using the above method, we can back-project the two endpoints of the 2D line section to the corresponding 3D line segment to locate the 3D line section intersecting the object silhouette in the view under consideration.

The ﬁnal step of the reconstruction process is to intersect the 3D line sections obtained from all views, as depicted in Fig. 4.

3. Dynamic line resolution adjustment

The line segments in the phase-one reconstruc-tion are uniformly distributed. If the line resolu-tion is not high enough, it may miss some details of the object. For this reason, we have to check any two horizontally adjacent line segments for possi-ble loss of the object details. If there is such a possibility, then a new line segment is inserted between the two original line segments (i.e., in-crease the line resolution locally) to capture the possible details of the object.

Two conditions for inserting a new line seg-ment:

(1) No vertical overlap: The object we reconstruct is supposed to be an integral part, so it has no isolated components. Under this assumption, the line sections of any two horizontally adja-cent line segments should have some degree of overlap in the vertical direction (i.e., y-direc-tion). If no vertical overlap is found, the line may be near the object boundary or has a steep object slope. To obtain high shape accuracy, a new line segment is inserted to detect the ﬁner object features. Fig. 5 shows an example of this case.

(2) A rapid change rate of vertical length: If the line sections of the two horizontally adjacent line segments are vertically overlapped, but the total lengths of these overlapping line

sec-tions have a large diﬀerence. Then it indicates that the object geometry may change dramati-cally between the two line segments. We will insert a new line segment to make the shape change smoothly.

To design the scheme for inserting the new line segments, we have to consider the 3D visualization of the reconstructed object at a later stage. As shall be seen, we need to divide the line segments into triplets for this object visualization. To do so, we proceed with the top view of the line-based model in which the line segments are projected to grid points, as shown in Fig. 6. We decompose each grid cell into two triangles along a diagonal line, then the line segments indicated by the three grid points of a triangle constitute a triplet we need. We should check each pair of line segments of the

Fig. 5. The line sections S2 and S3 have no vertical overlap. A new line segment must be inserted in between.

Fig. 6. Triplets of the line segments become grid points when seen from the top view of the line-based model.

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triplets to see if they satisfy the two conditions for a new line insertion, as mentioned above.

To insert the new line segments we have to maintain the uniformity of the triangle-pair data structure depicted in Fig. 6, so we insert the new line segments as follows.

In Fig. 7, if Liand Lj are two original line

seg-ments of a triplet that satisfy the condition of either having no vertical overlap or having a rapid length change rate, then a new line segment Lk

needs to be inserted. Depending on the spatial relationship between Li and Lj, there are three

cases to be considered:

Case (a): Li and Lj have a top-down spatial

rela-tionship.

As depicted in Fig. 7(a), the two grid cells on both sides of the triangle edge LiLj need to be equally subdivided into

eight smaller cells. These cells form 16 new triangles.

Case (b): Li and Lj have a left–right spatial

rela-tionship.

This is similar to case (a), as depicted in Fig. 7(b). Totally eight new line seg-ments are inserted together with the line segment Lk.

Fig. 7. (a)–(c) Three cases for inserting new line segments as seen from the top view of the line-based model. (Here indicates an inserted line segment.)

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Case (c): Li and Lj have a diagonal spatial

rela-tionship.

As depicted in Fig. 7(c), only the grid cell containing the diagonal edge LiLj is

subdivided. Totally four new line seg-ments are inserted together with Lk.

Using the above scheme, we can dynamically increase the local line resolution of the line-based model to reconstruct the details of the object, while maintaining the data structure uniformity.

The algorithm for dynamic line resolution ad-justment is summarized as follows:

(1) Each pair of line segments in the original trip-lets is checked to see if a new line segment needs to be inserted.

(2) When either new line insertion condition is sat-isﬁed, the relevant grid cell(s) is subdivided and new line segments in triples are inserted, as described in the above three cases.

(3) The line segments of the newly inserted triplets are checked in pairs for the need of new line insertion.

(4) Steps (1)–(3) are repeated until a user-speciﬁed maximum subdivision level is reached or until no new line insertion is needed.

4. The bounded triangular mesh geometric model in the triangular mesh form

The object line-based geometric model obtained above is a collection of line sections that is obvi-ously not bounded. An example is shown in Fig. 8. In order to visualize the 3D shape of the recon-structed object, this model needs to be converted to a bounded triangular mesh model.

The conversion involves two steps, as depicted in Fig. 9. In the ﬁrst step the non-bounded line-based model is converted to a solid prism model with bounded surfaces. In the second step the solid prism model is converted to a bounded triangular mesh model.

4.1. Conversion of the line-based model to the solid prism model

As described in Section 3, the line segments in the line-based model are organized in triplets. To construct a prism from the line sections of the line segments in a triplet, as shown in Fig. 10, we need ﬁrst to specify which line sections to use. If each line segment contains only one line section, then there is only one possibility to connect the three line sections, which leads to a unique prism.

Fig. 8. A non-bounded line-based geometric model obtained from a reconstruction for a riﬂe.

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However, when one or more line segments in the triplet are broken into line sections, then there is at least a portion of the line segment lying outside the object, called the open line section. The existence of open line section(s) means the object part under reconstruction contains a notch or a cut (see Figs. 11(c)–14(c)). Thus, all we need to do is to decide which open line sections deﬁne a cut. There are one to three open line sections in a cut. If two or three open line sections are in-volved in a cut, these sections are required to be vertically overlapped pairwise or through a tran-sitive manner. Therefore, there are three possible ways to form a cut: (In the following in order to depict the mutual connections between line sec-tions in constructing the cut or prisms in 2D space, we cut the constructed prisms in Figs. 11– 14 along line segment 3 and, then, unfold the prism surfaces to make the three line segments lie on a 2D plane).

Case 1: A cut containing one open line section be-longing to one line segment in the triplet. This is the case shown in Fig. 11. Without loss of generality, we assume line segment 2 has an open line section. To construct the cut and the prisms, we proceed as fol-lows:

1. Create one interior point 1M on line segment 1 that has an average height relative to the endpoints 2A and 2B, i.e. height of 1M¼ ðheight of 2A þ height of 2B)/2, and then connect point 1M to points 2A and 2B.

2. Similarly, create another interior point 3M on line segment 3 having the same height as point 1M. Then connect point 3M to points 2A and 2B.

3. Finally, connect points 1M and 3M to get two triangles with vertex sets ð1M;

Fig. 11. A cut containing only one open line section of a line segment in the triplet.

Fig. 10. The prisms at right are converted from the triplet of line segments 1, 2 and 3, shown in the top view of the line-based model at left.

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3M; 2AÞ and ð1M; 3M; 2BÞ, which form a cut as shown in Fig. 11(c).

Case 2: A cut containing two open line sections belonging to two line segments in the trip-let, as shown in Fig. 12. We assume the two open line sections reside on line segments 2 and 3. To handle this case, the following steps are taken:

4. The corresponding endpoints of line sections of line segments 2 and 3 are connected to each other. Thus, points 2A and 3A are connected, so are end-points 2B and 3B.

5. Create one interior point 1M having an average height with regard to points 2A, 2B, 3A and 3B.

6. Connect point 1M to points 2A and 3A and connect 1M to 2B and 3B. We get two triangles with vertex sets ð1M; 2A; 3AÞ and ð1M; 2B; 3BÞ, which form a cut as depicted in Fig. 12(c). Case 3: A cut containing three open line sections,

one from each line segment in the triplet. There are two sub-cases shown in Figs. 13 and 14.

In Fig. 13 the three open line sections are ver-tically overlapped pairwise. Thus, it is intuitive to connect them together to form a cut.

In Fig. 14, the open line sections belonging to line segments 1 and 2 are not vertically overlapped. However, the open line sections belonging to line segments 2 and 3 are vertically overlapped, so are the open line sections belonging to line segments 3 and 1. That is, the open line sections belonging to line segments 1 and 2 are linked to the same cut in a transitive manner. Thus, we can connect the corresponding endpoints of the three open line sections to form a cut.

In this way, we can convert the non-bounded line-based model to a bounded prism model. 4.2. Conversion of the prism model to a bounded triangular mesh model

After the solid prism model is obtained, it is a simple matter to convert it into a bounded trian-gular mesh model. Each prism consists of the top

Fig. 14. A cut containing three open line sections belonging to three line segments, which are linked through the vertical overlapping relation in a transitive manner.

Fig. 13. A cut containing three open line sections belonging to three line segments, which are vertically overlapped pairwise. Fig. 12. A cut consists of two open line sections of two line segments in a triplet.

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and bottom triangular faces, and three quadran-gles. If each quadrangle is cut along its diagonal to yield two triangles, then each prism can be con-verted to eight connected triangular polygons. By performing this conversion to each prism, we can construct the triangular mesh representation of the object. We can render the geometry of the object using any conventional graphic card. With the Z-buﬀer technique, the hidden surfaces can be re-moved and only the boundary surfaces are ren-dered.

5. Experimental results

We have implemented our line-based recon-struction method on a PC with an AMD Athlon 1.2 GHz CPU and 128 MB RAM. The 2D array dimension of the original line-based model for the phase-one reconstruction was chosen according to the physical size and details of the given object. The maximum subdivision level in the phase-two reconstruction is set to 2. For

abbre-viation, these model parameters are represented byðM N ; RÞ, indicating the 2D array dimension is M N and the maximum subdivision level in the dynamic line resolution scheme is R. If R¼ 0, the phase-two reconstruction is skipped and the model is reconstructed with a ﬁxed line resolu-tion.

We first applied our method to three different synthetic objects with an increasing geometric complexity: teapot, rifle, and flower. Thirty six views of each of these objects were taken with the object resting on a turntable that was rotated by 10° each time. In Fig. 15(a)–(c) three of the 36 views are shown. Notice the objects contain parts of varying detail. The qualities of the recon-structed objects with the listed model parameter settings are shown in Fig. 16(a)–(c). We can see the fine details of the objects were captured by our method, including teapotÕs flat handle, rifleÕs small trigger, and flowerÕs thin petals, etc.

To evaluate the performance of the dynamic line resolution scheme, the reconstruction qualities of the dynamic and the ﬁxed line resolution

Fig. 15. Three typical samples of the 36 input images corresponding to the turntable rotation angles of 0°, 90°, and 180° for (a) a teapot, (b) a riﬂe, and (c) a ﬂower.

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schemes are given in Fig. 17(a)–(c) and the pro-cessing time and memory space taken are tabu-lated in Table 2. From these comparisons, we can see the quality of reconstruction for the ð25 16; 0Þ low ﬁxed line resolution setting is not so good as that for the ð97 61; 0Þ high ﬁxed resolution setting, especially in the areas around the teapot handle and the lid knob. However, when changing the model parameter setting from ð25 16; 0Þ to ð97 61; 0Þ, the processing time increases roughly by 10 times and the memory space increases by some 16 times. If using the dy-namic line resolution scheme, the reconstruction quality for theð25 16; 2Þ setting is nearly as good

as that for the ð97 61; 0Þ setting, while the pro-cessing time taken is reduced to one third of that taken for theð97 61; 0Þ setting. This is because in the dynamic line resolution scheme the higher line resolution is only applied to the sparse object parts having high curvature or steep slope, while the lower line resolution to the major parts. Therefore, the dynamic line resolution scheme is a good compromise between the reconstruction quality and processing time, which is desirable in practical applications. (Note that we choose ð25 16; 2Þ and ð97 61; 0Þ for comparison. This is because the line spacing is the same in these two model parameter settings.)

Fig. 16. Two views generated from the reconstructed triangular mesh models obtained for: (a) the teapot using theð25 16; 2Þ setting, (b) the riﬂe using theð100 8; 2Þ setting, and (c) the ﬂower using the ð30 30; 2Þ setting.

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Fig. 17. Comparison between reconstruction qualities of the teapots obtained with the three given model parameter settings: (a) ð25 16; 0Þ, (b) ð25 16; 2Þ, and (c) ð97 61; 0Þ. The left column shows a reconstructed view, and the right column shows the dif-ference between the reconstructed view and the corresponding original view. The diﬀerence is marked by the black pixels whose number is given with respect to the total pixel number of the original view.

Table 2

Processing time and memory space for the synthetic object reconstruction experiments

Processing time (s) Memory space (Sections)

Low resolution Dynamic

resolution

High resolution Low resolution Dynamic resolution

High resolution

Teapot ð25 16; 0Þ 0.4 ð25 16; 2Þ 2 ð97 61; 0Þ 6 214 1078 3423

Riﬂe ð100 8; 0Þ 1 ð100 8; 2Þ 4 ð397 29; 0Þ 11 690 3019 9796

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Fig. 18. Three of the 36 input images corresponding to the turntable rotation angles of 0°, 90°, and 180° for (a) a dinosaur and (b) a pot plant.

Fig. 19. Reconstruction results for the dinosaur using theð81 25; 2Þ model parameter setting. (a) Two views generated from the reconstructed line-based model. (b) Two views generated from the reconstructed triangular mesh model. In (a) the gray lines (lying mostly at the outer surface of the object) indicate the line segments inserted when using the dynamic line resolution scheme; the white lines (lying mostly inside the object) are the original line segments with a ﬁxed resolution.

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We also apply our method to reconstruct the real objects. Figs. 18–20 and Table 3 show the reconstruction results and the time and space complexities for two real objects: dinosaur and pot plant. Here, again the turntable was rotated every 10° so that 36 images were taken for each object.

Generally speaking, the recovered shape of the object is in a fairly good agreement with the real shape except at the object spots where they are indented or too thin to be detected. These spots may be corrected using some post processing technique.

6. Conclusions

We have presented a rapid line-based method for constructing the 3D shape for complex objects. The method consists of two reconstruction phases; phase one is with a ﬁxed line resolution and phase two with a dynamic line resolution. A conversion from the line-based geometric model to the bounded triangular mesh model is given to render the 3D reconstructed object shape. It has been shown in the experiments that it is capable of capturing most of the ﬁne details of the object. The

Fig. 20. Reconstruction results for the pot plant using theð116 106; 2Þ model parameter setting. (a) Two views generated from the reconstructed line-based model. (b) Two views generated from the reconstructed triangular mesh model.

Table 3

Processing time and memory space for the real object reconstruction experiments

Processing time (s) Memory space (Sections)

Dynamic resolution Fixed resolution Dynamic resolution Fixed resolution

Dinosaur ð81 25; 2Þ 13 ð321 97; 0Þ 53 6113 14177

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statistics show that the method works fast and requires a relatively low memory space compared to the octree-based reconstruction method. If needed, we can refine the reconstruction result by adding additional views in the reconstruction process. The geometric model thus obtained is adequate for use in object recognition and pose determination. And it is also suitable for browsing if the texture mapping is added. Currently we are planning to do the post-processing refinement of the geometric model. We shall use the geometric model obtained by the current method to enhance the stereo matching between the feature points visible on the object surfaces. Once the corre-sponding point pairs are found, we can compute their new 3D coordinates by the triangula-tion technique. Afterwards the surface patches are modified accordingly.

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