Abstract:
Disclosed are data storing and processing methods or apparatuses which may efficiently perform a process of retrieving neighboring points between points stored in a leaf cell in a point-based 3D data expressing method. A computer-readable recording medium having stored thereon instructions for implementing a method of three-Dimensional (3D) data processing, the instructions including an instruction set of connecting each of a plurality of points with a leaf cell of a spatial tree structure; an instruction set of assigning a ball to correspond to each of the plurality of points; and an instruction set of connecting the ball with a leaf cell intersecting the ball.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of Korean Patent Application No. 10-2008-0016996, filed on Feb. 25, 2008, in the Korean Intellectual Property Office, the disclosure of which is incorporated herein by reference. 
     BACKGROUND 
     1. Field 
     The present invention relates to a three-dimensional (3D) image processing method, and more particularly, to data storing and processing methods (or apparatuses) which may efficiently perform a process of retrieving neighboring points between points stored in a leaf cell in a point-based 3D data expressing method. 
     2. Description of the Related Art 
     Currently, studies of a method for obtaining three-dimensional (3D) information through image matching after obtaining a depth image using a depth camera and then obtaining a color image using general Charge-Coupled Device (CCD)/Complementary Metal Oxide Semiconductor (CMOS) cameras have been attempted. 
     As examples of methods for representing the 3D information obtained as described above, a method using a triangular mesh, a method using point cloud, and the like may be given. The obtained 3D information may be transformed into an efficient representing structure after an appropriate modeling process is performed. In this manner, performance of the transformation or various processing and rendering after the transformation may significantly depend on efficiency of a data structure expressing 3D information. 
     In a mesh-based 3D information expressing method, neighboring pixels are connected using regularity of images and mesh connection information is generated to process images. According to the mesh based 3D information expressing method, noises, holes, and the like generated when obtaining the depth image may be relatively easily eliminated. However, the mesh based 3D information expressing method may require a stitching process with respect to overlapped portions of a plurality of pieces of image data, which is known as a significantly complex process, in order to generate a single 3D mesh model from the plurality of pieces of image data being different from a piece of image data. 
     Meanwhile, in a point-based 3D information expressing method, a relatively efficient 3D model may be generated in comparison with the mesh-based 3D information expressing method because 3D point data obtained from the plurality of pieces of the image is simply combined on a space. In this instance, the above described point data may be stored in a spatial data structure such as a kd-tree or an octree. The spatial data structure such as the octree used in a conventional point-based 3D information expressing method may not have connection information between points as in a polygon mesh method, and thereby retrieving of neighboring points required for performing various types of processing may require a great amount of time. 
     SUMMARY 
     Accordingly, it is an aspect of the present invention to provide a three-dimensional(3D) data storing method which may store 3D data obtained by a depth camera from an object using an improved method, thereby improving efficiency in storing and processing point-based 3D data. 
     It is another aspect of the present invention to provide a 3D data processing method and apparatus which may efficiently retrieve information on neighboring points from among points stored in the point-based 3D data structure. 
     Still another aspect of the present invention provides a data processing method such as a k Nearest Neighbors (kNN) query, a Spatial Ball Neighbors (SBN) query, a leaf cell division, and the like using a data structure obtained in the improved method. 
     Additional aspects and/or advantages will be set forth in part in the description which follows and, in part, will be apparent from the description, or may be learned by practice of the invention. 
     The foregoing and/or aspects may be achieved by providing a computer-readable recording medium having stored thereon instructions for implementing a method of three-dimensional (3D) data processing, the instructions including: an instruction set of connecting each of a plurality of points with a leaf cell of a spatial tree structure; an instruction set of assigning a ball to correspond to each of the plurality of points; and an instruction set of connecting the ball with a one of the leaf cells intersecting the ball. 
     In this instance, the computer-readable recording medium may further include an instruction set of retrieving, in the spatial tree structure, a first leaf cell associated with a first point from among the plurality of points; an instruction set of retrieving a second ball intersecting the first leaf cell; and an instruction set of retrieving a second point corresponding to the second ball. 
     Also, the computer-readable recording medium may further include an instruction set generating a Moving Least Square (MLS) surface based on first and second ones of the points. 
     Also, the instruction set of assigning the ball assigns a ball having an adaptive radius to correspond to each of the plurality of points. 
     According to another aspect of the present invention, there is provided a computer-readable recording medium having stored thereon instructions for implementing a method of data processing, the instructions including: an instruction set of connecting each of a plurality of points with at least one leaf cell of a quad tree structure; an instruction set of assigning a circle to correspond to each of the plurality of points; and connecting the circle with a one of the leaf cells intersecting the circle. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       These and/or other aspects, features, and advantages of the invention will become apparent and more readily appreciated from the following description of exemplary embodiments, taken in conjunction with the accompanying drawings of which: 
         FIG. 1  is a flowchart illustrating a method of storing location information of points composing a 3D object in a spatial tree structure according to an exemplary embodiment of the present invention; 
         FIG. 2  is a diagram illustrating a case where location information of points is stored in an octree according to an exemplary embodiment of the present invention; 
         FIG. 3  is a flowchart illustrating a method of generating a Moving Least Square (MLS) surface by retrieving a neighboring point according to an exemplary embodiment of the present invention; 
         FIG. 4  is a diagram illustrating an example of a three-dimensional (3D) image obtained using a MLS surface according to an exemplary embodiment of the present invention; 
         FIG. 5  is a flowchart illustrating a method of determining a ball corresponding to each of points according to an exemplary embodiment of the present invention; 
         FIG. 6  is a flowchart illustrating a method of inserting, in a leaf cell, information about a ball intersecting the leaf cell according to an exemplary embodiment of the present invention; 
         FIG. 7  is a flowchart illustrating a method of storing location information of points composing a 2D object in a quad tree structure according to an exemplary embodiment of the present invention; 
         FIG. 8  is a diagram illustrating a state where location information of points composing a 2D object is stored in a quad tree structure according to an exemplary embodiment of the present invention; and 
         FIG. 9  is a flowchart illustrating a method of generating a MLS surface by retrieving neighboring points of a quad tree structure according to an exemplary embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Reference will now be made in detail to the embodiments, examples of which are illustrated in the accompanying drawings, wherein like reference numerals refer to the like elements throughout. The embodiments are described below to explain the present invention by referring to the figures. 
       FIG. 1  is a flowchart illustrating a method of storing location information of points composing a 3D object in a spatial tree structure according to an exemplary embodiment of the present invention. 
     In operation S 110 , an image processing apparatus may retrieve a cell in which location information of points is inserted. According to the present exemplary embodiment of the present invention, location information of points composing a 3D object may be stored in a spatial tree structure. The spatial tree structure may be a data structure having a hierarchical structure and representing the location information of points composing the object, and for example, may be a kd-tree or an octree. Herein, the octree may be described in detail as an example of the spatial tree structure, however, the present invention is not limited thereto. A three-dimensional (3D) location information of points composing a 3D object may be obtained in various methods such as an image-based method using computer vision techniques, a method using a depth acquiring device, and the like. As an example, the 3D location information of points may be obtained using a depth camera such that the depth camera is located in multiple points of view with respect to a single object to thereby obtain a plurality of depth images according to the time points. The image processing apparatus may store the location information of points in the octree. In this case, the image processing apparatus may retrieve a cell of the octree in which the location information of points is inserted in operation S 110 . The octree is a spatial tree structure in which a cube on a space is split into 8 portions and a point intended to be stored is assigned to a descendant of a lower splitting step. 
     In operation S 120 , the image processing apparatus may connect each of a plurality of points with a leaf cell of a spatial tree structure. In order to store point-based 3D data, a cell may be assigned to each of the plurality of points. Operation S 120  may be implemented by dividing the cell within the spatial tree structure and storing points in a sub-cell. The sub-cell may be the leaf cell. For example, when inserting the points composing the 3D object to the octree, the image processing apparatus traverses the octree to find target leaf cell. Then, the image processing apparatus inserts the point to the leaf cell. 
       FIG. 2  is a diagram illustrating a case where location information of points is stored in an octree according to an exemplary embodiment of the present invention. 
     Referring to  FIG. 2 , a leaf cell is connected with each of points  210  and  230 . 
     In operation S 130 , the image processing apparatus may assign a ball to correspond to each of the plurality of points. Here, the ball is a sphere. The ball may be assigned to each of the plurality of points. For example, a first ball is assigned to a first point, and a second ball is assigned to a second point. The ball has a radius. 
     According to the present exemplary embodiment of the present invention, the ball is a sphere having the corresponding point acting as a center. According to the present exemplary embodiment of the present invention, the ball assigned to each of the points has a given radius. For example, a ball having an identical radius may be assigned with respect to all points. According to the present exemplary embodiment of the present invention, the ball assigned to each of the points has an adaptive radius. In this case, the ball assigned to each of the points may be variable. Also, the radius of the ball assigned to each of the points may be different from one another. For example, the radius of the ball may be determined according to a density of a point connected with a leaf cell. In this instance, the leaf cell may be connected with each of the plurality of points. The density may be an index indicating how many neighboring points exist. A method of determining the density is not limited to a single exemplary embodiment. According to the present exemplary embodiment of the present invention, a distance between the first point and a k-th nearest point from the first point may be determined as a radius of the ball. In this instance, ‘k’ is a natural number greater than ‘1’. In this case, as a number of neighboring points of the first point increases, a distance between the first point and the k-th nearest point from the first point decreases, and thereby a radius of the ball assigned to the first point is reduced. Referring to  FIG. 2 , a ball  220  is assigned to the point  210 , and a ball  240  is assigned to the point  230 . 
     In operation S 140 , the image processing apparatus may retrieve a cell intersecting the ball assigned to the point. The image processing apparatus may compare (1) the radius of the ball assigned to the point and (2) a distance between the point and a neighboring cell including the point, thereby retrieving the cell intersecting the ball assigned to the point. 
     In operation S 150 , the ball may be connected with a leaf cell intersecting the ball. The ball is assigned to the point, and connected with a cell connected with the point. For example, when the first point is stored in the first cell and the first ball is assigned to the first point, the first ball is connected with the first cell. Specifically, reference information of the first ball is stored in the first cell. Also, when another cell intersecting the ball exists, the ball is also connected with the other cell. For example, the other cell may also store the reference information of the ball. Thus, according to the present exemplary embodiment of the present invention, information about the balls intersecting the cell may be obtained based on the cell. 
     According to the present exemplary embodiment of the present invention, information about a leaf cell where the point is stored may be easily obtained from a single first point from among points composing a 3D object, and information about points corresponding to the balls intersecting the leaf cell may be also easily obtained from the leaf cell. The obtained points may be points located near the first point. Thus, according to the present exemplary embodiment of the present invention, information is obtained about other points located near an arbitrary point from among the points composing the 3D object. 
     In operation S 160 , according to the present exemplary embodiment of the present invention, the image processing apparatus may determine whether the leaf cell intersecting the ball is split. The spatial tree structure according to the present exemplary embodiment of the present invention may generate a hierarchical tree structure by repeatedly splitting the cell. In this case, when either (1) the tree depth is a given maximum tree depth or (2) the cell has a given minimum number of points and a length of a diagonal line of the cell is less than a mean ball size, the split of the cell is stopped and the leaf cell is determined. For example, when a length of a diagonal line of the leaf cell is greater than a mean diameter of the balls corresponding to each of the plurality of points, the leaf cell intersecting the ball is split. By splitting the leaf cell as described above, a smaller leaf cell capable of storing the points may be obtained along with an increase in the density of the points. 
     According to the present exemplary embodiment of the invention, the point may be inserted within the spatial tree structure for processes such as hole filling, and the like. The above point insertion may be divided into a point location information inserting process and a ball information inserting process. The point location information insertion process may include computing a location code of a target cell including a given point, retrieving the target cell by performing a tree traversal until the leaf cell is detected from a root cell, and determining whether the target cell is further split after inserting the location information of the point in the target cell. 
     Meanwhile, the ball information insertion process may include computing a bounding box of the ball, computing a location code of the first cell including the bounding box, performing a tree retrieval until either the leaf cell or the first cell is detected from the root cell, determining whether all cells of a sub-tree of the first cell intersect the ball to thereby store reference information of the ball only in a cell intersecting the ball, and determining whether a cell where the reference information is added is further split. According to the present exemplary embodiment of the present invention, determining whether the cell is further split during either the point location information inserting process or the ball information inserting process may be performed by a method for splitting the leaf cell intersecting the ball when the length of the diagonal line of the leaf cell where a new point is stored is greater than the mean diameter of the ball corresponding to each of the plurality of points within the spatial tree structure. 
     As illustrated in  FIG. 2 , according to the present exemplary embodiment of the present invention, 3D location information of points composing the 3D object may be stored in the octree structure. 
     The octree may be a spatial tree structure in which a cube on a space is split into 8 portions and a point intended to be stored is assigned to a descendant of a lower splitting operation. For example, as illustrated in  FIG. 2 , when the location information of additional points is required to be stored, a cell  235  may be split into 8 portions to thereby store location information of added points. 
     As illustrated in  FIG. 2 , the points  210  and  230  may be stored in each cell. Also, the balls  220  and  240  are assigned to correspond to each of the points  210  and  230 , respectively. In this instance, the balls  220  and  240  are a sphere. As illustrated in  FIG. 2 , the ball  240  intersects cells  231 ,  232 ,  233 ,  234 ,  235 ,  236 , and  237 . In this case, the cells  231 ,  232 ,  233 ,  234 ,  235 ,  236 , and  237  may store reference information about the ball  240 . The reference information about the ball  240  may be a pointer with respect to a memory space where information about the ball  240  is stored. 
     According to the present exemplary embodiment of the present invention, the balls assigned to the plurality of points do not yield the same size, and may have adaptive radii. For example, the radius of the ball may be determined according to a density of points connected with the leaf cell including points connected with the ball. 
       FIG. 3  is a diagram illustrating a method of generating a Moving Least Square (MLS) surface by retrieving a neighboring point according to an exemplary embodiment of the present invention. 
     In operation S 310 , the image processing apparatus may retrieve the first leaf cell, containing the location information of a query point. When the octree is used as the spatial tree structure, the image processing apparatus may perform a tree traversal with respect to the octree, thereby retrieving the first leaf cell, containing the query point. 
     In operation S 320 , the image processing apparatus may retrieve a second ball intersecting the first leaf cell. Through operations  140  and  150  of  FIG. 1 , the reference information of the ball intersecting the first leaf cell may be stored in the first leaf cell. Thus, the image processing apparatus may read the reference information of the ball intersecting the first leaf cell and stored in the first leaf cell in operation S 320 , thereby obtaining information about a second ball intersecting the first leaf cell. In case of a plurality of balls intersecting the first leaf cell, information about the plurality of balls intersecting the first leaf cell may be stored in the first leaf cell. 
     In operation S 330 , the image processing apparatus may retrieve a second point corresponding to the second ball. In operation S 130  of  FIG. 1 , a ball is assigned to correspond to each of points stored in the spatial tree structure. In operation S 330 , the image processing apparatus may retrieve the second point corresponding to the second ball using the relationship between the point and the ball. For example, each of the balls may store reference information of points corresponding to the balls. In this case, the image processing apparatus may obtain information about the points corresponding to the ball using the reference information of the points stored in the ball. 
     In operation S 340 , the image processing apparatus may determine whether another ball intersecting the first leaf cell exists. When the ball intersecting the first leaf cell is a plurality of balls, information about a plurality of points may be obtained to correspond to each of the plurality of balls. For example, when the ball intersecting the first leaf cell is two of a ball_A and a ball_B, the ball_A is first retrieved in operation S 320 , and then a point_A corresponding to the ball_A is retrieved in operation S 330 . Next, in operation S 340 , since another ball intersecting the first leaf cell exists, the ball_B is retrieved by returning to operation S 320 . Next, in operation S 330 , a point_B corresponding to the ball_B is retrieved. When information about points corresponding to all balls intersecting the first leaf cell is obtained, the image processing apparatus may advance to operation S 350 . 
     In operation S 350 , the image processing apparatus may generate a Moving Least Square (MLS) surface based on the first point and the second point. The MLS surface is a continuous and smooth surface generated using a MLS operator from points discretely distributed. Through operations  310  and  340 , the image processing apparatus may obtain information about a single point composing the 3D object and neighboring points of the single point. The image processing apparatus may generate the MLS surface by applying the MLS operator to the point. In generating the MLS surface, neighboring points are required to be retrieved and a distance with each of the points is required to be computed. According to the present exemplary embodiment of the present invention, efficiency of the computation may be increased. According to the present exemplary embodiment of the present invention, information about other points located near an arbitrary point from among points composing the 3D object may be readily obtained, and thus readily obtaining 3D image. Also, according to the present exemplary embodiment of the present invention, a projection surface may be generated using a projection operator other than the MLS operator. 
       FIG. 4  is a diagram illustrating an example of a 3D image obtained using a MLS surface according to an exemplary embodiment of the present invention. 
       FIG. 5  is a flowchart illustrating a method of determining a ball corresponding to each of points according to an exemplary embodiment of the present invention. 
     In operation S 510 , the image processing apparatus may retrieve a first leaf cell containing a query point. The image processing apparatus may retrieve the first leaf cell where location information of the query point is connected. When the octree is used as the spatial tree structure, the image processing apparatus may perform a tree traversal with respect to the octree, thereby retrieving the first leaf cell containing the query point. 
     In operation S 520 , the image processing apparatus may select k nearest neighboring points from among the points connected with the first leaf cell. According to the present exemplary embodiment of the present invention, ‘k’ is a given natural number (for example, k=3). 
     In operation S 530 , the image processing apparatus may determine a first distance from the query point to the farthest point among the k neighboring points. For example, when neighboring points of k=3 with respect to the query point are selected, the farthest point among the three neighboring points may be selected, and a distance between the query point and the farthest point may be determined as the first distance. 
     In operation S 540 , the image processing apparatus may compare (1) a distance between the query point and a bounding box of a second cell of the spatial tree structure and (2) the first distance. 
     According to the results of operation S 540 , when the distance between the query point and the bounding box is less than the first distance, the image processing apparatus may update the k nearest neighboring points according to a distance between a point connected with a leaf cell of a sub-tree of the second cell and the query point (operation  550 ). 
     In operation S 560 , the image processing apparatus may determine the ball corresponding to the query point based on the k nearest neighboring points. According to the present exemplary embodiment of the present invention, each size of the ball assigned to each of the points may differ. 
       FIG. 6  is a flowchart illustrating a method for inserting, in a leaf cell, information about a ball intersecting the leaf cell according to an exemplary embodiment of the present invention. 
     In operation S 610 , the image processing apparatus may determine a bounding box of the ball. The bounding box of the ball is a minimum sized-cube including the ball and a point corresponding to the ball acting as a center. For example, when a first ball is assigned to a first point, a minimum-sized cube including the first ball and the first point acting as the center is a bounding box of the first ball. 
     In operation S 620 , the image processing apparatus may determine a first cell including the bounding box in the spatial tree structure. The first cell including the bounding box is a cell of a lowest splitting step capable of including all portions of the bounding box. The first cell may include leaf cells of a plurality of sub-trees. 
     In operation S 630 , the image processing apparatus may determine whether the leaf cells of the plurality of sub-trees of the first cell intersect the ball. 
     In operation S 640 , the image processing apparatus may connect the leaf cell intersecting the ball with the ball. Also, the image processing apparatus may store reference information of the first ball in a leaf cell intersecting the first ball from among the leaf cells of the plurality of sub-trees, and may not store the reference information of the first ball in a leaf cell not intersecting the first ball from among the leaf cells of the plurality of sub-trees. 
     In operation S 650 , the image processing apparatus may determine whether a supplementary leaf cell included in the bounding box exists. For example, when a leaf cell not determined whether to intersect the ball, from among the leaf cells of the plurality of sub-trees of the first cell, exists, the image processing apparatus may advance to operation S 630 . 
       FIG. 7  is a flowchart illustrating a method of storing location information of points composing a 2D object in a quad tree structure according to an exemplary embodiment of the present invention. 
     In operation S 710 , the image processing apparatus may retrieve a cell in which location information of points composing a 2D object is inserted. According to the present exemplary embodiment of the present invention, the location information of the points may be stored in a quad tree structure. The image processing apparatus may retrieve a cell of the quad tree in which the location information of the points is inserted. The quad tree is a tree data structure in which each node has a maximum of four children nodes. 
     In operation S 720 , the image processing apparatus may connect each of the plurality of points with a leaf cell of the quad tree structure. The image processing apparatus may assign a cell to each of the plurality of points in order to store point based 3D data. The image processing apparatus may store the location information of the points in the retrieved leaf cell of the quad tree. 
     In operation S 730 , the image processing apparatus may assign a circle to each of the plurality of points. The circle may be assigned one by one to correspond to each of the plurality of points. For example, a first circle is assigned to a first point, and a second circle is assigned to a second point. The circle has a radius. 
     According to the present exemplary embodiment of the present invention, the circle assigned to each of the plurality of points may have a given radius. For example, circles having an identical radius may be assigned to all the plurality of points. According to another exemplary embodiment of the present invention, the circle assigned to each of the plurality of points may have an adaptive radius. In this case, the radius of the circle assigned to each of the plurality of points may be variable. Also, the radius of the circle may differ. For example, the radius of the circle may be determined according to a density of points connected with the leaf cell connected with each of the plurality of points. According to another exemplary embodiment of the present invention, a distance with a k-th nearest point from the first point may be determined as the radius of the circle. In this instance, ‘k’ is a natural number. In this case, as a number of points near from the first point increases, the distance with the k-th nearest point from the first point may decrease, and accordingly the radius of the circle assigned to the first point is reduced. 
     In operation S 740 , the image processing apparatus may retrieve a cell intersecting the circle assigned to the point. The image processing apparatus may compare (1) the radius of the circle assigned to the point (2) a distance between the point and a neighboring cell of the point, thereby retrieving a cell intersecting the circle assigned to the point. 
     In operation S 750 , the image processing apparatus may connect the circle with a leaf cell intersecting the circle. The circle is assigned to the point, however, is required to be first connected with a cell connected with the point. For example, when a first point is stored in a first cell and a first circle is assigned to the first point, the first circle is connected with the first cell. Specifically, reference information of the first circle is stored in the first cell. Also, when another cell intersecting the circle exists, the circle is also connected with the other cell. For example, the other cell may store the reference information of the circle. Thus, according to the present exemplary embodiment of the present invention, information about circles intersecting the cell may be obtained based on the cell. 
     According to the present exemplary embodiment of the present invention, information about a leaf cell where the point is stored may be obtained from a single first point from among the points composing the 2D object, and information about points corresponding to the circles intersecting the leaf cell may be easily obtained from the leaf cell. The obtained points are points located near the first point. Thus, according to the present exemplary embodiment of the invention, information about other points located near an arbitrary point from among the points composing the 2D object may be readily obtained. 
     In operation S 760 , according to the present exemplary embodiment of the present invention, the image processing apparatus may determine whether a leaf cell intersecting the circle is split. The quad tree structure according to the present exemplary embodiment of the present invention may generate a hierarchical tree structure by repeatedly splitting the cell. In this case, when either (1) the tree depth is a given maximum tree depth or (2) the cell has a given minimum number of points and a length of a diagonal line of the cell is smaller than a mean diameter of the circle, the split of the cell is stopped and the leaf cell is determined. 
       FIG. 8  is a diagram illustrating a state where location information of points composing a 3D object is stored in a quad tree structure according to an exemplary embodiment of the present invention. 
     As illustrated in  FIG. 8 , according to the present exemplary embodiment of the present invention, location information of the points composing the 3D object is stored in the quad tree structure. As illustrated in  FIG. 8 , a circle is assigned to correspond to a point  830 . The circle of  FIG. 8  intersects cells  831 ,  832 ,  833 ,  834 ,  835 ,  836 ,  837 . Reference information of the circle may be pointer with respect to a memory space where information about the ball is stored. 
       FIG. 9  is a flowchart illustrating a method of generating a MLS surface by retrieving neighboring points of a quad tree structure according to an exemplary embodiment of the present invention. 
     In operation S 910 , the image processing apparatus may retrieve, in the quad tree structure, a first leaf cell connected with a first point from among the plurality of points. The image processing apparatus may retrieve the first leaf cell where location information of the first point from among the plurality of points is stored. The image processing apparatus may perform a tree traversal with respect to the quad tree, thereby retrieving the first leaf cell where the first point is stored. 
     In operation S 920 , the image processing apparatus may retrieve a second circle intersecting the first leaf cell. Through operations  740  and  750  of  FIG. 7 , the reference information of the circle intersecting the first leaf cell is stored in the first leaf cell. Thus, the image processing apparatus may read the reference information of the circle intersecting the first leaf cell and stored in the first leaf cell in operation S 920 , thereby obtaining information about a second circle intersecting the first leaf cell. The circle intersecting the first leaf cell may be a plurality of circles. In this case, information about the plurality of circles intersecting the first leaf cell may be stored in the first leaf cell. 
     In operation S 930 , the image processing apparatus may retrieve a second point corresponding to the second circle. In operation S 730  of  FIG. 7 , a circle is assigned to correspond to each of the plurality of points stored in the quad tree structure. In operation S 930 , the image processing apparatus may retrieve a second point corresponding to the second circle using relation between the point and the circle in operation S 730 . For example, each of the circles may store reference information of points corresponding to each of the circles. In this case, the image processing apparatus may obtain information about the point corresponding to the circle using the reference information of the point stored in the circle. 
     In operation S 940 , the image processing apparatus may determine whether any other circle intersecting the first leaf cell exists. When the circle intersecting the first leaf cell is a plurality of circles, information about a plurality of points to correspond to each of the plurality of circles may be obtained. For example, when two circles of a circle_A and a circle_B each intersecting the first leaf cell exist, the circle_A is first retrieved in operation S 920 , and then a point_A corresponding to the circle_A is retrieved in operation S 930 . Next, in operation S 940 , since another circle intersecting the first leaf cell exists, the circle_B is retrieved returning to operation S 920 . Next, in operation S 930 , a point_B corresponding to the circle_B is retrieved. When information about points corresponding to all circles intersecting the first leaf cell is obtained, the image processing apparatus may advance to operation S 950 . 
     In operation S 950 , the image processing apparatus may generate a MLS surface based on the first point and the second point. Through operations  910  to  940 , the image processing apparatus may obtain information about a single point composing a 2D object and information about neighboring points of the single point. The image processing apparatus may generate the MLS surface by applying a MLS operator to the point. Also, according to the present exemplary embodiment of the present invention, a project surface may be generated using another project operator other than the MLS operator. 
     The embodiments can be implemented in computing hardware (computing apparatus) and/or software, such as (in a non-limiting example) any computer that can store, retrieve, process and/or output data and/or communicate with other computers. The results produced can be displayed on a display of the computing hardware. A program/software implementing the embodiments may be recorded on computer-readable media comprising computer-readable recording media. The program/software implementing the embodiments may also be transmitted over transmission communication media. Examples of the computer-readable recording media include a magnetic recording apparatus, an optical disk, a magneto-optical disk, and/or a semiconductor memory (for example, RAM, ROM, etc.). Examples of the magnetic recording apparatus include a hard disk device (HDD), a flexible disk (FD), and a magnetic tape (MT). Examples of the optical disk include a DVD (Digital Versatile Disc), a DVD-RAM, a CD-ROM (Compact Disc-Read Only Memory), and a CD-R (Recordable)/RW. An example of communication media includes a carrier-wave signal. 
     Although a few embodiments have been shown and described, it would be appreciated by those skilled in the art that changes may be made in these embodiments without departing from the principles and spirit of the invention, the scope of which is defined in the claims and their equivalents.