Construction of 3D Model of a Rectangular Parallelopiped Object from Its 2D Images

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1 Construction of 3D Model of a Rectangular Parallelopiped Object from Its 2D Images Md. Aminul Islam, Md. Nadim Saker, Asad Ibne Moin, and Rajib Kundu Computer Science & Engineering Discipline, KHULNA University, Khulna 9208, Bangladesh s: cseku@khulna.bangla.net, sadbt29@hotmail.com, raziv31@yahoo.com Abstract: 3D models and animation add a new era in the field of computer graphics. Most 3D models are built using high level CAD software. In this point of view, constructing 3D model from its images obviously an important approach. This paper presents an algorithm for constructing 3D object from its 2D images. The algorithm analysis a pair of images that cover all six planes of the rectangular parallelopiped object and construct the 3D model of that object with its real color values. Basically the algorithm finds out the corner points that help to calculate the length, width and height of the object. Using length, width and height, 3D models can be built. Two new approaches for finding points and the size of the object have been proposed in this paper. Keywords: 3D Model, 2D image, WCS, Rectangular parallelopiped, Center of projection. 1. INTRODUCTION Graphics provides one of the most natural means of communicating with computer. It is very interesting to construct a 3D model from 2D images of an object. Many research works have been done for recognition of 3D objects using parameterized volumetric models [1], arbitrary 3D shapes from range images [2], 3D face recognition and face segmentation into a limited number of analytical patches from stereo images [3]. This proposed approach is concerned with building the model of the object form its 2D images. This work constructs the 3D model of a rectangular parallelopiped object from two images that cover all six sides of that object. The background color of the image has to be different from the object colors. Though techniques for detecting the edges of an object placed on a background of the same grayscale intensity are developed [4] but it is easier to recognize the object from a background that has other color than the object colors. Figure 1. Construction of 3D Model from a pair of images. So it becomes possible to fill the object surface with its real color values. Figure 1 briefly presents the whole process to derive 3D model from a pair of images. 2. IMAGE ACQUISITION TECHNIQUE The images of the object have to be taken very carefully by following some rules. Background color will be different from the foreground or object colors. In daytime it is suggested to ensure that the object will produce minimum shade on the background. When daylight is not available or if it is in a room then it is suggested to ensure proper lighting. It is advised not to use the flash of the camera as it gives extra light to the image and the color of the image may lose its real color values. Normally a 3D object gets 2D view in an image, so it becomes critical to identify the corner points and the actual size of the object. To accomplish the job of seeking the corners, this algorithm provides an easier way. The most important and crucial part of this paper is to calculate the length, width and height of the object from its 2D viewing picture. The algorithm calculates the actual length, width and height of the object using corner points. Then it becomes easier to construct the 3D model of the object. Each pixels of the image is read respectively and there color values are stored.

2 Figure 2. (a) Image covering top, left, front Figure 2. (b) Image covering back, right, bottom Images can be taken by digital or normal camera. In both cases all the pictures should be in same resolution. It gives good results using black color as the background at daytime as it can absorb more light. While taking images inside a room with proper lighting any dark color can be used but it should be different from the object color. For taking the snap up of the object, it is necessary to maintain same distance from the camera to the object. Two snap ups of the object have to take so that each picture can cover three different planes of the object (Figure 2(a), (b)). So, from only two images, the total view of all the surfaces of the object can be acquired. 3. READING IMAGE FILE Each bitmap file contains a bitmap-file header, a bitmapinformation header, a color table, and an array of bytes that defines the bitmap bits [5]. The color table, defined as an array of RGBQUAD structures, contains as many elements as there are colors in the bitmap. Each element has an index number. Every pixel has three values- X coordinate, Y co-ordinate and an index number denotes the color table. Each pixel has been read and the color values of each pixel other than background color are stored. These values are needed for filling the surfaces of the object. 4. ALGORITHM PROPOSED The total work is designed into three parts background color detection, recognition of the object and construction of the 3D object. 4.1 Background color detection A new algorithm is proposed here for determining the background color of the image. A threshold value is used for making difference with the object from the background [6]. In a background, the color values of the pixels vary within a short range from pixel to pixel. So, the maximum value of the background color is measured as a threshold value. A good number of continuous pixels, which have values other than the threshold value, will be treated as object. This algorithm returns the threshold value as background color. Algorithm: Background color detection max_value := c_value; //c_value be the color index at the pixel (x, y) //max_value be the maximum background color value repeat Take a pixel at point (x, y) and assign its color value to c_value If ( max_value is less than c_value) then max_value := c_value until the areas defined by f (?) are not traversed return max_value end This algorithm uses a function, f (?), which defines the area to traverse. f (?)= f (? 1 )+ f (? 2 )+ f (? 3 )+ f (? 4 ), where, f (? 1 )= area between (x 1, y 1 ) and (x 2, y 1 + ), f (? 2 )= area between (x 1, y 2 - ) and (x 2, y 2 ), f (? 3 )= area between (x 1, y 1 + ) and (x 1 +, y 2 - ), f (? 4 )=area between (x 2, y 1 + ) and (x 2 -, y 2 - ) = (x 2 -x 1 )/ρ, and ρ=f (x 2, x1). Here, ρ is a function of x 2, and x 1, ρ is proportional to the value of x 2 -x 1. If x 2 -x 1 increases then increased ρ and if x 2 - x 1 decreased then decreases ρ will give good results. 4.2 Recognition of the object Here two major tasks - finding out the corner points and determining the length, width and height of the object, have been done Finding out the corner points of the object This algorithm analyzes two images. From the first image, (Figure 2(a)) the length and width is calculated. The algorithm returns three points A (x 1, y1), B (x 2, y2) and C (x 3, y3). Point A and B is used for calculating arbitrary length and A and C for arbitrary width. Analyzing second image, Image2 (Figure 2(b)), two points are returned also. Point D (x 1, y1) and E (x 2, y2) are used for calculating height. It is possible to find out the length, width and height from a single image. But at least two images are needed for covering all the surfaces of the object. This algorithm starts scanning pixels from the leftmost x coordinate and bottommost y co-ordinate position. Algorithm: Point Detection assign leftmost x co-ordinate value of the image to left_x assign topmost y co-ordinate value of the image to top_y assign rightmost x co-ordinate value of the image to right_x assign bottommost y co-ordinate value of the image to bottom_y

3 assign background color value to back_color_value // use background color detection algorithm for i: = bottom_y to top_y step-1 do for j: = left_x to right_x do //traverse the whole image starts from bottomleft corner if pixel_color_value is different from back_color_value then // detect pointa for Image1 or Image2 if pointa is not found in Image1 or Image2 then assign j to x 1 and i to y 1 // store x and y for point A(x 1,y 1 ) end for end. assign x 1 to previous_x and y 1 to previous_y // previous_x and previous_y store previous x and y co-ordinate // detect pointc for only Image1 if PointC is not found only in Image1 then if previous_x is greater than present x co-ordinate value and previous_y is the greater than present y co-ordinate value then // j is the present x co-ordinate and i is the present y co-ordinate assign j to previous_x and i to previous_y //assign present x and y to previous x and y if previous_x is equal to present x coordinate value then assign previous_x to x 3 and previous_y to y 3 //store x and y for point C(x 3,y 3 ) //detect pointb for both Image1 and Image2 if pointb is not found in Image1 or Image2 then if previous_x is less than present x co-ordinate value and previous_y is greater than present y co- -ordinate value then assign j to previous_x and i to previous_y //assign present x and y to previous x and y if previous_x is equal to present x co-ordinate value then assign previous_x to x 2 and previous_y to y 2 // store x and y for point C(x 3,y 3 ) 3 (a) 3 (b) Figure 3. (a) Calculating actual length and width. (b) Finding out the threshold value Determining actual length, width and height Figure 3(a) shows an object of an image where A, B and C are three points that has already determined. l ar and w ar are the arbitrary length and width of the object. θ be the angle between AB and a straight line parallel to x-axis. AB= l ar, AC= w ar,slope, m=tanθ= (y 2 -y 1 )/(x 2 -x 1 ) Table 1. Sample Data for calculating length and slope. Image No First point Second point Length l Slope m x 1 y 1 x 2 y l-m Curve (a)

4 4 (b) Figure 4. (a) l-m curve. (b) Comparison between l-m. A function f(m, l ar ) returns actual length of the object. It is obvious that m is inversely proportional to l ar and if it becomes zero, the resultant length will be the actual length. If slope m increases with a small amount such as dm, then small length dl decreases (Figure 4(b)). Rate of change of slope w.r.t. length is dm/dl If l add be the additional length, then total change of slope is (dm/dl)l add.if l ac be the actual length, then, l ac = l ar +l add. Since, the total change of slope is equal to m, m-(dm/dl).l add =0 (1) From (1), l add = m.(dl/dm) l add = f(m, l) The equation of the curve (Figure 4(b)) is, y=ax 2 +bx+c (2) At point (m, l), l= am 2 +bm+c and dl/dm=2am+b. So, f(m,l)= m(2am+b), where a and b are constant. Now, actual length, l ac = l ar + f(m, l) = l ar + m(2am+b) Same calculation is done for width and height. Equation (2) is derived from the l-m curve (Figure 4(a)). Table 1 shows some sample data collected from practical experiments. Images have been taken for various θ. Then l and m is calculated (Table 1) and l-m curve is plotted (Figure 4(a)). By using polynomial regression and Gauss Elimination technique [7], the equation y=ax 2 +bx+c has been derived, where a and b are the co-efficient of x 2 and x, c is the constant. The value of a, b and c are approximately same for the data of different images. Algorithm: Actual_size_calculation (ar_length, ar_width, ar_height, m) // ar_length, ar_width, ar_height are the arbitrary length, width and height. Compute ac_length := ar_length+ m (2 a m+b), //actual length ac_width :=ar_width + m*(2*a*m+b), //actual width ac_height:=ar_height+ m*(2*a*m+b), //actual height end Figure 5. Construction of 3D models 4.3 Construction of the 3D object After getting the actual length, height and width, rectangular parallelopiped is constructed by its eight corner points. If one corner point is on the origin of the World Co-ordinate System (WCS) and x, y, z be the origin, then other seven points are (x+l, y, z) (x+l, y, z+h) (x, y, z+h) (x, y+w, z) (x+l, y+w, z) (x+l, y+w, z+h) (x, y+w, z+h), where l, w and h are length, width and height. Now, the 3D object consisting eight points can be projected on a 2D view plane and various types of 3D transformation from any center of projection and view plane can be applied to this object by synchronization between center of projection and view plane for 3D graphics modeling [8]. 5. CONCLUSION This algorithm works for rectangular parallelopiped shape objects. Using this algorithm, model of any regular shaped cubic object can also be built. While reading the image, the algorithm stores the real color values of the object, so the surface of the object can be filled using existing surface filling algorithm [9]. As a result the model becomes more realistic. Once a model is constructed, it is possible to apply any geometric transformation [10] on the object. It is also possible to work with irregular cubic shape objects by making some little change in the algorithm. REFERENCES [1] Ch.Schiitz and H.Hiigli, Recognition of 3D objects with a closest point matching Algorithm, in ISPRS, Inter Commission Workshop From Pixels to Sequences, Vol. 30, Zurich, March 1995.

5 [2] D. Leandro Borges, 3D Recognition by Parts: A Complete Solution using Parameterized Volumetric Models, IX SIBGRAPI, pp , [3] R.Lengagne, J.P. Tarel, O. Monga, From 2D images to 3D Face Geometry, IEEE Second International Conference on Automatic Face and Gesture Recognitoin (FG 96), Killington, USA, October [4] R.Mani Maran, Leen pao Meng, Ong Kok Loon, Edge Detection in a homogeneous Background with the aid of sinusoidally-coded structured lighting, in ROVPIA, vol. 1, pp , Malaysia, July [5] Steve Rimmer. Supercharged Bit mapped Graphics, 1 st ed., Windcrest Books, 1992, pp [6] Rafael C. Gonzalez, Digital Image Processing, 5 th ed., Addison-Wesley, 2001, pp [7] Steven C.Chapra, Raymond P.Canale. Numerical Methods for Engineers, 3 rd ed., newdellhi, McGraw- Hill Book Company, [8] Md. Aminul Islam, S.M. Rafizul Haque, Synchronization Between Center of Projection and View Plane For 3D Graphics Modeling, ROVPIA, Vol. 2, Malaysia, July 16-18,1999, pp [9] Alan Watt and Mark Watt, Advanced Animation and Rendering Techniques, 2 nd ed., New York, Addison- Wesley, 1992, pp [10] Z.Xiang, R.Plastock. Theory and problems of Computer Graphics, 2 nd ed., Schaums s Outline, 2001, pp