A Mathematical model for the determination of distance of an object in a 2D image

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1 A Mathematical model for the determination of distance of an object in a 2D image Deepu R 1, Murali S 2,Vikram Raju 3 Maharaja Institute of Technology Mysore, Karnataka, India rdeepusingh@mitmysore.in 1, murali@mitmysore.in 2, vikramraju17@gmail.com 3 Abstract-Distance measurement in real world has always been one of the challenging tasks to be performed in the field of computer vision. Photographic images are two dimensional depiction of three-dimensional real space. Various methods have been developed, researched and improvised over the years depending on the focus of the object, defocus of the background, the vanishing point of an image and other information derived from images. The impact of various features of the camera are also considered when the above problem is under study like, focal length of the camera, ISO levels, white balance, resolution etc. The effect of the resolution of an object in an image, its actual size, the focal length of the camera, the camera resolution, all contribute in determining the distance between the camera and the object. In this paper we have proposed a model for the relationship between the resolution of an object of interest and the distance from the camera as a growth series. Relationship between the resolution of an object of interest and focal length is also expressed as a growth series model. The object of interest is segmented out and its pixels account to its resolution, which is standardized and used to determine either the distance or the focal length of the camera or the vice versa. This model, thus, very easily helps in tracking the object using a camera with minimal error rate, provided the image is along the optical axis of the camera. Keywords: Resolution, Focal Length, Camera and object distance, growth series, Pixels per inch, Geometric Progression. 1 Introduction The knowledge about the distance of an object in an image has its applications in many fields of science and technology and research, like, human motion estimation, web conferencing, gaming industry, surveillance, security systems, robotics, medical systems and imaging. Etc. The major factors in a photo that cannot be changed once the photo has been clicked are Aperture of the lens, focal length, number of pixels or resolution and shutter speed. A method which allows refocusing is a potential powerful tool for digital image editing. Once the depth map has been obtained, one can de-blur the image in order to acquire an all focus image or blur the image even more to create certain visual effects [1]. The depth map can also apply to the task such as automatic scene segmentation, post exposure refocusing and rerendering of the scene from an alternative view point. By analyzing the depth of field, the coarse depth map of scene can be recovered [2]. Depth from defocus and depth from focus are the two methods to estimate the 3D geometry of the scene by exploiting image focus [3]. In adaptive depth from focus, depth estimation method for images in narrow depth of field setting, the segments produced by mean shift segmentation as windows to analyze the focus measure and employ a hierarchical Markov Random Field to infer from the depth map [4]. Ina non-linear approach, the precise sparse depth map extraction, the noisy focus measurements are replaced with estimated values, which in turn helps to accurately compute 3D shapes while preserving edges of objects [5].In paper [6],authors have proposed a real-time method which can measure distance using a modified camera. The camera s image sensor is inclined by a certain angle. Thus, the image projected on the sensor is defocused differently at different areas. The area where the image is focused best is just at projection plane. The position of the projection plane can be obtained after finding out this area. The distance between the projection plane and the lens of the camera is image distance. Object distance can be obtained by applying image distance to lens formula. Although the results of the above methods are very promising, it is computationally expensive to employ Markov Random Fields or any of the procedures given in above papers. In real world applications it is necessary to provide real-time computations. By studying the model or rate at which features change in an image like size, distance, pixels of an object, focal length, give us a better approximation and a simpler computationally efficient method to determine the relationship among them.

The resolution of an image or the number of pixels of an image is a constant, which depends on the camera. Once an image is captured, its focal length is also constant.

2 The resolution of an image or the number of pixels of an image is a constant, which depends on the camera. Once an image is captured, its focal length is also constant. When an image is captured by a camera, the distance of the image from the camera or the viewer, thus can be determined using the above variables. The resolution also directly depends on the focal length, since it has a proportional relationship with the image size. The camera, like the human eye, works on perspective vision, where, the change in image size with variation of Focal Length and Distance is not linear. To study the relationship among the above, the following methodology is proposed. 2 Method With an objective of designing a robust model, sequence of images at different distances D from the object and for different focal lengths are taken along the optical axis. Every image is of size ix j and it is acquired for objects of various shapes and sizes. The total number of pixels P, in the image is thus given by = (1) The set of the pixels in an object P o,is a subset of P. The image is converted to grey scale using a basic thresholding algorithm to obtain the object of interest and the number of pixels in the object is obtained. This becomes the set of pixels in an object, and stored in a set P o. The behavior of the relationship between the distance with respect to the resolution factor R f and that of the focal length and the resolution factor is sought. Figure 1: Experimental setup varying distance from camera to objects. Once the images are captured, the background is segmented, to get only the pixels of the object of interest. Image preprocessing is necessary at times, when noise is found in the images, when the automatic thresholding algorithm fails to extract the object clearly. Manual image cleaning was used to remove unwanted noise in the image. This can be avoided using a better efficient object extraction algorithm The image segmentation and cleaning was carried out using MATLAB. To build the mathematical model, IBM SPSS was used. To determine the resolution factor, first, the Resolution per Inch, R In, of the object is determined. It is then multiplied by the resolution of the camera and later natural log is taken to scale down the data. This is so done in order to standardize the factor arising due to the resolution of the images and of the objects in them, thus making the method applicable to any kind of camera and of any resolution. In a way, it makes the method independent of the resolution itself. = ln ( ) (2) Figure1 shows the variation of size of the image with the changes in distance from the camera. Figure 2: Test Images (left) and Objects extracted from the image (right)

3 Results The results can be analyzed in two folds, one model with resolution factor and distance of object in an image and the other with resolution factor and the focal length of the camera.

Table 1 gives the difference between expected distance and obtained distance for circle object where as Table2 and Tables3 specify the same for square and triangle objects.

in Table 4: Model Summary and Parameter Estimates for Distance and Resolution Factor for a focal length of 18mm.

9% significance to the model, which is a good fit for the data set observed. Table 2:Observed results square object of area 41.69 sq.in Table 3: Observed results triangle object of area 20 sq.

3 3 Results The results can be analyzed in two folds, one model with resolution factor and distance of object in an image and the other with resolution factor and the focal length of the camera. Following results were tabulated for 3 different objects. Table 1 gives the difference between expected distance and obtained distance for circle object where as Table2 and Tables3 specify the same for square and triangle objects. We can also note that there is a relationship between pixels in an object to distance which is expressed by our model. Table 1:Observed results for circle object of area sq.in Table 4: Model Summary and Parameter Estimates for Distance and Resolution Factor for a focal length of 18mm. Equation Dependent Variable: Distance R Square Model Summary Parameter Estimates F df1 df2 Sig. Constant b1 Growth The R Square value shows a 99.9% significance to the model, which is a good fit for the data set observed. Table 2:Observed results square object of area sq.in Table 3: Observed results triangle object of area 20 sq.in Figure 3: The growth series of distance and resolution factor of object in image The model agrees completely with the growth series and gives us the following equation to determine the distance of the object from the camera: = (3) Part I: Resolution Factor and Distance The distance is considered as the dependent variable in the analysis of the model and the resolution factor is taken as the independent variable. In each case of the focal length the model is tested. Where, D is the distance of the object from the camera, A and B are constants in the growth series, which are determined by a geometric series, and R f is the resolution factor. The r in the geometric ratio is 1.01.The geometric series gives the values of the constants from the above equation.

Table 5: The geometric series to determine the constants of the growth series model. Focal Length A B 18mm 15.727-0.485 24mm 15.727r -0.485r 30mm 15.727r 2-0.485r 2 36mm 15.727r 3-0.485r 3 42mm 15.

4 Table 5: The geometric series to determine the constants of the growth series model. Focal Length A B 18mm mm r r 30mm r r 2 36mm r r 3 42mm r r 4 48mm r r 5 54mm r r 6 60mm r r 7 66mm r r 8 The geometric series gives the values of the constants from the above equation. The model was tested on random image samples, which agreed with the growth series model, with very minimal error. The error in determining the distance was in the range of percent. Part II: Resolution Factor and Focal Length The focal length is considered the dependent variable in the analysis of the model and the resolution factor is taken as the independent variable. In each case of the distance of the object from the camera, the model is tested. Table 6: Model Summary and Parameter Estimates for Distance and Resolution Factor at 51 inches from the camera. Equation Dependent Variable: Focal Length R Square Model Summary Parameter Estimates F df1 df2 Sig. Constant b1 Growth The R Square value shows a 99.1% significance to the model, which is a good fit for the data set observed. The model agrees completely with the growth series and gives us the following equation to determine the focal length: = (4) Where, F is the focal length of the camera, A and B are constants in the growth series, which are determined by a geometric series, and R f is the resolution factor. The r in the geometric ratio is Figure 4: The growth series of focal length and resolution factor of object in an image The model was tested on random image samples, which agreed with the growth series model, with very minimal error. The error in determining the distance was in the range of 1-2 percent. Table 7: The geometric series to determine the constants of the growth series model. Distance A B 51in in r 0.543r 71in r r 2 81in r r 3 91in r r 4 101in r r 5 111in r r 6 121in r r 7 131in r r 8 141in r r 9 4 Applications The model finds its use in many applications like obstacle detection in robotics, detection of road humps, navigation systems for the blind, unknown territory exploration, 3D game development are a few to name.

5 5 Future Work The distance of the object from the camera and focal length model is applicable only when the object is head-on or in the line of the optical axis of the camera. Future work can be done on studying the effect of resolution and distance for images not in the axis of the camera, considering the evolution of perspective vision of an image. It can also be used to create 3D scenes by determining the distance from the camera and rebuilding another scene with the same distance. 6 Conclusions The model shows promising enhancement on further development and an efficient model to determine the distance of an object in an image from the camera making use of the data in the resolution of an image, which cannot change once an image is taken. Building the model for varied focal lengths enables the model to work with cameras of any focal length. The obtained growth series enables to work with distance and focal length. With knowing one of the two, the other can be calculated and be easily modeled in the system. Computationally the calculation of the growth series works efficiently than the iterative approach. 7 Acknowledgements The work presented in this paper was supported by VTU research grants (Ref. No. VTU/ Aca./ /A-9/13551). We thank Visvesvaraya Technological University, Belgaum for this. 8 References [1] Potmesil, M., and Chakravarty, I., A lens and aperture camera model for synthetic image generation.in proc. Siggraph, [2] Pentland, A. P., A new sense for depth of field.ieee Trans. Pattern anal. Mach. Intell. 9, 4, [3] Hasinoff S. W., Kutulakos K. N., A layerbased restoration framework for variableaperture photography, F, 2007 [C], IEEE. [4] Bing-Zhong Jing, Daniel S. Yeung, Recovering depth from images using adaptive depth from focus. IEEE /12, [5] Muhammad Tariq Mahmood, Ikhyun Lee, Wook-Jin Choi, Tae-Sun Choi, A non-linear approach for depth from focus for digital cameras. IEEE /11, [6] Liu Xiaoming, Qin Tian, Chen Wanchen, Yin Xingliang, Real Time Distance Measurment Using a Modified Camera.IEEE /09.

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