Univerität Augburg à ÊÇÅÍÆ ËÀǼ Approximating Optimal Viual Senor Placement E. Hörter, R. Lienhart Report 2006-01 Januar 2006 Intitut für Informatik D-86135 Augburg
Copyright c E. Hörter, R. Lienhart Intitut für Informatik Univerität Augburg D 86135 Augburg, Germany http://www.informatik.uni-augburg.de all right reerved
APPROXIMATING OPTIMAL VISUAL SENSOR PLACEMENT E. Hörter Multimedia Computing Lab Univerity of Augburg Augburg, Germany R. Lienhart Multimedia Computing Lab Univerity of Augburg Augburg, Germany ABSTRACT Many novel multimedia application ue viual enor array. In thi paper we addre the problem of optimally placing multiple viual enor in a given pace. Our linear programming approach determine the minimum number of camera needed to cover the pace completely at a given ampling frequency. Simultaneouly it determine the optimal poition and poe of the viual enor. We alo how how to account for viual enor with different propertie and cot if more than one kind are available, and report performance reult. 1. INTRODUCTION Viual enor array are ued in many novel multimedia application uch a video urveillance, ening room, or mart conference room. An important iue in deigning enor array i the appropriate placement of the viual enor uch that they achieve a predefined goal. Our goal it to get complete coverage of a given pace at a predefined ampling rate guaranteeing that an object in the pace will be imaged at a minimum reolution (ee Section 2 for a precie definition). Currently deigner of multi-camera ytem place camera by hand a there exit no theoretical reearch on planning viual enor placement. A video enor array are getting larger, efficient camera placement trategie need to be developed. Often everal different type of camera are available. They differ in the range of view, intrinic parameter, image enor reolution, optic, and cot. Minimizing the cot of a viual enor array while maintaining the required reolution (i.e., minimal ampling frequency ) i alo an important iue we conider. Fig. 1 how one ineffective etup that not even achieve coverage. Although ignificant amount of reearch exit in deigning and calibrating video enor array, automated viual enor placement in general ha not been addreed. There i ome work in the area of grid coverage problem with enor ening event that occur within a ditance r (the ening range of the enor) [1, 2, 3]. Our work i baed on thoe approache, but differ in the enor model, ince camera do not poe circular ening range. Fig. 1. Example of an inefficient etup we deire to optimize The enor placement problem i alo cloely related to the guard placement problem (AGP) the problem of determining the minimum number of guard required to cover the interior of an art gallery. It i addreed by the art gallery theorem [4]. Our camera placement problem differ from the AGP in two way: (1)in the retriction of the field-of-view of camera in our enor model due to reolution and enor propertie; (2) in conidering camera with different fieldof-view at different level of cot. In AGP all guard are aumed to have imilar capabilitie. The paper i organized a follow. In Section 2 the problem i formulated. Our approach i preented in Section 3, before Section 4 give implementation detail and report reult. Section 5 conclude the paper. 2. PROBLEM STATEMENT Definition: In the following the term pace denote a phyical 2D or 3D room, which we want to cover by our viual enor. Currently, we aume that the room i rectangular. Coverage mean that every point of a given pace i ened with a pecified minimal reolution. In thi work, the minimal reolution i atified if a given point in pace i imaged by at leat one pixel of a camera that doe not aggregate more than x cm 2 of a urface parallel to the imaging plane through that point. x i expreed in term of the ampling frequency f and converted into the field-of-view of a camera. The fieldof-view i defined a a the area in which a pixel aggregate no 1 more that f cm 2 of a urface parallel to the imaging plane. Thu an object that appear in the camera field-of-view i imaged with at leat thi reolution auming the object ha
a planar urface orthogonal to the optical axi 1. Occluion are not conidered. To implify the derivation we conider only the 2D problem in thi paper; however, the preented approach can be extended eaily to the third dimenion. Problem Statement: Given a pace to be covered at a ampling frequency f by viual enor, we are intereted in the following two camera placement problem: Determine the minimum number of viual enor of a certain type a well a their poition and poe in the pace uch that coverage i achieved. Given different type of viual enor determine how to obtain coverage while minimizing the total cot of the enor array. 2.1. Modeling a camera field-of-view The field-of-view of a camera can be decribed by a triangle a hown in Fig. 2. The parameter of thi triangle can be eaily calculated given the (intrinic) camera parameter and the ampling frequency f uing well known geometric relation. Defining the field-of-view by a triangle enable u to decribe Fig. 2. Deriving the model of a camera field-of-view the area covered by a camera at poition (c x, c y ) and poe (ϕ) linearly. Therefore a camera field-of-view i firt tranlated to the origin of the coordinate ytem (Fig. 2 left): x = x c x, y = y c y (1) Then we rotate the field-of-view, o that the optical axi become parallel to the x-axi (Fig. 2 center): x = co(ϕ) x + in(ϕ) y (2) y = in(ϕ) x + co(ϕ) y (3) The reulting area covered by the triangle (Fig. 2 right) can now be decribed by three line equation l 1, l 2, l 3 : l2 : l1 : x d (4) y a 2d x (5) l3 : y a 2d x (6) 1 Clearly the reolution i maller if the urface i not orthogonal. Thu by ubtitution the following three equation define the area covered by the field-of-view of a certain camera: co(ϕ) (x c x ) + in(ϕ) (y c y ) d (7) in(ϕ) (x c x ) + co(ϕ) (y c y ) a 2d (co(ϕ) (x c x) + in(ϕ) (y c y )) (8) in(ϕ) (x c x ) + co(ϕ) (y c y ) a 2d (co(ϕ) (x c x) + in(ϕ) (y c y )) (9) 2.2. Modeling pace In the ideal cae camera can be placed continuouly in the pace, i.e. the variable c x, c y and ϕ that define a camera poition and poe are continuou variable. A we are not able to olve our problem for the continuou cae we approximate the pace by a two-dimenional grid of point. The minimum ditance between two grid point in the x- and y- direction i determined by the patial ampling frequency f a : = 1/f a. Camera can only be placed at thee dicrete grid point, and coverage i enured only for thee grid point. Thu our problem turn into a grid coverage problem. For f a our approximated olution converge to the continuou-cae olution. We only conider rectangular pace of width w and height h with no obtacle in the room contricting the field-of-view of our viual enor. Remark about more complex pace are made in Section 3.1. 3. LINEAR PROGRAMMING Conidering only one type of camera, i.e., only camera with the ame field-of-view, we formulate our camera placement problem in term of minimizing the number of camera needed ubject to the coverage contraint. We aume that our pace conit of x and y grid point in the x and y dimenion repectively 2. Viual enor location are retricted to thee grid point. Similarily we dicretize the angle ϕ defining a camera poe to ϕ different poe only. A camera at poition (c x, c y ) with orientation ϕ cover a grid point (x, y) if and only if Eq. 7 to 9 are atified. Thu, we can tate the optimization problem a follow: Problem 1: Given a et of grid point and a camera model, minimize the total number of camera (by optimally aigning camera to grid point and angel) while enuring that every grid point i covered by at leat one camera. In the following we derive an integer programming (ILP) model 2 Given a rectangular pace x and y can be eaily calculated given the room dimenion and the patial ampling rate f a.
to olve thi viual enor placement problem. Our approach i baed on the algorithm preented in [2]. Let a binary variable x ijϕ be defined by: 1 if a camera i placed at grid point (i, j) x ijϕ = with orientation ϕ 0 otherwie (10) The total number of camera N i then given by N = x ijϕ (11) Furthermore we define a binary variable c(i1, j1, ϕ1, i2, j2): c(i1, j1, ϕ1, i2, j2) = 1 if a camera placed at grid point (i1, j1) with orientation ϕ1 cover grid point (i2, j2) 0 otherwie (12) c(i1, j1, ϕ1, i2, j2) can be calculated in advance and tored in a table. Our enor deployment problem can now be formulated a an IPL model: ubject to ϕ1=0 i1=0 j1=0 min x ijϕ (13) x i1,j1,ϕ1 c(i1, j1, ϕ1, i2, j2) 1 (14) 0 i2 ( x 1), 0 j2 ( y 1) The contraint repreented by Eq. 14 enure that each grid point i covered by at leat one camera. The minimum number of camera that hould cover each grid point can be eaily changed to e.g. two or more camera. To enure that only one camera i located at each grid point, we can add the contraint: x ijϕ 1 (15) 0 i ( x 1), 0 j ( y 1) The number of variable x ijϕ in our ILP i x y ϕ. Thu, if we increae the number of grid point to achieve a better approximation of the continuou cae, the number of variable and contraint in our IPL increae accordingly. 3.1. Different type of camera A very imilar problem arie if everal type of camera with different enor reolution and optic (i.e. focal length) are available. Then we have for each type of camera k different field-of-view parameter d k and a k (ee Fig. 2) and a cot K k. Our objective i to find the configuration of camera that minimize the total cot of the viual enor while enuring coverage. The optimization problem i formulated a follow: Problem 2: Given a et of grid point and k type of viual enor with cot K k and field-of-view parameter d k, a k minimize the total cot of the enor array (by optimally aigning camera to grid point and angle) while enuring coverage for every grid point. To olve thi problem we need to modify the previou olution. We define a binary variable x ijϕk a follow: x ijϕk = 1 if a camera of type k i placed at grid point (i, j) with orientation ϕ 0 otherwie The total cot C of the viual enor i then calculated by: C = k K k ( k=1 ϕ 1 (16) x ijϕk ) (17) Similar to Eq. 12 a binary variable c k (i1, j1, ϕ1, i2, j2) i defined for each camera type k by: 1 if a camera of type k with orientation ϕ1 i placed at c k (i1, j1, ϕ1, i2, j)) = grid point (i1, j1) and cover grid point (i2, j2) 0 otherwie (18) Again the value of all variable c k (i1, j1, ϕ1, i2, j2) can be calculated in advance. Like in Problem 1 the olution to our viual enor placement problem i found by olving the following ILP model: Minimize the cot function ubject to k C = k K k ( k=1 ϕ 1 x i1,j1,ϕ1,k1 k1=1 ϕ1=0 i1=0 j1=0 x ijϕk ) (19) c k (i1, j1, ϕ1, k1, i2, j2) 1 (20) 0 i2 ( x 1), 0 j2 ( y 1) To enure that at each grid point only one camera i located we need to add the contraint: k x ijϕk 1 (21) k=1 0 i ( x 1), 0 j ( y 1)
Fig. 3. Reult of the ILP minimization for identical camera of two ample configuratation (a) and (b) Fig. 4. Reult of minimizing the total cot of the enor array for two different parameter etting The number of variable x ijϕk in our ILP i k x y ϕ, i.e. the number of variable increae with the number of grid point and available camera type a do the contraint. More complex pace: We have only conidered rectangular pace o far. More complex room can be eaily included a linear contraint into our framework a long a they are convex. Additionally, the ILP extenion to the third dimenion i traightforward, but ha been excluded in the dicuion due to pace limitation. 4. EXPERIMENTAL RESULTS The ILP have been implemented in C++ uing the lpolve package [5]. Firt we evaluated our propoed olution to Problem 1: Given a ingle camera type minimize the total number of camera ubject to the coverage contraint. Fig. 4 how two reult with two ample configuration. Blue dot mark camera location; red line mark the field-of-view of the camera. The ampling frequency f and the camera type were et equally in both experiment, the other parameter were et a follow: Experiment (a): room: 120 120, f a = 1 10, ϕ = 2 Experiment (b): room: 160 120, f a = 1 20, ϕ = 16 It can be oberved in Fig. 4(b) that ome area are uncovered. Thi i due to the low patial ampling frequency f a. Coverage i only aured for dicrete grid point and the ditance between two grid point depend on f a. To enure near complete coverage, f a need to be increaed. We alo evaluated the cot minimization ILP for two camera type with different field-of-view. A cot of $70 wa aumed for the camera with the larger field-of-view, while only $50 were aumed for the camera with the maller fieldof-view. Fig. 4 how two reult. Blue dot denote camera location, while blue and green line mark the field-of-view of the two different camera type. The ize of the room, the ampling and patial ampling frequency f and f a were contant for both experiment. In the firt and econd experiment we choe ϕ = 2 (Fig. 4(a)) and ϕ = 8 (Fig. 4(a)), repectively. The total cot of the enor array in Fig. 4(a) wa $440 and $420 in Fig. 4(b). The above preented ILP problem are practically olvable for only a mall number of grid point. For a large number Chakrabarty et al. [2] propoe a divide-and-conquer algorithm. 5. CONCLUSION AND FUTURE WORK We have preented an approach to optimally place viual enor in a given pace uch that coverage i achieved for dicrete grid point. We have propoed two different algorithm. The firt one minimize the number of camera covering a ening field at a given ampling frequency. The econd algorithm minimize the total cot of the enor array while covering a pace with enor of different propertie. Future work will include a lack variable for coverage, i.e., only a predefined percentage of grid point need to be covered. Additionally it ha to be invetigated how to handle a large number of grid point. 6. REFERENCES [1] S. Sahni and X. Xu, Algorithm for wirele enor network, Intl. Journal on Ditributed Senor Network, vol. 1(1), pp. 35 56, 2005. [2] H. Qi K. Chakrabarty, S. S. Iyengar and E. Cho, Grid coverage for urveillance and target location in ditributed enor network, IEEE Tranaction on Computer, vol. 51(12), pp. 1448 1453, 2002. [3] J. Wang and N. Zhong, Efficient point coverage in wirele enor network, Journal of Combinatorial Optimization, to appear. [4] J. O Rourke, Art Gallery Theorem and Algorithm, Oxford Univerity Pre, New York, 1987. [5] P. Notebaert M. Berkelaar and K. Eikland, lpolve: Open ouce (mixed-integer) linear programming ytem, Eindhoven U. of Technology, http://group.yahoo.com/group/lp olve/file/verion5.5/.