Tutorial of Motion Estimation Based on Horn-Schunk Optical Flow Algorithm in MATLAB
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1 AU J.T. 1(1): 8-16 (Jul. 011) Tutorial of Motion Estimation Based on Horn-Schun Optical Flow Algorithm in MATLAB Darun Kesrarat 1 and Vorapoj Patanavijit 1 Department of Information Technolog, Facult of Science and Technolog Department of Computer and Networ Engineering, Facult of Engineering Assumption Universit, Bango, Thailand <darun@scitech.au.edu; patanavijit@ahoo.com> Abstract The Horn-Schun algorithm (HS) is one of the classical algorithms in optical flow due to its reasonable performance and simplicit of the algorithm. This article presents in detail the process of the HS algorithm and its step b step coding in MATLAB. The HS algorithm is a technique used to identif the image velocit or motion vector based on Spatial Temporal Gradient Technique which computes the image velocit from spatiotemporal derivatives of image intensit. Firstl, the original estimated intensit for gradient constraint on the image sequence is obtained b using the gradient constraint with a global smoothness. Then, iterative equations are solved to minimize the sum of the errors for the rate of change of image gradient intensit and obtain the image velocit. Kewords: Spatial Temporal Gradient Technique, gradient intensit, motion vector, image velocit. Introduction Optical flow is a technique used for man particular fields, such as motion estimation to predict the motion vector of a moving object, video compression and reconstruction for reducing temporal redundanc present in frame sequences and allowing a better compression of video material, and image segmentation for tracing a moving object. In conventional predictive methods for motion estimation, the difference between the current frame and the predicted frame, based on a previous frame (motion vector, or MV), is coded and transmitted; then it is used to reconstruct a higher resolution still image or video sequence from a sequence of low resolution images in achieving super-resolution (Kesrarat and Patanavijit 011). This article concentrates on a classical optical flow algorithm, namel the Horn-Schunc algorithm (HS) proposed b Horn and Schunc (1981). The HS algorithms are the most popular differential algorithms which have been applied for man applications and have been referenced for man performance evaluation models. Barron, Fleet and Beauchemin (BFB) adjusted the ernel model for performance evaluation over HS algorithms, but its focus is on the densit of velocit (Barron et al. 1994). This article eplains the motion estimation algorithm based on the Horn-Schun optical flow algorithm (HS), which applies the ernel of BFB, and also represents the algorithm step b step in MATLAB. Horn-Schun Algorithm (HS) This algorithm is based on a differential technique computed b using a gradient constraint (brightness constanc) with a global smoothness to obtain an estimated velocit field (Horn and Schun 1981). There are two main processes for the implementation of the HS algorithm. The first one is an estimation of partial derivatives, and the second one is a minimization of the sum of the errors b an iterative process to present the final motion vector. Review Article 8
2 AU J.T. 1(1): 8-16 (Jul. 011) Estimation of classical partial derivatives This section presents the estimation process of the classical derivatives of image intensit or brightness from the image sequence. The brightness of each piel is constant along its motion trajector in the image sequence. The relationship in continuous images sequence will be taen into account to estimate the original intensit for a gradient constraint. Let I(,,t) denote the gradient intensit (brightness) of point (,) in the images at time t. In each image sequence, I, I, and I t are computed for each piel (see also Figs. 1-): I = ¼ {I,+1,t - I,,t + I +1,+1,t - I +1,,t + I,+1,t+1 - I,,t+1 + I +1,+1,t+1 - I +1,,t+1 }, I = ¼{I +1,,t - I,,t +I +1,+1,t - I,+1,t +I +1,,t+1 - I,,t+1 + I +1,+1,t+1 - I,+1,t+1 }, I t = ¼ {I,,t+1 - I,,t + I +1,,t+1 - I +1,,t + I,+1,t+1 - I,+1,t + I +1,+1,t+1 - I +1,+1,t }. (1) is the core functional of the HS algorithms. The gradient estimation ernel of the BFB model uses 4-point central differences for differentiation defined as (see also Figs. 3-7): I = 1/1 {-1 I,- + 8 I, I, - 8 I, I,+ }, I = 1/1 {-1 I -, + 8 I -1, + 0 I, - 8 I +1, + 1 I +, }, I t = 1/1 {-1 I,,t- + 8 I,,t I,,t - 8 I,,t I,,t+ }. () It is nown that in most situations of motion estimation one ma use onl two frames to calculate I t as shown in Fig. 6. As a result of using the BFB ernel, it presents more stabilit of original gradient estimation than the result of performance evaluation in Barron et al. (1994). % im1 is gra image sequence at time t % im is gra image sequence at t+1 I = conv(im1,0.* [-1 1; -1 1],'same') + conv(im, 0.*[-1 1; -1 1],'same'); 30,, I = conv(im1, 0.*[-1-1; 1 1], 'same') + conv(im, 0.*[-1-1; 1 1], 'same'); t+1 +1, 0 t +1, It = conv(im1, 0.*ones(),'same') + conv(im, -0.*ones(),'same'); Fig.. MATLAB code for gradient estimation with original partial derivatives. 1/ I = ((10 ) + (18 0) + ( 3 0) + (180 0)) / 4 = I = ((0 ) + (18 10) + (0 30) + (180 )) / 4 = 31. I t = ((30 ) + (0 0) + ( 10) + (180 18)) / 4 = 3.7 Fig. 1. The partial derivatives of image brightness at point (, ). Estimation of partial derivatives on BFB ernel Barron et al. (1994) proposed performance evaluation over man algorithms of optical flow and modification of some of the variant variables. We focus on the ernel of mas coefficient for gradient estimation which Fig. 3. The ernel coefficient of BFB. % im1 is gra image sequence at time t % im is gra image sequence at t+1 I= conv(im1,(1/1)*[ ], 'same'); I= conv(im1,(1/1)*[ ]', 'same'); It = conv(im1, ones(1),'same') + conv(im, -ones(1),'same'); % modif ft because there are onl frames which are used for calculation Fig. 4. MATLAB code for gradient estimation with BFB ernel of -image sequence. Review Article 9
3 AU J.T. 1(1): 8-16 (Jul. 011) Image Image , , / /1 BFB Kernal BFB Kernal 1 I (,) = ( ) / 1 = Fig.. An illustration of gradient calculation for I of BFB ernel coefficient. I (,) = ( ) / 1 = Fig. 6. An illustration of gradient calculation for I of BFB ernel coefficient Frame (t+), Current Frame (t) Frame (t-1) Frame (t-) /1 Frame (t+1) BFB Kernal I t (,) = ( ) / 1 = Fig. 7. An illustration of gradient calculation for I t of BFB ernel coefficient. Review Article 10
4 AU J.T. 1(1): 8-16 (Jul. 011) Minimization In practice, the image intensit or brightness measurement ma be corrupted b quantization or noise. According to the equation for the rate of change of image brightness: I(,,t) = I(+u, +v, t+1), (3) ε = u I + v I + I t = 0, (4) where u and v are the horizontal and vertical motion vectors of optical flow, respectivel, one cannot epect ε to be zero. The problem is to minimize the sum of errors in the equation for the rate of change of image brightness as near as 0. So, the smoothness weight (α) is iterativel presented as (see also Fig. 8): u 1 v 1 u v I I I u I I I u I I v I v I I t I t,, () I I It Set of calculated partial derivative of image brightness Calculate Equation () of u 1 u 1/1 1/6 1/1 1/6-1 1/6 1/1 1/6 1/1 Calculate Equation () of u On Iterative process 3 Average of u initial as zero Calculate neighborhood average of u from weighting average ernal Calculated Result of Process Calculated Result of Process Calculated Result of Process Fig. 8. An illustration of the iterative minimization process for u where steps and 3 are iterative. Review Article 11
5 AU J.T. 1(1): 8-16 (Jul. 011) where u and v denote horizontal and vertical neighborhood averages ( u and v ), which initiall are set to zero and then the weighted average of the value at neighboring points based on the ernel in Fig. 8 is applied for further iterations using Eq. () as illustrated in Figs The smoothness weight (α) plas an important role where the brightness gradient is small, for which the suitable value should be determined. 1/1 1/6 1/1 1/6-1 1/6 1/1 1/6 1/1 Fig. 9. Weighted average ernel at neighboring points. Conclusion According to the characteristic of the HS algorithm, when applied with the BFB ernel it provides simplicit in the algorithm with reasonable performance and better qualit, but the value of the smoothing weight (α) cannot be defined eactl because the suitable value is varing upon different image sequences. As a consequence, the suitable iteration times also cannot be defined for the best outcome, which impacts the processing time for the best motion vector at the output. Figures show the results for the motion vector of the HS algorithm from the partial representation of Foreman, Coastgurad, Aio and Container video sequences, respectivel. The results of the motion vector identif the direction of piel movement from the two different consecutive frames under the smoothing weight α = 10 and iterations of the minimizing process. % Set initial value of u and v to zero u = 0; v = 0; % Weighted Average ernel ernel=[1/1 1/6 1/1;1/6 0 1/6;1/1 1/6 1/1]; %Minimizing Iterations ( times) for i=1: %Compute local averages of the vectors uavg=conv(u,ernel,'same'); vavg=conv(v,ernel,'same'); %Compute flow vectors constrained b the local averages and the optical flow constraints, where alpha is the smoothing weight u = uavg - ( I.* ( ( I.* uavg ) + ( I.* vavg ) + It ) )./ ( alpha^ + I.^ + I.^); v = vavg - ( I.* ( ( I.* uavg ) + ( I.* vavg ) + It ) )./ ( alpha^ + I.^ + I.^); References Barron, J.L.; Fleet, D.J.; and Beauchemin, S.S Performance of optical flow techniques. International Journal of Computer Vision 1(1): Horn, B.K.P.; and Schunc, B.G Determining optical flow. Artificial Intelligence 17(1-3): Kesrarat, D.; and Patanavijit, V Performance analsis on weighting factor (α) on spatial temporal gradient technique and high confidence reliabilit with sub-piel displacement. Proc. 8 th International Conference on Electrical Engineering/ Electronics, Computer, Telecommunications and Information Technolog (ECTI-CON 011), Khon Kaen, Thailand, Ma 011, pp. 1, end Fig. 10. MATLAB code of the iterative minimization process. Review Article 1
6 AU J.T. 1(1): 8-16 (Jul. 011) Fig. 11. Eamples of the motion vector of the HS algorithm from a part of FOREMAN sequence. Review Article 13
7 AU J.T. 1(1): 8-16 (Jul. 011) Fig. 1. Eamples of the motion vector of the HS algorithm from a part of COASTGUARD sequence. Review Article 14
8 AU J.T. 1(1): 8-16 (Jul. 011) Fig. 13. Eamples of the motion vector of the HS algorithm from a part of AKIYO sequence. Review Article 1
9 AU J.T. 1(1): 8-16 (Jul. 011) Fig. 14. Eamples of the motion vector of the HS algorithm from a part of CONTAINER sequence. Review Article 16
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