Kalman-Filter-Object-Tracking-master.zip_lungsk5w_matlab_trackin

#Object Tracking Using Kalman Filter ##Shahin Khobahi ###I. Introduction In this project, we are proposing an adaptive filter approach to track a moving object in a video. Currently, object tracking is an important issue in many applications such as video survelance, traffic management, video indexing, machine learning, artificial intelligence and many other related fields. As we will discuss in the following sections, moving object tracking can be interpreted as an estimation problem. Kalman filter is a powerful algorithm that can be used in the state estimation problems and that is the reason we used this method to estimate and predict the position of a moving object. In the first stage of this project we use the background subtraction method to detect the moving object in the video and then we use the Kalman filter ro predict and estimate the next state of the object. <a href="" id="x1-3r2"></a> ###II. Problem Formulation <a href="" id="x1-4r1"></a> ####A. Object Detection Using Background Subtraction A video is composed of a series of frames each which can be considered as a 2D signal. So, a video can be seen as a two-dimensional (2D) signal through time. Moreover, there are two types of objects in a video: <span class="ptmri7t-">steady and moving objects</span>. Steady objects do not change from frame to frame and can be considered as the background scene. The goal is to detect a moving objects from the steady ones. First let us provide an example: consider that you are looking at a wall and suddenly a birdy fly over this wall. The steady object in this scene is the wall and the moving object is the birds. The bird is in fact disturbing your observation of the wall(background), so in the context of signal processing, this bird(moving object) can be seen as a noise to that background. In other words, in a video, a moving object is like a noise to the background scene which is a fixed signal, and this moving object is adding noise to our observation of that background. Consequently, each frame of the video can be interpreted as a noisy observation of the background. Therefore, the problem is just simply the noise detection in a signal. The following model can be used for our problem: <table> <colgroup> <col width="50%" /> <col width="50%" /> </colgroup> <tbody> <tr class="odd"> <td><a href="" id="x1-5r1"></a> <img src="./Images/document0x.png" alt="y = x + v " class="math-display" /></td> <td>(1)</td> </tr> </tbody> </table> Where <span class="cmmi-10">y </span>is our noisy measurement of <span class="cmmi-10">x </span>(background signal), and <span class="cmmi-10">v </span>denotes the disturbance which in fact is our moving object disturbing that background signal <span class="cmmi-10">x</span>. As it was mentioned earlier, we need to extract the noise <span class="cmmi-10">v </span>from our noisy signal <span class="cmmi-10">y </span>(the video). Each frame of the video is a noisy realization of the signal <span class="cmmi-10">y </span>and we refer to the i-th frame of the video as <span class="cmmi-10">u</span>_{<span class="cmmi-7">i</span>}. Further we assume that the video has <span class="cmmi-10">N </span>frames. Our approach of extracting the noise from the observations <span class="cmmi-10">u</span>_{<span class="cmmi-7">i</span>} is to first obtain an <span class="ptmri7t-">estimation </span>of the background signal <img src="./Images/document1x.png" alt="ˆx" class="circ" />, then we subtract each observation <span class="cmmi-10">u</span>_{<span class="cmmi-7">i</span>} from the estimated signal <img src="./Images/document2x.png" alt="ˆx" class="circ" /> to obtain an estimation of the noise at each frame: <table> <colgroup> <col width="50%" /> <col width="50%" /> </colgroup> <tbody> <tr class="odd"> <td><a href="" id="x1-6r2"></a> <img src="./Images/document3x.png" alt="ˆvi = ui − xˆ " class="math-display" /></td> <td>(2)</td> </tr> </tbody> </table> Given two deterministic random variables <span class="cmsy-10">{</span><span class="cmbx-10">x</span><span class="cmmi-10">,</span><span class="cmbx-10">y</span><span class="cmsy-10">}</span>, we define the least-mean-squares estimator (l.m.s.e) of <span class="cmmi-10">x </span>given <span class="cmmi-10">y </span>as the conditional expectation of <span class="cmbx-10">x </span>given <span class="cmbx-10">y</span>: <table> <colgroup> <col width="50%" /> <col width="50%" /> </colgroup> <tbody> <tr class="odd"> <td><a href="" id="x1-7r3"></a> <img src="./Images/document4x.png" alt="xˆ= E (x |y) = E[x|u0,u2,...,uN −1] " class="math-display" /></td> <td>(3)</td> </tr> </tbody> </table> For simplicity, we assume that we model <span class="cmmi-10">x </span>as an unknown constant instead of a random variable. Further we assume that we are given <span class="cmmi-10">N </span>frames of the video and that is all the information we have. Now, we model our problem as follows: <table> <colgroup> <col width="50%" /> <col width="50%" /> </colgroup> <tbody> <tr class="odd"> <td><a href="" id="x1-8r4"></a> <img src="./Images/document5x.png" alt="y(i) = x + v(i), i = 0,1,...,N − 1 " class="math-display" /></td> <td>(4)</td> </tr> </tbody> </table> and we define the column vector <span class="dsrom-10">𝟙</span> <span class="cmr-10">= \[1</span><span class="cmmi-10">,</span><span class="cmr-10">1</span><span class="cmmi-10">,</span><span class="cmmi-10">…</span><span class="cmmi-10">,</span><span class="cmr-10">1\]</span>^{<span class="cmmi-7">T</span>}. Then, <table> <colgroup> <col width="50%" /> <col width="50%" /> </colgroup> <tbody> <tr class="odd"> <td><a href="" id="x1-9r5"></a> <img src="./Images/document6x.png" alt="y = 𝟙x +v " class="math-display" /></td> <td>(5)</td> </tr> </tbody> </table> if this is the case, according to Gauss-Markov theorem, the optimal linear estimator (m.v.u.e) of <span class="cmmi-10">x </span>is: <table> <colgroup> <col width="50%" /> <col width="50%" /> </colgroup> <tbody> <tr class="odd"> <td><a href="" id="x1-10r6"></a> <img src="./Images/document7x.png" alt=" 1 N∑−1 1 N∑−1 ˆxmvue = -- y(i) =-- ui N i=0 N i=0 " class="math-display" /></td> <td>(6)</td> </tr> </tbody> </table> Namely, (6) means that the optimal linear estimator of x given <span class="cmsy-10">{</span><span class="cmbx-10">y</span><span class="cmr-10">(</span><span class="cmmi-10">i</span><span class="cmr-10">)</span><span class="cmsy-10">}</span>, is simply the mean of the samples(measurements). So, in order to obtain an estimation of the background of the video, we take the average of all frames and store it as the background scene. Fig. 1 illustrates 4 frames of a sample video, these samples are in fact 4 noisy measurements of our signal, and that yellow ball which we are trying to track acts as the disturbance to the background of the video(the door and the wall). Fig. 2 provides the background scene that is the result of averaging over all of the frames. Please note that in this project we are assuming that the background does not change, so sample-mean estimator is a good estimation of the background. However, in the case of real-time tracking where the background is not changing, one can feed the average estimator as the frames arrives; evidently, in this case, estimation improves with time. Now that we obtained <img src="./Images/document8x.png" alt="ˆx" class="circ" />, we can extract the noise from the signal(video) by subtracting each frame from the background. Fig. 3 provides four realization of the noise(moving object) at different frames. Due to the fact that in our problem we do not care about the energy of the noise - the gray level of an image(pixels) is proportional to the energy or the amount of information it contains (entropy of the image) - we can use <span class="ptmri7t-">one-bit Compressed Sensing </span>method to store the noise. That is, we use the following model to store the noise