G-SURF Mid-Presentation Presentation Date : Presented by : Kyuewang Lee in CVL Laboratory, GIST > ( Focused on Tracking & Detection Technology )

Theme #01. Filters in Tracking We are talking about filters in “Signal Processing” It is a device that removes some components or features from a signal We can classify filters in various aspects and these classifying properties are related to the property of the signal 01. Linear X Non-Linear 02. Time-Invariant X Time-Variant 03. Causal X Not-Causal 04. Analog X Digital 05. Discrete-Time X Continuous-Time ETC. So why “FILTERS” on (T & D) ???

Theme #01. Filters in Tracking #01. Linear Filter Terminologies Low-pass Filter High-pass Filter Band-pass Filter Band-stop Filter( Notch Filter ) Comb Filter All-pass Filter( Only phase is modified ) Low-pass Filters are important in erasing “White Noises” White Noise  Noises mix together!! ※ Even if the object is non-linear, we can approximate it into piecewise linear objects Think of Taylor’s Approximation!

Theme #01. Filters in Tracking So, we usually use Low-pass Filter( LPF ) in order to cut high-frequency noises on the camera images, data, etc. But this still does not explain us why we use filters when we implement detection.

Theme #01. Filters in Tracking Batch Filter & Recursive Filter  Batch System : Uses Empirical Method  Recursive System : Uses Deductive Method ※ I’ll explain this in more detail

Theme #01. Filters in Tracking System Difference  You have a data of n people’s math grade But you forgot to count the last person’s grade In this situation, Batch System has to Add up All Grades respectively and divide the sum by n+1. However, Recursive System only needs the previous average value, last person’s grade and n to derive present average value. Needs at least n+1 memories in the system  If n is big, memories are being wasted!!! Has better memory usage  Only wastes at least 3 memories ( 3 terms are in the calculation!! )

Theme #01. Filters in Tracking Recursive Filters are beneficial in memory  As filter iteration goes higher, filter speed does not drastically fall Also in the aspect of algorithm, Recursive Filter’s Computational Complexity is better than that of Batch Filter.  We’ll check it fast right away now Let’s compare C codes /** 평균 다시 구하기 **/ for (i=0;i<n+1;i++) { Tsum += vArray[i]; } Tsum += tval; Tmean = Tsum/(n+1); /** 평균 다시 구하기 **/ double alpha = 1-(1/n); Tmean = alpha*Tmean + (1-alpha)*tval; “For” Statement in the Batch system increases the time complexity at least to its iteration number We Just Saw Computational Complexity is MUCH MORE ECONOMICAL in the Recursive Code!!

Theme #01. Filters in Tracking Okay, then we know understand the importance of recursive filter in both aspects 1. Memory 2. Speed Recursive Filters have another important property  Since it is recursive, it uses the result value of the last iteration.  It means that it can remember the last data. Why it is important to remember the last data?! The most important reason is that we can let the filter system to STUDY ITSELF about the object’s motion

Theme #01. Filters in Tracking ※ One example of LPF  Exponentially Weighted Moving Average Filter Firstly, this filter became the fundamental notion for the famous “Kalman Filter” Also, classified as 1 st ordered Low-pass Filter and this is used in various types of studies, especially in Economics. You can recognize that this equation is similar to setting a division point between two points. This is the reason that we call this “Weighted Filter” We can determine whether to grant more importance to the previous data prediction value or to the present data measurement value

Theme #01. Filters in Tracking When α is close to 1  Filter gets more sensitive to previous predicted value  Present predicted value becomes less noisy When α is close to 0  Filter gets more sensitive to present measured value  Present predicted value becomes more noisy Result : α value determines the prediction graph’s form

Theme #02. Kalman Filter Measurement Value : z(k) Kalman Filter Predicted Value : Xx(k) 5 Equations consisting the Kalman Filter

Theme #02. Kalman Filter PurposeVariables Measurementz(k) EstimatedXx(k) System ModelA, H, Q, R Internal CalculationXx_(k), P_(k), P(k), K(k)

Theme #02. Kalman Filter Measurement z(k) The “NOISY” input signals that the filter feeds in Can be approximated into “Ground_Truth + N(0,σ)” ( Signals that have standard deviation of σ from ground truth ) N(0,σ) is assumed to be the noise of this measurement value. ( e.g. Noise is assumed to follow the normal distribution of mean value 0 )

Theme #02. Kalman Filter Estimated Value Xx(k) The estimated value of one iteration of Kalman Filter This value is feed-backed into the filter again This is also called “Estimated Status Value” On the first iteration, we usually set this value as the original state of the object unless there is no condition for starting point

Theme #02. Kalman Filter Error Covariance P(k) It shows us how much the predicted value is accurate ( e.g.  If P(k) is small, the prediction error is big ) Diagonally symmetric matrix since P(k) talks about covariances  P(k)(ij) = P(k)(ji) K(k) = P_(k)*H’*inv( H*P_(k)*H’+R ) Xx(k) = Xx_(k) + K(k)*(z(k)-H*z_(k))

Theme #02. Kalman Filter System Models A : State Transition Model applied to the previous state x(k-1)  This determines the dynamic motion state of the object. Therefore if the object moves in uniformly accelerated motion of straight line, the matrix A would be different from that which moves in uniform circular motion. H : Observation Model which maps the true state space x(k) into the observed space z(k) Q : Covariance Matrix for the process noise( State Variable Noise )  w(k) ~ N( 0, Q(k) ) R : Covariance Matrix for the zero mean Gaussian white noise( Measurement Noise )  v(k) ~ N( 0, R(k) ) B : Control-Input Model which considers external factors System Modeling x(k) = A*x(k-1) + B*u(k) + w(k) z(k) = H*x(k) + v(k) u(k) is the Control Vector

Theme #02. Kalman Filter 1. Prediction Process  Filter predicts the state value and the error covariance 2. Correction Process  Filter draws “Kalman Gain” and calculates the estimated state value and estimated error covariance

Theme #02. Kalman Filter Correction Process K(k) = P_(k)*H’*inv( H*P_(k)*H’+R )  Kalman Gain K(k) Xx(k) = Xx_(k) + K(k)*( z(k)-H*x_(k) )  Estimation Xx(k) P(k) = P_(k) – K(k)*H*P_(k)  Error Covariance P(k) What if H is I ( Identity Matrix ) ?? The equation becomes like below form Xx(k) = ( I-K(k)*H )*Xx_(k) + K(k)*z(k)  Xx(k) = ( I-K(k) )*Xx_(k) + K(k)*z(k) These two equation are in same form Thus the method of solving estimation value is in actual, solving this exponentially weighted equation K(k) = P_(k)*H’*inv( H*P_(k)*H’+R )  Kalman Gain The weight factor( Kalman Gain ) differs every iteration  This filter can balance the importance by itself!!

Theme #02. Kalman Filter Prediction Process Xx_(k) = A*Xx(k-1) + B*u(k) P_(k) = A*P(k-1)*A’ + Q Linear Modeling This process makes the prediction values for each variables and these are to be used in the correction process to yield estimation values. This is the singular iteration and we do it ‘k’ times.

Theme #02. Kalman Filter Actual Implementation 01. Sine function tracking Implemented with MATLAB Ground Truth Estimated // Ground Truth Measured // Estimated Estimated // Ground Truth Nonlinear Linear

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