KALMAN FILTER Applications in Image processing

Introduction

• The kalman filter is a recursive state space model based estimation algorithm.

• This filter is named after Rudolph E. Kalman, who in 1960 published his famous paper describing a recursive solution to the discrete data linear filtering problem (Kalman 1960).

• This algorithm was basically developed for single dimensional and real valued signals which are associated with the linear systems assuming the system is corrupted with linear additive white Gaussian noise.

• The Kalman filter addresses the general problem of trying to estimate the state x ∈ ℜn of a discrete-time controlled process that is governed by the linear difference equation

xk = Axk – 1 + Buk – 1 + wk – 1

• with a measurement z that is

zk = Hxk + vk

• The random variables wk and vk represent the process noise and measurement noise respectively.

• The nxn matrix A in the previous difference equation relates the state at the previous time step k-1 to the state at the current step k , in the absence of either a driving function or process noise.

• The nxl matrix B relates the optional control input u to the state x.

• The mxn matrix H in the measurement equation relates the state to the measurement zk.

The Computational Origins of the Filter :

• We define𝑥 𝑘−∈ℜn to be our a priori state

estimate at step k given knowledge of the process prior to step k , and 𝑥 𝑘∈ℜn to be our a posteriori state estimate at step k given measurement zk.

• We can then define a priori and a posteriori estimate errors as

𝑒𝑘− ≡ 𝑥𝑘 − 𝑥 𝑘

− & 𝑒𝑘 ≡ 𝑥𝑘 − 𝑥 𝑘

• The a priori estimate error covariance is then

𝑃𝑘− = 𝐸 𝑒𝑘

−𝑒𝑘−𝑇

&

• the a posteriori estimate error covariance is

𝑃𝑘 = 𝐸 𝑒𝑘𝑒𝑘𝑇

• The posteriori state estimate 𝑥 𝑘 is written as a linear combination of an a priori estimate 𝑥 𝑘

− and a weighted difference between an actual measurement zk & a measurement prediction H 𝑥 𝑘

−.

. 𝑥 𝑘 = 𝑥 𝑘

− + 𝐾 𝑧𝑘 − 𝐻𝑥 𝑘−

• The difference 𝑧𝑘 − 𝐻𝑥 𝑘

− is called the measurement innovation, or the residual.

• The nxm matrix K is chosen to be the gain or blending factor that minimizes the a posteriori error covariance.

• Substituting 𝑥 𝑘 in 𝑒𝑘 and 𝑒𝑘 in Pk , and performing minimization, we get

𝐾𝑘 =𝑃𝑘

−𝐻𝑇

𝐻𝑃𝑘−𝐻𝑇 + 𝑅

= 𝑃𝑘−𝐻𝑇 𝐻𝑃𝑘

−𝐻𝑇 + 𝑅 −1

Kalman filter algorithm

• The Kalman filter estimates a process by using a form of feedback control: the filter estimates the process state at some time and then obtains feedback in the form of (noisy) measurements.

• As such, the equations for the Kalman filter fall into two groups: time update equations and measurement update equations.

• The time update equations are responsible for projecting forward (in time) the current state and error covariance estimates to obtain the a priori estimates for the next time step.

• The measurement update equations are responsible for the feedback—i.e. for incorporating a new measurement into the a priori estimate to obtain an improved a posteriori estimate.

• The time update equations can also be thought of as predictor equations, while the measurement update equations can be thought of as corrector equations.

• The final estimation algorithm resembles that of a predictor-corrector algorithm.

Discrete Kalman filter time update equations:

𝑥 𝑘− = 𝐴𝑥 𝑘−1 +𝐵𝑢𝑘−1

𝑃𝑘− = 𝐴𝑃𝑘−1𝐴

𝑇 + 𝑄

𝐾𝑘 = 𝑃𝑘−𝐻𝑇 𝐻𝑃𝑘

−𝐻𝑇 + 𝑅 −1

𝑥 𝑘 = 𝑥 𝑘− + 𝐾𝑘 𝑧𝑘 − 𝐻𝑥 𝑘

−

𝑃𝑘 = 𝐼 − 𝐾𝑘𝐻 𝑃𝑘−

Discrete Kalman filter measurement update equations:

Prediction

Correction

A complete picture of the operation of the Kalman filter

Implementation

• The image process is modelled as an auto regressive(AR) process driven by a white gaussian noise (Wn) with variance Q described by

• Mathematically it can be written as

𝑦 𝑖, 𝑗= 𝑎1𝑦 𝑖, 𝑗 − 1 + 𝑎2𝑦 𝑖 − 1, 𝑗+ 𝑎3𝑦 𝑖 − 1, 𝑗 − 1 + 𝑎4𝑦(𝑖 − 1, 𝑗 + 1)

• The state space model for this system can be written as

𝑋𝑛+1 = 𝐴𝑋𝑛 + 𝑉𝑛 𝑍𝑛 = 𝐻𝑋𝑛 + 𝑊𝑛

where

𝐴 =

𝑎1 𝑎2 𝑎3 𝑎41 0 0 00 1 0 00 0 1 0

& 𝑋𝑛 =

𝑦(𝑖, 𝑗 − 1)𝑦(𝑖 − 1, 𝑗)

𝑦(𝑖 − 1, 𝑗 − 1)𝑦(𝑖 − 1, 𝑗 + 1

Results

Original Images Measured Images Corrected Images

Extended Kalman Filter

• An extended Kalman filter is used if the process to be estimated and (or) the measurement relationship to the process is non-linear.

• Here 𝑥𝑘 = 𝑓 𝑥𝑘−1, 𝑢𝑘−1, 𝑤𝑘−1

• with measurement z that is

𝑧𝑘 = ℎ 𝑥𝑘, 𝑣𝑘

• Similar to the Kalman filter, the time and measurement equations for EKF can be written as below:

• EKF time update equations:

• EKF measurement update eqns:

A complete picture of the operation of the extended Kalman filter

Results

Original Image Measured Image Corrected Image

COMPLEX KALMAN FILTERING

• In complex Kalman filtering, image model is represented in complex form as real and imaginary values represented as real and imaginary part of the complex number.

Y= Real+(imag)i

• where, Y is complex image model.

• Complex valued Kalman filters have been used extensively in a variety of applications, including frequency estimation of time-varying signals, training of neural networks etc.

Properties of Kalman filter

• Kalman filter is a time-varying filter as Kalman gain changes with n.

• The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown.

• In the Kalman filter, prediction acts like the prior information about the state at time n before we observe the data at time n.

Refernces

• Natasha Devroye. Estimation: parts of Chapters 12-13,

Wiener and Kalman Filtering.

• Greg Welch and Gary Bishop. An Introduction to the

Kalman Filter, Monday, July 24, 2006.

• R. E. KALMAN. A New Approach to Linear Filtering and

Prediction Problems.

• http://www.cs.unc.edu/~welch/kalman/