Implementation of Discrete Kalman filter for object tracking purposes

Table of Contents

• About
• How to use
• Main algorithm and equations
• Support
• Dependencies
• License
• Devs
• References

About

The Kalman filter estimates the state of a system at time $k$ via the linear stochastic difference equation considering the state of a system at time $k$ is evolved from the previous state at time $k-1$. See the ref. https://en.wikipedia.org/wiki/Kalman_filter

In other words, the purpose of Kalman filter is to predict the next state via using prior knowledge of the current state.

In this repository Hybrid Kalman filter is implemented considering continuous-time model while discrete-time measurements. See the ref. - https://en.wikipedia.org/wiki/Kalman_filter#Hybrid_Kalman_filter

You can find version for Rust programming language also - link

Kalman 1D	Kalman 2D

A showcase how to visualize Kalman filter works

How to use

Simply add dependency into your project:

go get github.com/LdDl/kalman-filter

Start using it, e.g. Kalman2D:

package main

import (
	"encoding/csv"
	"fmt"
	"os"

	kalman_filter "github.com/LdDl/kalman-filter"
)

func main() {
	dt := 0.04 // 1/25 = 25 fps - just an example
	ux := 1.0
	uy := 1.0
	stdDevA := 2.0
	stdDevMx := 0.1
	stdDevMy := 0.1

	// Sample measurements
	// Note: in this example Y-axis going from up to down
	xs := []float64{311, 312, 313, 311, 311, 312, 312, 313, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 311, 311, 311, 311, 311, 310, 311, 311, 311, 310, 310, 308, 307, 308, 308, 308, 307, 307, 307, 308, 307, 307, 307, 307, 307, 308, 307, 309, 306, 307, 306, 307, 308, 306, 306, 306, 305, 307, 307, 307, 306, 306, 306, 307, 307, 308, 307, 307, 308, 307, 306, 308, 309, 309, 309, 309, 308, 309, 309, 309, 308, 311, 311, 307, 311, 307, 313, 311, 307, 311, 311, 306, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312}
	ys := []float64{5, 6, 8, 10, 11, 12, 12, 13, 16, 16, 18, 18, 19, 19, 20, 20, 22, 22, 23, 23, 24, 24, 28, 30, 32, 35, 39, 42, 44, 46, 56, 58, 70, 60, 52, 64, 51, 70, 70, 70, 66, 83, 80, 85, 80, 98, 79, 98, 61, 94, 101, 94, 104, 94, 107, 112, 108, 108, 109, 109, 121, 108, 108, 120, 122, 122, 128, 130, 122, 140, 122, 122, 140, 122, 134, 141, 136, 136, 154, 155, 155, 150, 161, 162, 169, 171, 181, 175, 175, 163, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178}

	// Assume that initial X,Y coordinates match the first measurement
	ix := xs[0] // Initial state for X
	iy := ys[0] // Initial state for Y

	kalman := kalman_filter.NewKalman2D(dt, ux, uy, stdDevA, stdDevMx, stdDevMy, kalman_filter.WithState2D(ix, iy))

	predictions := make([][]float64, 0, len(xs))
	updatedStates := make([][]float64, 0, len(xs))
	for i := 0; i < len(xs); i++ {
		// Considering that the measurements are noisy
		mx := xs[i]
		my := ys[i]

		// Predict stage
		kalman.Predict()
		state := kalman.GetVectorState()
		predictions = append(predictions, []float64{state.At(0, 0), state.At(1, 0)})

		// Update stage
		err := kalman.Update(mx, my)
		if err != nil {
			fmt.Println(err)
			return
		}
		updatedState := kalman.GetVectorState()
		updatedStates = append(updatedStates, []float64{updatedState.At(0, 0), updatedState.At(1, 0)})
	}

	file, err := os.Create("kalman-2d.csv")
	if err != nil {
		fmt.Println(err)
		return
	}
	defer file.Close()

	writer := csv.NewWriter(file)
	defer writer.Flush()
	writer.Comma = ';'

	err = writer.Write([]string{"measurement X", "measurement Y", "prediction X", "prediction Y", "updated X", "updated Y"})
	if err != nil {
		fmt.Println(err)
		return
	}
	for i := 0; i < len(xs); i++ {
		err = writer.Write([]string{
			fmt.Sprintf("%f", xs[i]),
			fmt.Sprintf("%f", ys[i]),
			fmt.Sprintf("%f", predictions[i][0]),
			fmt.Sprintf("%f", predictions[i][1]),
			fmt.Sprintf("%f", updatedStates[i][0]),
			fmt.Sprintf("%f", updatedStates[i][1]),
		})
		if err != nil {
			fmt.Println(err)
			return
		}
	}
}

Main algorithm and equations

Define mentioned linear stochastic difference equation:

$$\chi_{k} = A⋅\chi_{k-1} + B⋅u_{k-1} + w_{k-1} \tag{1}$$

Define measurement model: $$z_{k} = H⋅\chi_{k} + v_{k}\tag{2}$$

Let's denote variables:

• $A$ (sometimes it's written as $F$, but I prefer to stick with $A$) - Transition matrix of size $n \times n$ relating state $k-1$ to state $k$
• $B$ - Control input matrix of size $n \times l$ which is applied to optional control input $u_{k-1}$
• $H$ - Transformation (observation) matrix of size $m \times n$.
• $u_{k}$ - Control input
• $w_{k}$ - Process noise vector with covariance $Q$. Gaussian noise with the normal probability distribution: $$w(t) \sim N(0, Q) \tag{3}$$
• $v_{k}$ - Measurement noise vector (uncertainty) with covariance $R$. Gaussian noise with the normal probability distribution: $$v(t) \sim N(0, R) \tag{4}$$

Prediction

Let's use the dash sign " $-$ " as superscript to indicate the a priory state.

A priory state in matrix notation is defined as

$$\hat{\chi}^-_ {k} = A⋅\hat{\chi}_ {k-1} + B⋅u_ {k-1} \tag{5}$$

$$\text{, where $\hat{\chi}^-_ {k}$ - a priory state (a.k.a. predicted), $\hat{\chi}_ {k-1}$ - a posteriory state (a.k.a. previous)} $$

Note: A posteriory state $\hat{\chi}_{k-1}$ on 0-th time step (initial) should be guessed

Error covariance matrix $P^-$ is defined as

$$P^-_ {k} = A⋅P_ {k-1}⋅A^{T} + Q \tag{6}$$

$$\text{, where $P_ {k-1}$ - previously estimated error covariance matrix of size $n \times n$ (should match transition matrix dimensions), } $$ $$\text{Q - process noise covariance}$$

Note: $P_ {k-1}$ on 0-th time step (initial) should be guessed

Correction

The Kalman gain (which minimizes the estimate variance) in matrix notation is defined as:

$$K_ {k} = P^-_ {k}⋅H^{T}⋅(H⋅P^-_ {k}⋅H^{T}+R)^{-1} \tag{7}$$

$$\text{, where H - transformation matrix, R - measurement noise covariance}$$

After evaluating the Kalman gain we need to update a priory state $\hat{\chi}^-_ {k}$. In order to do that we need to calculate measurement residual:

$$r_ {k} = z_ {k} - H⋅\hat{\chi}^-_ {k} \tag{8}$$

$$\text{, where $z_ {k}$ - true measurement, $H⋅\hat{\chi}^-_ {k}$ - previously estimated measurement}$$

Then we can update predicted state $\hat{\chi}_ {k}$:

$$\hat{\chi}_ {k} = \hat{\chi}^-_ {k} + K_{k}⋅r_{k}$$

$$\text{or} \tag{9}$$

$$\hat{\chi}_ {k} = \hat{\chi}^-_ {k} + K_{k}⋅(z_{k} - H⋅\hat{\chi}^-_{k})$$

After that we should update error covariance matrix $P_{k}$ which will be used in next time stap (an so on): $$P_{k} = (I - K_{k}⋅H)⋅P^-_{k}\tag{10}$$ $$\text{, where $I$ - identity matrix (square matrix with ones on the main diagonal and zeros elsewhere)}$$

Overall

The whole algorithm can be described as high-level diagram:

Fig 1. Operation of the Kalman filter. Welch & Bishop, 'An Introduction to the Kalman Filter'

1-D Kalman filter

Considering acceleration motion let's write down its equations:

Velocity: $$v = v_{0} + at \tag{11}$$ $$v(t) = x'(t) $$ $$a(t) = v'(t) = x''(t)$$

Position: $$x = x_{0} + v_{0}t + \frac{at^2}{2} \tag{12}$$

Let's write $(11)$ and $(12)$ in Lagrange form:

$$x'_ {k} = x'_ {k-1} + x''_{k-1}\Delta t \tag{13}$$

$$x_{k} = x_{k-1} + x'_ {k-1}\Delta t + \frac{x''_{k-1}(\Delta t^2)}{2} \tag{14}$$

State vector $\chi_{k}$ looks like:

$$\chi_{k} = \begin{bmatrix} x_{k} \\ x'_ {k} \end{bmatrix} = \begin{bmatrix} x_{k-1} + x'_ {k-1}\Delta t + \frac{x''_ {k-1}(\Delta t^2)}{2} \\ x'_ {k-1} + x''_{k-1}\Delta t \end{bmatrix} \tag{15}$$

Matrix form of $\chi_{k}$ :

$$\chi_{k} = \begin{bmatrix} x_{k} \\ x'_ {k} \end{bmatrix} = \begin{bmatrix} 1 & \Delta t \\ 0 & 1\end{bmatrix} ⋅ \begin{bmatrix} x_{k-1} \\ x'_ {k-1} \end{bmatrix} + \begin{bmatrix} \frac{\Delta t^2}{2} \\ \Delta t \end{bmatrix} ⋅ x''_ {k-1} = \begin{bmatrix} 1 & \Delta t \\ 0 & 1\end{bmatrix} ⋅ \chi_{k-1} + \begin{bmatrix} \frac{\Delta t^2}{2} \\ \Delta t \end{bmatrix} ⋅ x''_{k-1} \tag{16}$$

Taking close look on $(16)$ and $(1)$ we can write transition matrix $A$ and control input matrix $B$ as follows:

$$A = \begin{bmatrix} 1 & \Delta t \\ 0 & 1\end{bmatrix} \tag{17}$$

$$B = \begin{bmatrix} \frac{\Delta t^2}{2} \\ \Delta t \end{bmatrix} \tag{18}$$

Let's find transformation matrix $H$. According to $(2)$:

$$z_{k} = H⋅\chi_{k} + v_{k} = \begin{bmatrix} 1 & 0 \end{bmatrix} ⋅\begin{bmatrix} x_{k} \\ {x'_ {k}} \end{bmatrix} + v_{k} \tag{19}$$

$$ H = \begin{bmatrix} 1 & 0 \end{bmatrix} \tag{20}$$

Notice: $v_{k}$ in $(19)$ - is not speed, but measurement noise! Don't be confused with notation. E.g.:

$$ \text{$ \chi_{k} = \begin{bmatrix} 375.74 \\ 0 - \text{assume zero velocity} \end{bmatrix} $, $ v_{k} = 2.64 => $} $$

$$ \text{$ => z_{k} = \begin{bmatrix} 1 & 0 \end{bmatrix} ⋅\begin{bmatrix} 375.74 \\ 0 \end{bmatrix} + 2.64 = \begin{bmatrix} 378.38 & 2.64 \end{bmatrix} $ - you can see that first vector argument it is just noise $v_{k}$ added}$$

$$ \text{to observation $x_{k}$ and the second argument is noise $v_{k}$ itself.}$$

Process noise covariance matrix $Q$:

$$Q = \begin{matrix} & \begin{matrix}x && x'\end{matrix} \\ \begin{matrix}x \\ x'\end{matrix} & \begin{bmatrix} \sigma^2_{x} & \sigma_{x} \sigma_{x'} \\ \sigma_{x'} \sigma_{x} & \sigma^2_{x'}\end{bmatrix} \\\ \end{matrix} \tag{21}$$

$$\text{, where} $$

$$ \text{$\sigma_{x}$ - standart deviation of position} $$

$$ \text{$\sigma_{x'}$ - standart deviation of velocity} $$

Since we know about $(14)$ we can define $\sigma_{x}$ and $\sigma_{x'}$ as:

$$ \sigma_{x} = \sigma_{x''} \frac{\Delta t^2}{2} \tag{22}$$

$$ \sigma_{x'} = \sigma_{x''} \Delta t \tag{23}$$

$$\text{, where $\sigma_{x''}$ - standart deviation of acceleration (tuned value)} $$

And now process noise covariance matrix $Q$ could be defined as:

$$ Q = \begin{bmatrix} (\sigma_{x''} \frac{\Delta t^2}{2})^2 & \sigma_{x''} \frac{\Delta t^2}{2} \sigma_{x''} \Delta t \\ \sigma_{x''} \Delta t \sigma_{x''} \frac{\Delta t^2}{2} & (\sigma_{x''} \Delta t)^2 \end{bmatrix} = $$

$$ = \begin{bmatrix} (\sigma_{x''} \frac{\Delta t^2}{2})^2 & (\sigma_{x''})^2 \frac{\Delta t^2}{2} \Delta t \\ (\sigma_{x''})^2 \Delta t \frac{\Delta t^2}{2} & (\sigma_{x''} \Delta t)^2 \end{bmatrix} = \begin{bmatrix} (\frac{\Delta t^2}{2})^2 & \frac{\Delta t^2}{2} \Delta t \\ \Delta t \frac{\Delta t^2}{2} & \Delta t^2 \end{bmatrix} \sigma^2_{x''}$$

$$ = \begin{bmatrix} \frac{\Delta t^4}{4} & \frac{\Delta t^3}{2} \\ \frac{\Delta t^3}{2} & \Delta t^2 \end{bmatrix} \sigma^2_{x''} \tag{24}$$

$$ \text{Assuming that $x''$ - is acceleration $a$, $Q = \begin{bmatrix} \frac{\Delta t^4}{4} & \frac{\Delta t^3}{2} \\ \frac{\Delta t^3}{2} & \Delta t^2 \end{bmatrix} \sigma^2_{a}$} \tag{25}$$

Covariance of measurement noise $R$ is scalar (matrix of size $1 \times 1$) and it is defined as variance of the measurement noise:

$$R = \begin{matrix} \begin{matrix}& x\end{matrix} \\ \begin{matrix}x\end{matrix} \begin{bmatrix}\sigma^2_{z}\end{bmatrix} \\\ \end{matrix} = \sigma^2_{z} \tag{26}$$

Golang implementation is here

Example of usage:

rand.Seed(1337)
dt := 0.1
u := 2.0
stdDevA := 0.25
stdDevM := 1.2

n := 100
iters := int(float64(n) / dt)
track := make([]struct {
    t float64
    x float64
}, iters)
v := 0.0
for i := 0; i < iters; i++ {
    track[i] = struct {
        t float64
        x float64
    }{
        t: v,
        x: dt * (v*v - v),
    }
    v += dt
}
kalman := kalman_filter.NewKalman1D(dt, u, stdDevA, stdDevM)

measurements := make([]float64, 0, iters)
predictions := make([]float64, 0, iters)
for _, val := range track {
    // tm := val.t
    x := val.x

    // Add some noise to perfect track
    noise := rand.Float64()*100 - 50
    z := kalman.H.At(0, 0)*x + noise
    measurements = append(measurements, z)

    // Predict stage
    kalman.Predict()
    state := kalman.GetVectorState()
    predictions = append(predictions, state.At(0, 0))

    // Update stage
    err := kalman.Update(z)
    if err != nil {
        fmt.Println(err)
        return
    }
}
fmt.Println("time;perfect;measurement;prediction")
for i := 0; i < len(track); i++ {
	fmt.Printf("%f;%f;%f;%f\n", track[i].t, track[i].x, measurements[i], predictions[i])
}

How exported chart does look like:

2-D Kalman filter

Considering acceleration motion again let's write down its equations:

Considering the same physical model as in $(13)$ - $(14)$ let's write down state vector $\chi_{k}$:

$$\chi_{k} = \begin{bmatrix} x_{k} \\ y_{k} \\ x'_ {k} \\ y'_ {k} \end{bmatrix} = \begin{bmatrix} x_{k-1} + x'_ {k-1}\Delta t + \frac{x''_ {k-1}(\Delta t^2)}{2} \\ y_{k-1} + y'_ {k-1}\Delta t + \frac{y''_ {k-1}(\Delta t^2)}{2} \\ x'_ {k-1} + x''_ {k-1}\Delta t \\ y'_ {k-1} + y''_ {k-1}\Delta t \end{bmatrix} \tag{27}$$

Matrix form of $\chi_{k}$ :

$$\chi_{k} = \begin{bmatrix} x_{k} \\ y_{k} \\ x'_ {k} \\ y'_ {k} \end{bmatrix} = \begin{bmatrix} 1 & 0 & \Delta t & 0 \\ 0 & 1 & 0 & \Delta t \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} ⋅ \begin{bmatrix} x_{k-1} \\ y_{k-1} \\ x'_ {k-1} \\ y'_ {k-1} \end{bmatrix} + \begin{bmatrix} \frac{\Delta t^2}{2} & 0 \\ 0 & \frac{\Delta t^2}{2} \\ \Delta t & 0 \\ 0 & \Delta t \end{bmatrix} ⋅ \begin{bmatrix} x''_ {k-1} \\ y''_ {k-1} \end{bmatrix} = $$ $$ = \begin{bmatrix} 1 & 0 & \Delta t & 0 \\ 0 & 1 & 0 & \Delta t \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} ⋅ \chi_{k-1} + \begin{bmatrix} \frac{\Delta t^2}{2} & 0 \\ 0 & \frac{\Delta t^2}{2} \\ \Delta t & 0 \\ 0 & \Delta t \end{bmatrix} ⋅ \begin{bmatrix} x''_ {k-1} \\ y''_{k-1} \end{bmatrix} \tag{28}$$

$$ \text{Assuming that $x''$ and $y''$ - is acceleration $a$, }$$

$$ a_{k-1} = \begin{bmatrix} x''_ {k-1} \\ y''_{k-1} \end{bmatrix} \tag{29}$$

$$\chi_{k} = \begin{bmatrix} x_{k} \\ y_{k} \\ x'_ {k} \\ y'_ {k} \end{bmatrix} = \begin{bmatrix} 1 & 0 & \Delta t & 0 \\ 0 & 1 & 0 & \Delta t \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} ⋅ \chi_{k-1} + \begin{bmatrix} \frac{\Delta t^2}{2} & 0 \\ 0 & \frac{\Delta t^2}{2} \\ \Delta t & 0 \\ 0 & \Delta t \end{bmatrix} ⋅ a_{k-1} \tag{30}$$

Taking close look on $(16)$ and $(1)$ we can write transition matrix $A$ and control input matrix $B$ as follows:

$$A = \begin{bmatrix} 1 & 0 & \Delta t & 0 \\ 0 & 1 & 0 & \Delta t \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{31}$$

$$B = \begin{bmatrix} \frac{\Delta t^2}{2} & 0 \\ 0 & \frac{\Delta t^2}{2} \\ \Delta t & 0 \\ 0 & \Delta t \end{bmatrix} \tag{32}$$

Let's find transformation matrix $H$. According to $(2)$ and $(19)$ - $(20)$:

$$z_{k} = H⋅\chi_{k} + v_{k} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} ⋅\begin{bmatrix} x_{k} \\ y_{k} \\ {x'_ {k}} \\ {y'_ {k}} \end{bmatrix} + v_{k} \tag{33}$$

$$ H = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \tag{34}$$

Process noise covariance matrix $Q$:

$$Q = \begin{matrix} & \begin{matrix}x && y && x' && y'\end{matrix} \\ \begin{matrix}x \\ y \\ x' \\ y'\end{matrix} & \begin{bmatrix} \sigma^2_{x} & 0 & \sigma_{x} \sigma_{x'} & 0 \\ 0 & \sigma^2_{y} & 0 & \sigma_{y} \sigma_{y'} \\ \sigma_{x'} \sigma_{x} & 0 & \sigma^2_{x'} & 0 \\ 0 & \sigma_{y'} \sigma_{y} & 0 & \sigma^2_{y'}\end{bmatrix} \\\ \end{matrix} \tag{35}$$

$$\text{, where} $$

$$ \text{$\sigma_{x}$ - standart deviation of position for $x$ component} $$

$$ \text{$\sigma_{y}$ - standart deviation of position for $y$ component} $$

$$ \text{$\sigma_{x'}$ - standart deviation of velocity for $x$ component} $$

$$ \text{$\sigma_{y'}$ - standart deviation of velocity for $y$ component} $$

Since we know about $(14)$ we can define $\sigma_{x}$, $\sigma_{y}$, $\sigma_{x'}$ and $\sigma_{y'}$ as:

$$ \sigma_{x} = \sigma_{x''} \frac{\Delta t^2}{2} \tag{36}$$

$$ \sigma_{y} = \sigma_{y''} \frac{\Delta t^2}{2} \tag{37}$$

$$ \sigma_{x'} = \sigma_{x''} \Delta t \tag{38}$$

$$ \sigma_{y'} = \sigma_{y''} \Delta t \tag{39}$$

$$\text{, where $\sigma_{x''}$ and $\sigma_{y''}$ - standart deviation of acceleration (tuned values)} $$

And now process noise covariance matrix $Q$ could be defined as:

$$ Q = \begin{bmatrix} (\sigma_{x''} \frac{\Delta t^2}{2})^2 & 0 & \sigma_{x''} \frac{\Delta t^2}{2} \sigma_{x''} \Delta t & 0 \\ 0 & (\sigma_{y''} \frac{\Delta t^2}{2})^2 & 0 & \sigma_{y''} \frac{\Delta t^2}{2} \sigma_{y''} \Delta t \\ \sigma_{x''} \frac{\Delta t^2}{2} \sigma_{x''} \Delta t & 0 & (\sigma_{x''} \Delta t)^2 & 0 \\ 0 & \sigma_{y''} \frac{\Delta t^2}{2} \sigma_{y''} \Delta t & 0 & (\sigma_{y''} \Delta t)^2 \end{bmatrix} = $$

$$ = \begin{bmatrix} (\sigma_{x''} \frac{\Delta t^2}{2})^2 & 0 & (\sigma_{x''})^2 \frac{\Delta t^2}{2} \Delta t & 0 \\ 0 & (\sigma_{y''} \frac{\Delta t^2}{2})^2 & 0 & (\sigma_{y''})^2 \frac{\Delta t^2}{2} \Delta t \\ (\sigma_{x''})^2 \frac{\Delta t^2}{2} \Delta t & 0 & (\sigma_{x''} \Delta t)^2 & 0 \\ 0 & (\sigma_{y''})^2 \frac{\Delta t^2}{2}\Delta t & 0 & (\sigma_{y''} \Delta t)^2 \end{bmatrix} = \text{| Knowing that $x''$ and $y''$ - acceleration|} = $$ $$ = \begin{bmatrix} (\frac{\Delta t^2}{2})^2 & 0 & \frac{\Delta t^2}{2} \Delta t & 0 \\ 0 & (\frac{\Delta t^2}{2})^2 & 0 & \frac{\Delta t^2}{2} \Delta t \\ \frac{\Delta t^2}{2} \Delta t & 0 & \Delta t^2 & 0 \\ 0 & \Delta t \frac{\Delta t^2}{2} & 0 & \Delta t^2 \end{bmatrix} \sigma^2_{a''}$$

$$ = \begin{bmatrix} \frac{\Delta t^4}{4} & 0 & \frac{\Delta t^3}{2} & 0 \\ 0 & \frac{\Delta t^4}{4} & 0 & \frac{\Delta t^3}{2} \\ \frac{\Delta t^3}{2} & 0 & \Delta t^2 & 0 \\ 0 & \frac{\Delta t^3}{2} & 0 & \Delta t^2 \end{bmatrix} \sigma^2_{a''} \tag{40}$$

Covariance of measurement noise $R$ is matrix of size $2 \times 2$ (since there are two components - $x$ and $y$) and it is defined as variance of the measurement noise:

$$R = \begin{matrix} \begin{matrix}& x & y\end{matrix} \\ \begin{matrix}x \\ y \end{matrix} \begin{bmatrix}\sigma^2_{x} & 0 \\ 0 & \sigma^2_{y} \end{bmatrix} \\\ \end{matrix} = \begin{bmatrix}\sigma^2_{x} & 0 \\ 0 & \sigma^2_{y} \end{bmatrix} \tag{41}$$

Golang implementation is here

Example of usage:

dt := 0.04 // 1/25 = 25 fps - just an example
ux := 1.0
uy := 1.0
stdDevA := 2.0
stdDevMx := 0.1
stdDevMy := 0.1

// Sample measurements
// Note: in this example Y-axis going from up to down
xs := []float64{311, 312, 313, 311, 311, 312, 312, 313, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 311, 311, 311, 311, 311, 310, 311, 311, 311, 310, 310, 308, 307, 308, 308, 308, 307, 307, 307, 308, 307, 307, 307, 307, 307, 308, 307, 309, 306, 307, 306, 307, 308, 306, 306, 306, 305, 307, 307, 307, 306, 306, 306, 307, 307, 308, 307, 307, 308, 307, 306, 308, 309, 309, 309, 309, 308, 309, 309, 309, 308, 311, 311, 307, 311, 307, 313, 311, 307, 311, 311, 306, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312, 312}
ys := []float64{5, 6, 8, 10, 11, 12, 12, 13, 16, 16, 18, 18, 19, 19, 20, 20, 22, 22, 23, 23, 24, 24, 28, 30, 32, 35, 39, 42, 44, 46, 56, 58, 70, 60, 52, 64, 51, 70, 70, 70, 66, 83, 80, 85, 80, 98, 79, 98, 61, 94, 101, 94, 104, 94, 107, 112, 108, 108, 109, 109, 121, 108, 108, 120, 122, 122, 128, 130, 122, 140, 122, 122, 140, 122, 134, 141, 136, 136, 154, 155, 155, 150, 161, 162, 169, 171, 181, 175, 175, 163, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178, 178}

// Assume that initial X,Y coordinates match the first measurement
ix := xs[0] // Initial state for X
iy := ys[0] // Initial state for Y

kalman := kalman_filter.NewKalman2D(dt, ux, uy, stdDevA, stdDevMx, stdDevMy, kalman_filter.WithState2D(ix, iy))

predictions := make([][]float64, 0, len(xs))
updatedStates := make([][]float64, 0, len(xs))
for i := 0; i < len(xs); i++ {
    // Considering that the measurements are noisy
    mx := xs[i]
    my := ys[i]

    // Predict stage
    kalman.Predict()
    state := kalman.GetVectorState()
    predictions = append(predictions, []float64{state.At(0, 0), state.At(1, 0)})

    // Update stage
    err := kalman.Update(mx, my)
    if err != nil {
        fmt.Println(err)
        return
    }
    updatedState := kalman.GetVectorState()
    updatedStates = append(updatedStates, []float64{updatedState.At(0, 0), updatedState.At(1, 0)})
}
fmt.Println("measurement X;measurement Y;prediction X;prediction Y;updated X;updated Y")
for i := 0; i < len(xs); i++ {
    fmt.Printf("%f;%f;%f;%f;%f;%f\n", xs[i], ys[i], predictions[i][0], predictions[i][1], updatedStates[i][0], updatedStates[i][1])
}

How exported chart does look like:

Support and contribution

If you have troubles or questions please open an issue.

PR's are welcome.

Dependencies

• Matrix computations - gonum. License is BSD 3-Clause "New" or "Revised" License. Link
• Errors wraping - errors. License is BSD 2-Clause "Simplified" License. Link

License

License of this library is MIT.

You can check it here

Developers

LdDl https://github.com/LdDl

Pavel7824 https://github.com/Pavel7824

References

• Greg Welch and Gary Bishop, ‘An Introduction to the Kalman Filter’, July 24, 2006
• Introducion to the Kalman Filter by Alex Becker
• Kalman filter on wikipedia
• State-transition matrix
• Python implementation by Rahmad Sadli

P.S.

I did struggle on displaying matrices in GitHub's MathJax markdown. If you know better way to do it you are welcome