Examples of Error Propagation

Error-propagation theory is important in computer vision because basic feature-extraction operations affected by noise are common, such as measuring color intensity or the position of a particular feature in an image, and it is important to understand how much this noise affects subsequent computations.

Measurement error due to additive noise is formalized as $x = \hat{x} + \varepsilon$, where $x$ is the observed value, $\hat{x}$ is the true value, and $\varepsilon$ is the additive noise, for example white Gaussian noise with variance $\sigma^{2}_{x}$.

In vision, it may be useful to estimate how the error generated by the imprecise observation of a point in an image propagates through the system. In this case, the observed variables are $(x,y)$ image coordinates, both affected by localization errors with variances $\sigma^{2}_{x}$ and $\sigma^{2}_{y}$, respectively, and normally (at least to a first approximation) uncorrelated with each other.

Using the result of equation (2.37), the generic function $z(x,y)$, a function of two random variables, can be approximated to first order through a Taylor-series expansion as

$$z(x', y') \approx z(x, y) + \left. \frac{\partial z}{\partia... ...eft. \frac{\partial z}{\partial y} \right\vert _{x,y} (y' - y)$$ (2.40)

from which the uncertainty in the value of $z$ can be estimated by applying the variance-propagation result presented in the previous section, obtaining

$$\sigma^{2}_{z} = \left( \frac{\partial z}{\partial x} \righ... ...left( \frac{\partial z}{\partial y} \right)^{2} \sigma^{2}_{y}$$ (2.41)

where the derivative has been evaluated at $(x,y)$.

With this formulation, several examples can be presented:

Example 1

The error propagation of $z = x + y$ is

$$\sigma^{2}_{z} = \sigma^2_{x} + \sigma^2_{y}$$ (2.42)

the result already obtained above.

Example 2

The error propagation of $z = x y$ is

$$\sigma^{2}_{z} = y^2 \sigma^2_{x} + x^2 \sigma^2_{y}$$ (2.43)

Example 3

The error propagation of $z = \frac{1}{x \pm y}$ is

$$\sigma^{2}_{z} = \frac{ \sigma^{2}_{x} + \sigma^{2}_{y} } { (x \pm y)^4 }$$ (2.44)

Example 4

The error propagation of $z = \frac{x}{y}$ is

$$\sigma^{2}_{z} = \frac{1}{y^{2}} \sigma^{2}_{x} + \frac{x^{2}}{y^{4}} \sigma^{2}_{y}$$ (2.45)

Example 5

The error propagation of $z = \sqrt{x^2 + y^2}$ is

$$\sigma^{2}_{z} = \frac{x^2 \sigma^{2}_{x} + y^2 \sigma^{2}_{y}}{x^2 + y^{2}}$$ (2.46)

It is interesting to note from these equations how the absolute values assumed by the variables ($x$ and $y$ in the examples) directly affect the error estimate for the final variable $z$: some variables produce results with lower variance as their magnitude increases, whereas others may exhibit the opposite behavior. For these reasons, depending on the transformation and therefore on the model estimate to be obtained, some image points may be more important to observe than others.