Segmentation of Stochastic Images using ... - Jacobs University

Chapter 2 Image Segmentation and Limitations
Figure 2.10: A test pattern corrupted by uniform (left), Gaussian (middle), and speckle noise (right).
It is possible to reduce some of the noise sources by averaging the image values over a period.
When a signal is available for a period L, the expected value of a pixel is
1
E(a) = lim
L→∞ L
∫ L
0
a(x)dx . (2.44)
When the probability density function (PDF) of the process a is known the integral reduces to
E(a) =
∫ ∞
where ρ is the PDF. The variance of the stochastic process is (cf. [44])
σ 2 =
∫ ∞
With these quantities, the signal-to-noise ratio (SNR) [44] is
−∞
−∞
aρ(a)da , (2.45)
(a − E(a)) 2 ρ(a)da . (2.46)
SNR = |E(a)|2
σ 2 . (2.47)
One divides the noise sources into additive and multiplicative noise sources. Fig. 2.10 shows three
noise models. Additive noise is modeled via
g(x) = f (x) + n(x) , (2.48)
where g is the measured signal, f the true signal and n the noise. Multiplicative noise is modeled via
g(x) = f (x) + n(x) f (x) . (2.49)
The multiplicative noise depends on the image value. In what follows, we use the additive noise
model, because we are not directly interested in the noise modeling, but need a noise model as input
for the stochastic image processing framework. Once the noise is characterized, the noise is no
longer a free parameter and (2.49) can be expressed as
g(x) = f (x) + ñ(x) , (2.50)
and it is possible to use the additive model.
All these sources of noise influence the image quality and it is not well understood how the noise
influences the segmentation result, e.g. how the image noise influences the segmented object volume.
This is due to the construction of typical segmentation algorithms. They have no knowledge about the
noise that corrupted the image to segment and it is impossible to apply the segmentation algorithm
on noise realizations, apart from artificial test data corrupted with a known noise model. In the
next two sections, we present two problems related to image noise that cannot be investigated with
deterministic segmentation models, apart from using a sampling based approach.
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