Estimating Degradation Function

The degradation function $h(x,y)$ for image restoration is usually estimated using 3 ways:

1. Observation
   • infer blur from a strong, interpretable image region
2. Experimentation
   • create an image of a known impulse under the same settings
3. Modelling
   • derive $H$ from optics, motion, or atmospheric physics

I. Observation

$$H_s(u,v)=\frac{G_s(u,v)}{{\hat{F}}_s(u,v)}$$

• $g_s(x,y)$ is a sub image where the signal content is strong (we need to find this)
• ${\hat{f}}_s(x,y)$ is what you construct that subimage to look like when undegraded (basically our assumption of what that subsample is supposed to look like)
• The subscript s marks that both come from the subimage, not the whole image

My understanding

Ignoring noise, $G_s=H_s{F}_s$. Meaning that we have 2 unknowns $H$ and $F$... so this method is basically estimating $F_s$ so that we can find $H_s$

II. Experimentation

$$H(u,v)=\frac{G(u,v)}{A}$$

• $G(u,v)$ is the Fourier transform of the observed image of a small dot of light
• $A$ is a constant denoting the strength of the impulse

III. Mathematical Modelling

$$H(u,v)=e^{-k{\left(u^2+v^2\right)}^{5/6}}$$

$k$ depends on the nature of the turbulence, and a larger $k$ gives more blur.

Motion Blur

Motion blur can by caused by relative motion between the scene and the sensor during image acquisition. Suppose the motion can be described by $x_0(t)$ and $y_0(t)$ in the $x$ and $y$ directions respectively. The blurred image can be modelled as:

$$g(x,y)={\int }_0^{T}f\left(x-x_0(t),y-y_0(t)\right)dt$$

where $T$ is the duration of the exposure.

Assume $x_0(t)=at/T$ and $y_0(t)=bt/T$. Then the degradation function is:

$$H(u,v)=\frac{G(u,v)}{F(u,v)}=\frac{T\sin \left(\pi (au+bv)\right)}{\pi (au+bv)}{e}^{-j\pi (au+bv)}$$