*2.1.2 Image restoration*

In many imaging systems, such as visual cameras, microscopes, telescopes or Infra Red cameras, due to some limitations such as limited aperture or limited resolution, the forward problem can be approximated by a 2D convolution equation:

$$\mathbf{g}(\mathbf{x}, \mathbf{y}) = \iint f(\mathbf{x}', \mathbf{y}') h(\mathbf{x} - \mathbf{x}', \mathbf{y} - \mathbf{y}') \, \mathbf{dx} \, \mathbf{dy}.\tag{4}$$

The corresponding inverse problem is called image deconvolution or more often *image restoration*. The example given in **Figure 4**, is the case of satellite imaging [6, 7].

**Figure 2.** *Signal deconvolution problem.*

**Figure 3.** *Image deconvolution or restoration inverse problem in sattelite imaging.*

**Figure 4.**

*Image reconstruction in CT. On the left, the projections g r*ð Þ , *ϕ and on the right the object f (x, y).*

*2.1.3 Image reconstruction in X ray computed tomography (CT)*

In X-ray CT, assuming parallel geometry, where a ray is characterized by its angle *ϕ* and its distance *r* from the center of the object *f* (*x*, *y*) the relation between the data *g r*ð Þ , *ϕ* , called projections at angle *ϕ* and the function *f* (*x*, *y*), called object, is given by the Radon transform:

$$\log(r,\phi) = \iint f(\mathbf{x},\mathbf{y})\delta(r-\mathbf{x}\cos\phi-\mathbf{y}\sin\phi)\,\mathrm{d}\mathbf{x}\,\mathrm{d}\mathbf{y}.\tag{5}$$

The inverse problem here is called *Image reconstruction*. A simulated example is shown in **Figure 4**.
