Restoration

Picture restoration deals with images recorded in the presence of one or more sources of degradation Many sources of degradation are present in imaging systems.Some types affect only the gray levels at the individual picture points without introducing spatial blur. They are called point degradation. Other types which do involve blur are spatial degradations. Blurring is a form of bandwidth reduction of an ideal image caused by imperfect image formation process.

Point and spatial degradation occur in a variety of applications eg, In aerial mapping, astronomy, remote sensing, pictures obtained through atmosphere turbulence, aberration of optical systems relative motion between camera and object.

Degradation Model

Given an ideal picture and a degraded picture

We will assume g and f are related by,

	(6.1.1)

where is the degradation function and is random noise.

In the absence of noise

1) The degraded image of a point source described by would be given by . Therefore is a PSF which in general is dependent on position of the point in the ideal picture.

Note: The spatially continuum PSF of any blur satisfies three constraints

• takes on non -ve values only because of the physics of the underlying image formulation process.
• When real valued images are dealt with the PSF is real valued too.
• the imperfections in the image forming process are modeled as "passive" operations on data ie. no "energy" is absorbed or generated. This means PSF is constrained to satisfy

2) The assumption that is a linear function of is approximately correct only over a small dynamic range of gray levels for eg. in a photographic system what is recorded is usually a non linear function of . If however the non linear characteristic of the film is known, it can be used to recover from what is recorded on the film over a large dynamic range of gray levels.

3) The assumption noise is additive is also subject too criticism. Many noise sources may be individually modelled as additive. When however additive noise is followed by a non linear transformation its effect on function can be assumed additive only over a small dynamic range. However as the assumption of additivity of noise makes the problem mathematically tractable, it is common to most work on picture restoration

If except for translation, the degraded image of a point is independent of the position of the point, then the PSF takes the form and

	(6.1.2)

In this case degradation is shift invariant. In our discussion we will restrict ourselves to pictures that have suffered this type of degradation i.e, point degradation.

In the absence of noise

	(6.1.3)

Fourier transforming both sides

	(6.1.4)

where is transfer function of system that transfers ideal picture f to degrade picture g .

A-priori knowledge required in restoration:

From a mathematical view point, given the model described by (6.2) and the degrad picture the aim of picture restoration is to make as good an estimate as possible of the original picture or scene . Evidently any such estimation would require some form of knowledge concerning the degradation function in (6.1.2)

1. In some cases, the physical phenomenon underlying the degradation can be used to determine -- examples such as degradation caused in photography by relative motion between camera and the scene.

2. In other situations it may be possible to determine from the degrad picture itself. eg. when it is known a priori that a certain portion of the degraded picture is the image of a point, line or edge in the original pictures. eg., will be discussed.

We also consider the a-priori information about noise that is needed for restoration.