# What is the difference between image restoration and image reconstruction?

I am new to image processing. I don't know whether this is the right place to ask, but what is the difference between image restoration and image reconstruction?

The use might depend on authors. From one of my colleagues, Jean-Christophe Pesquet, supported by the book Image Reconstruction: Algorithms and Analysis by Fessler, especially Chapter 1: Image Restoration, one can say that:

There is not universal terminology for image recovery problems.

In the past, it was common to distinguish between:

• image enhancement : a subjective process, aiming at a good-looking result,
• image restoration : a more objective process, using a given, known or assumed, degradation model.

What seems more or less common (I am open to discussion) is the following, with notations $x$ (data), $n$ (noise), $\bigstar$ (a pixel-by-pixel operation, like a sum or a product), $H$ (linear blur), $\mathcal{N}$ (nonlinear operator):

• denoising: only noise, no blur, a simple case of restoration (e.g. $x\bigstar n$);
• deconvolution: (sometimes) only convolutive blur (assuming negligible noise),
• restoration: blur and noise, a concept close to deconvolution, when the degrading operator is a convolution, or at least locally (admitting space-varying linear kernels), with some noise (additive or not). It may encompass the (e.g. $Hx\bigstar n$);
• reconstruction: when the measured data cannot be directly interpreted as an image, like a sinogram in tomography, or extended to the case when the degrading operator is not a convolution, for instance a sigmoid that emulates a saturation (e.g. $\mathcal{N}(x)$);
• recovery: restoration+reconstruction (e.g. $\mathcal{N}(Hx)\bigstar n$).

You may notice that I left the order of operators very vague, on purpose, as practices differ, but usually the noise is "combined" at the end, after the other operators.

Sometimes, "deconvolution" is used in place of "restoration" (as above, linear blur and noise), and "restoration" may include generic degradations $y=\mathcal{D}(x)\bigstar n$), where $x$ and $y$ belong to the same domain ($\mathbb{R}^2$ or $\mathbb{N}^2$), while keeping "reconstruction" for $x$ and $y$ in different domains.

The introduction of this paper explains the difference and gives an example. In short:

Image restoration techniques presume that data are acquired in the image space; that is, the raw data represent a corrupted version of the image scene. In contrast, images are not directly observed in reconstruction problems. Instead, projections of an image are obtained from two or more angles, and the image scene is pieced together from these projections.