Difficulties of Digital Image Restoration

1. The ill-posed nature of image restoration

The classical linear inverse problem formulation of image restoration has the form

                                                                      y = Ax + n
where y denotes the observed data, x is the clean image to be estimated, A is the observation matrix, and n is noise. Generally speaking, these restoration problems are ill-posed, i.e., there is no solution, or the solution is not unique, because A is not invertible or the computation of its inverse is prone to large numerical errors.

2. lack of knowledge

Even though we are facing the challenge of ill-posed nature of image restoration, we have found a way to deal with this ill-posed problem. That is to introduce a regularizing term incorporating prior knowledge of the imaging process. Prior knowledge is the knowledge we already has before we meet the observed image. For example, in a natural image, it is easy to find some similar patches from different locations (Fig. 1). This property is called self-similarity, which has been applied in the state-of-art denoising algorithms to remove the most of noise in images.

Fig. 1. self-similarity property of a natural image.

However in realistic imaging scenario, images are corrupted due to complex factors, for instance, the combination of optical blur and mixed noise. We need prior knowledge for each process step.  Fig. 2 breaks down the three key areas that benefit from prior knowledge in the identification, estimation, and restoration processes. These are knowledge of the degradation, knowledge about the original image, and the knowledge about the noise.

Fig. 2. Piro knowledege used in restoration processes [1].

Image restoration has benefited a lot from prior knowledge, but still needs more prior knowledge. Researchers are now attempting to improve the models used in restoration by incorporating better prior knowledge into the problem. But the fact is that we still have not understood the imaging process and corrupting process completely. We are still lack of prior knowledge of images.

3. incomplete model

 It has been shown that digital restoration may fail when incomplete system models are used. For example, most current denoising approaches are assuming that noise in the image are additive Gaussian and independent and identically distributed.  But this is a simplified observation model. In real case, imaging process is usually influenced not only by Gaussian noise, but also by Poissionian noise. Also, linear imaging model is widely adopted for restoration problems. However, linear model is clearly not sufficient to model the complex imaging process in real situation.

We are likely to obtain better image restoration though making use of more complete models. However, these complete models require more complex than classical approaches to solve it, which brings another challenging to image restoration.

4. Cross-channel correlations and heavy time consume in multichannel image

Fig. 3. Thirteen measurements from Ultraviolet (UV) to Infrared (IR) of the image [2].

In image restoration, multichannel Images (such as color image, multispectral image, and hyperspectral image) present a unique difficulty in that the multiple channels are related. Thus, cross-channel correlations need to be exploited in order to achieve optimal restoration results. Furthermore, multichannel sometimes results in heavy computation. For example, in multispectral image and hyperspectral image, the number of channels can be hundreds and thousands. Obviously, the process of these multichannel images needs much more time than that single channel images do. This time consuming process disable multichannel images from real time application, such as earthquake emergency monitoring and forest fire monitoring.

References:
[1] Banham, M. R., and Aggelos K. K. "Digital image restoration." IEEE signal processing magazine 14.2 (1997): 24-41.
[2] http://www.lumiere-technology.com/Pages/Services/services3.htm