What is the image in the mathematical sense?

The structure of image data for image processing applications

Digital images usually are comprised of a set of dots called pixels which are the smallest discernable detail in a digital image. These pixels in digital cameras are received by sensors. For a pixel in the image the sensor measures the number of photons and turns them into an electrical signal which is finally represented as a number after some process in the camera such as quantization.  For grayscale images, this process leads to one number for each pixel, and for color images it may lead to three numbers which represents red, green and blue values of the corresponding pixel.  So, for each pixel, there is number which roughly represents the number of photons received by the sensor of camera. Usually in the digital image processing tasks, we deal with these set of numbers and often represent them as a matrix. The matrix is two dimensional for grayscale images and three dimensional for color images. The figure below shows an example of how a 4*4 part of an image (which we call it a patch hereafter) can be represented mathematically in a color image.

Displaying a color image patch data stored in a 4*4*3 matrix

The effect of quantization leads to a set of discrete possible numbers for a pixel value. For example, in a stored 8-bit grayscale image the value of each pixel is an integer between 0 and 255 in which 0 means the pure black color and 255 represents the pure white color.

Why is it important to research this topic?

Digital Image Restoration plays a vital role in almost all modern scientific branch. To better understand the importance of the topic, we glance over some of them. Successful applications of image restoration concepts are found in the field of astronomy, medical imaging, defence, biology, industry and many other areas.

Astronomy

The field of digital image restoration has a very long history starting from the space programmes in early 1950s. The images of the celestial objects such as Earth, Moon, and Mars, captured under big technical difficulties, were degraded and are of very poor resolution. Motivated from the need to retrieve information from these poor quality image, people began to think about two dimensional signal processing algorithms, which in turn results in a new field "Digital Image Restoration"

The launching of the $2000 million Hubble Space Telescope (HST) in 1990, was an important event in the history of astronomy where Digital Image Restoration played an unavoidable role. The mirror of the telescope had a serious problem of spherical aberration as it was polished and checked with faulty devices which lead to wrong curvature! A substantial amount of work had to be done by the image restoration experts to correct the aberrant HST images.  Over the decades the restoration techniques and algorithms are improved enormously. Nowadays, Digital Image Restoration is widely used in almost all astronomical observations.

Raw image of planet Saturn obtained with the WF/PC camera of the Hubble Space Telescope. 

Restored image of Saturn using Richardson-Lucy algorithm (Don’t bother about the algorithm now: D). 

Medical Imaging System

A very important emerging application of Digital Image Restoration is Medical Imaging. Modern medical diagnostics are based on the images captured by the medical imaging system such as Magnetic Resonance Imaging (MRI), Computer Assisted Tomography (CAT/CT), Positron Emission Tomography (PET), Single Photon Emission Computed Tomography (SPECT). As in any other imaging system, due to the physical mechanisms of these acquisition systems, the acquired images are often affected by certain degradation. It can be either a due to noise or blur. Image restoration is essential in medical imaging applications in order to enhance and recover anatomical details that may be hidden in the data and plays an important role in improving their usability and extending the application of medical imaging devices.

MRI Restoration [2]

Remote sensing

"Remote sensing is the acquisition of information about an object or phenomenon without making physical contact with the object and thus in contrast to on-site observation" [1] . Capturing the images of Earth or a particular terrain by satellite or high-flying aircraft to extract some information is a typical example. Remote sensing is used in numerous fields, including geography and most Earth Science disciplines (for example, hydrology, ecology, oceanography, glaciology, geology); it also has military, intelligence, commercial, economic, planning, and humanitarian applications [1].

"During the image acquisition process of the remote sensing camera, aberration of the optical system, performance of CCD sensors, motion of the satellite platform and atmospheric turbulence will cause image degradation. The degradation results in image blur, affecting identification and extraction of the useful information in the images. The cost to develop a remote sensing camera is huge, thus the degradation phenomenon of the acquired images causes serious economic loss. Therefore, restoring the degraded images is an urgent task in order to expand uses of the image which signifies the role of Digital Image Restoration in remote sensing" [5].

©2009 Google – Imagery c©2009 Digital Globe, Sanborn, Cnes/Spot Image GeoEye, U.S. Geological Survey

Restoration of hyperspectral image [4]

We saw the importance of digital image restoration in astronomy, medical imaging system and remote sensing. But the list does not end here. You might have understood by now that in any imaging system, the captured images are prone to degradation which can be caused from the instruments used for acquisition or can be due to the motion of the instruments/object or many other reasons. This reminds us the significance of the field Digital Image Restoration

References

[1] https://en.wikipedia.org/wiki/Remote_sensing

[2] R. Molina, J. Nunez, F. J. Cortijo and J. Mateos, "Image restoration in astronomy: a Bayesian perspective," in IEEE Signal Processing Magazine, vol. 18, no. 2, pp. 11-29, Mar 2001.

[3] José V. Manjón, José Carbonell-Caballero, Juan J. Lull, Gracián García-Martí, Luís Martí-Bonmatí, Montserrat Robles, MRI denoising using Non-Local Means, Medical Image Analysis, Volume 12, Issue 4, August   2008, Pages 514-523, ISSN 1361-8415

[4] Shen, H.; Zhao, W.; Yuan, Q.; Zhang, L. Blind Restoration of Remote Sensing Images by a Combination of Automatic Knife-Edge Detection and Alternating Minimization. Remote Sens. 2014, 6, 7491-7521.

[5] Lihong Yang and Jianyue Ren, "Remote sensing image restoration using estimated point spread function," 2010 International Conference on Information, Networking and Automation (ICINA), Kunming, 2010, pp. V1-48-V1-52.