Digital images
Digital images are representations of visual information encoded as binary data, requiring decoding to be viewed on screens as a grid of pixels. The origins of digital images trace back to the early 1960s, driven by developments in fields like medicine and satellite imagery, particularly during significant historical contexts such as the Cold War. The introduction of microprocessors in the 1970s led to advanced imaging technologies, including computerized axial tomography (CAT) scans, which incorporated complex mathematical principles into medical imaging. By the mid-1990s, digital cameras began to replace analog film, with innovations like charge-coupled devices (CCDs) revolutionizing photography.
Digital images can be categorized into bitmap graphics, where each pixel is defined numerically, and vector graphics, which use mathematical equations to describe images. Resolution in digital images is determined by the number of pixels present, impacting both display quality and file size. Various file formats exist for storing bitmap graphics, with compression techniques allowing for file size reduction, sometimes at the cost of image accuracy. Additionally, image reconstruction methods are employed in medical imaging to estimate internal structures from limited data, showcasing the intricate relationship between mathematics and digital imaging across multiple disciplines.
Digital images
Summary: Digital images are recorded as a binary account of pixels, which algorithms may compress.
Digital images are not images at all but rather are visual information encoded as binary data. Viewing a digital image requires a computer to decode binary information and display it on a screen in the form of an array of discrete lights called “picture elements” or “pixels.” The first computer-generated digital images were produced in the early 1960s. The needs of the Cold War, medicine, and the space race drove many developments in digital imagery, some of which were achieved in the context of projects on satellite imagery, medical imaging, optical character recognition, and photo enhancement. The advent of microprocessors in the 1970s and advances in digital storage and display technologies made possible sophisticated imaging tools, like computerized axial tomography (CAT scanning).
The degree of mathematical sophistication that CAT scans introduced into medical imaging, such as integral geometry, optimal sampling, and transport equations, was unheard of at that time. It is reflected in further advances such as magnetic resonance imaging as well as developments in other fields that use similar imaging techniques, like seismic and electron microscopy. At the same time, scanners to digitize analog images began to be used in a diverse array of fields, such as archaeology and law enforcement. The first fully digital camera was released in 1995, and by the end of the twentieth century, charge-coupled devices (CCDs) largely displaced analog film and tape for photography and videography. Willard Boyle and George Smith shared the 2009 Nobel Prize in Physics for their invention of the CCD, an idea they first brainstormed at Bell Labs in 1969. Improved computing power also allowed for production of near-photorealistic images. All areas of digital imagery (creation, compression, restoration, recognition, and display) involve mathematics. In the twenty-first century, digital images are regularly used in both mathematics research and teaching.
Bitmap Graphics
In most digital images, each pixel has been defined numerically and this number has been converted into a string of “1”s or “0”s. This system is the approach of “bitmap graphics” (also known as “raster graphics”), and it is how digital cameras work. Depending on the number of bits used to represent each pixel, more or less color information is given. For example, a one-bit system would allow only a black or white pixel, as the only choices would be a “0” or a “1.” A two-bit system would gives four choices per pixel, “00” (black), “01” (dark grey), “10” (light grey), and “11” (white). Typically, in photo editing programs of the early twenty-first century, each pixel is described by 24 bits of information, yielding more than 16 million possible colors.
Resolution
Bitmap images contain information for a given number of pixels. The larger the pixel number, the more information is in the image and the higher the resolution; typically, this also results in a bigger file. Screens are all made of pixels, whether they are on computers or cell phones; if an image is viewed at full size, each pixel in the image will show up as one pixel on the screen. However, if a viewer zooms in beyond this point, the pixels in the file are actually represented by big blocks of pixels on the screen, and the image is said to become “pixelated.”
Thus, if an image is to be viewed on a screen, it will ideally have the same number of pixels as the size one wants it on the screen; any more than that is wasted file space, and any fewer will result in an image that appears pixelated. If images are going to be printed, however, more pixels will translate into sharper pictures, limited only by the resolution of the printer. Again, the larger the print, the more pixels you will need for a sharp print.
File Types and Compression
Bitmap graphics can be stored in a variety of file formats depending on how they will be used. Raw files, which store all the raw data for the light that hits each CCD pixel, are commonly used by photographers who wish to have maximum flexibility and are not worried about file size. In order to make files smaller, computers use mathematical algorithms to compress the files. For example, instead of recording values for each pixel, the values for some could be calculated by the difference between a pixel and its surroundings, thus yielding substantial file size savings where blocks of pixels are the same as their neighbors. Some kinds of compression are considered “lossless,” because all the information from the original can be re-created when the file is decompressed. However, there are a number of compression schemes such as the popular jpeg format in which the mathematical approximations do not quite match the original. In these cases, accuracy is sacrificed in order to save file size, and these approaches are said to be “lossy.” However, the algorithms used to compress and decompress files are generally so good as to be unnoticeable in many cases. The JPEG 2000 image compression standard for both lossless and lossy compression uses biorthogonal wavelets, which extends from the work of mathematician Ingrid Daubechies, known as the “mother of wavelets.”
Vector Graphics
Certain kinds of images, especially those created in computer graphics programs, use a different method for describing the content of the image. Instead of denoting each pixel with a number, these vector graphics are described mathematically as a set of equations representing the lines and curves that make them up. When a viewer zooms in on a vector graphic, the image does not become pixelated, because the computer recalculates the curve or line based on the new image size. While vector graphics are not appropriate for photographs, photo editing programs may use them when overlaying text or graphics on a digital image.
Image Reconstruction
The basic problem of image reconstruction is to build a “best-guess” object out of averaged data and then estimate how close the reconstruction is to the actual object. For example, in a single-angle X-ray of a person, the amount of radiation going in and coming out the other side can be measured and visualized on X-ray film. The difference between the values is how much was absorbed, but there is limited information about the inner structures that blocked the radiation. This limitation can make diagnoses difficult. However, if the same person is X-rayed from several directions and angles, the resulting information can be compiled, averaged, or mathematically modeled to estimate what the internal structure looks like.
Bibliography
Alsina, Claudi. Math Made Visual: Creating Images for Understanding Mathematics. Washington, DC: The Mathematical Association of America, 2006.
Hoggar, S. G. Mathematics of Digital Images: Creation, Compression, Restoration, Recognition. Cambridge, UK: Cambridge University Press, 2006.