1
Digital Image

An image is a two‐dimensional (2D) light intensity function , where is a coordinate system of interest. Without loss of generality, and to simplify our discussions, the rest of the book will concentrate on the case of 2D Cartesian coordinate system. The value of at the coordinates is proportional to the brightness of the image at that point. While digital images can be generated/acquired by a number of methods, primarily, the image is converted to a digital image through cameras using a 2D image sensor array. These sensors are typically constructed with charge‐coupled devices (CCD) and complementary metal oxide semiconductor (CMOS) technologies. Camera constructed with CCD or CMOS works in a similar fashion, where the light reflected from an object will impinge onto the face of the sensor array, such that each sensor element in the array will generate an electrical signal (for which the coordinate can be considered to be the digitized coordinate of ). Figure 1.1 illustrates the construction of a color digital camera which is used to capture the Sculpture image. The light bounced off the Sculpture will be focused onto the sensor array through the lens. Consider a sensor array with ‐rows and ‐columns, the output of the sensor array will be a matrix with , and . As a result, the arrangement of the image sensor array is also known as the sampling grid, where the intersection of a row and a column will be assigned with an integer coordinate in the discrete Cartesian coordinate system. The output of each sensor element represents the number of photons that react with the sensor at location . The output of the sensor array is not a digital image yet. The subsequent analog‐to‐digital converter (A/D converter) accomplishes the quantization processes of the light intensity at all locations to generate the digital image. The sampled image obtained from the sampling and quantization process, as shown in Figure 1.2(a), is the discrete image which forms a matrix . Each entry in this array, , records the number of photons sensed by the corresponding sensor in the arrays and is termed a pixel. Thus, a digital image obtained by a digital camera will look like

Figure 1.1 Illustration of capturing an image by digital camera.

The values assigned to every pixel are the brightness recorded by the image sensor, which is also interpreted as the pixel intensity (also known as the gray‐level or grayscale).1 To store, transmit, and visualize the discrete image, the pixel intensity of the discrete image will be rounded to the nearest integer value within different gray levels through the quantization process performed within the A/D converter. This process will produce the digital image, which can be visualized as a shade of gray denoted as the grayscale or ‐level value ranging from black (0) to white (), such that the higher the intensity value, the brighter the image pixel. Figure 1.2(b) shows the pixel values of an extract from the image .

Figure 1.2 Representation of the digital image Sculpture : (a) a grayscale printout of Sculpture, which is described by an 2D array within the computation system with each matrix element representing the intensity of a pixel taking a value in the quantizer (in this case, it is [0,255] as Sculpture is an 8‐bit quantized image); (b) a pixel intensity map of the selected region in the image, where the pixel intensity at [62,369] is 211; and the intensity variation across the complete image by viewing (c) the 2D vector mesh of the image on a plane with the height (‐axis) being the pixel intensity or through (d) the contour map, where the pixel with the same intensities are located to the same contour lines.

The discrete image is arranged with each pixel being located at the th row and th column starting from the top‐left image origin (as shown in Figure 1.2(a)) with respect to the MATLAB convention. For simplification, we shall also use the vector to represent the pixel location, such that . Now, the readers may have already noticed from Figure 1.2(a) that the matrix indices in the figure are different from those in Equation 1.1. This is one of the irritating features of MATLAB. Notwithstanding the similarity between the arithmetic and the language of MATLAB, all matrices within MATLAB are indexed with the top left‐hand entry as instead of , and hence the discrepancy between Figure 1.2(a) and Equation 1.1. The rest of the book will assume this difference to be natural and will no longer discuss the difference between the MATLAB implementation and the analytical analysis with respect to the indexing problem.

1.1 Color Image

As pointed out by Sir Isaac Newton, color is perceived by the mind to resolve the interaction of light sources, objects, and the visual system, which adds a subjective layer on top of the underlying objective physical properties – the wavelength of the electromagnetic radiation carried by color signal. The color signal is received by light‐sensitive cells in human eye. Hering's experimental results and the discovery of three different types of photosensitive molecules in human eyes [52] led us to the modern color perception theory, where color is perceived through a luminance (grayscale) and two chrominance (color) components. This is the basis of trichromacy, the ability to match any color with a mixture of three suitably chosen primaries. The basic principle of color additivity has led to a number of useful trichromatic descriptions of color, which is also known as the color space.

Among various color spaces, the RGB, and the YCrCb are the most popular. In particular, the RGB color space has been widely employed in digital cameras and monitors to capture and display digital color images. This is because the RGB space conveniently corresponds to the three primary colors which are mixed for display on a monitor or similar devices. A digital color image in the RGB space is similar to a digital monochrome (grayscale) image, except that it requires a three‐dimensional vector to represent each pixel, and thus three arrays are required to represent the whole image. Each of these array represents one of the RED, GREEN, and BLUE primitive color components. The RED, GREEN, and BLUE components of an RGB image can be viewed separately as a monochrome image by considering the corresponding array alone, as shown in Figure 1.3. When the three color components are superposed, it produces the rightmost color image in Figure 1.3. As a result, if each component image is encoded with the data type uint8 in MATLAB, the total number of bits required to represent each pixel will be 8 bits  3 = 24 bits. This is also the default representation adopted by MATLAB for the three color triplets, and such type of image is known as the True Color image. Disregarding the digital color image format, the MATLAB function imread can be used to import the image directly from the image file stored in the hard disk, as shown in Listing 1.1.1.

Figure 1.3 Three separate RED, GREEN, and BLUE channels are combined to create a final, full‐color image.

Although only grayscale image denoising algorithms are discussed in this book, the algorithms can be easily extended to color images by treating the spectral components of the color images as independent grayscale images. Actually the grayscale image contains a lot of information, and this is the reason why black‐and‐white television receivers have been perfectly acceptable to the public for many years, and black‐and‐white photographs are still popular. Nevertheless, color is an important property, and so we shall examine its role in this section.

1.1.1 Color Filter Array and Demosaicing

To capture a digital image in color, three sensors with each sensor measuring one of the three colors, respectively, are required to capture the RED, GREEN, and BLUE component images. A cheaper alternative to the three‐sensors camera system is to have one sensor only. In this case, each photo sensor in the sensor array is made to be sensitive to one of the three colors (ranges of wavelengths). This can be done in a number of different ways. A popular method in modern camera is to cover the photo sensor array with a Bayer pattern color filter array (CFA) [3], as shown in Figure 1.1. Besides the Bayer pattern CFA, the readers may have also noticed that there is a color demosaicing block by the end of the camera in Figure 1.1. These two modules are essential in capturing images with color. To be more precise, Bayer pattern CFA is a typical construction of CFA, which is commonly applied to the photo sensor arrays in modern cameras. The filter is arranged in Bayer pattern which is a combination of RED, GREEN, and BLUE filters in checkerboard format. The size of the CFA is identical to that of the sensor array and each color filter has a narrow passband, and will only allow the light component with the same color tone as that of the filter to pass through. Therefore, each pixel in the “digitized color filtered...