Rendering

Rendering is a process that takes an input a set of objects and produces as its output an array of pixels (image) each of which stores information about the color of the image at a particular point in a grid (determined by the target width and height)

An image can be abstracted as a function

\[ I(x,y): R \rightarrow V \]

Where \(R \in \mathbb{R}^2\) is a rectangular area and \(V\) is a set with the possible pixel values, the following are examples of the set \(V\)

• \(V = \mathbb{R}^+\) (non-negative reals) for grayscale images, each pixels represents only brightness and no color
• \(V = (\mathbb{R}^+)^3\) (combinations of 3 sets of non-negative reals), which is a color image with red/green/blue values for each pixel

Pixels

A pixel from a camera or scanner is a measurement of the average color of the image in the surrounding area near the pixel

If an image has \(n_x\) columns and \(n_y\) rows a common convention is to count rows and columns from the bottom left, the bottom left pixel is \((0,0)\) and the top-right is pixel \((n_x - 1, n_y - 1)\)

Note that because of the definition gave to a pixel the coordinate \((0,0)\) is mapped to the center of the pixel \((0,0)\), therefore half-pixel will exist in both the \(-\mathbf{x}\)-axis and the \(-\mathbf{y}\)-axis

So the domain of a \(n_x \times n_y\) image is

\[ R = [0.5, n_x - 0.5] \times [0.5, n_y - 0.5] \]

Pixel values

The value of a pixel depends on the precision and range of value needed, for example high dynamic range (HDR) images store floating-point numbers allowing a wide range of values, low dynamic range (LDR) images are instead stored with integers, the following pixel-values are used in a variety of applications

• 1-bit grayscale per pixel - images where intermediate grays are not needed e.g. text
• 8-bit grayscale per pixel - images with intermediate grays, it can store a total of 256 gray values e.g. a grayscale photo
• 8-bit red, green and blue (RGB), 24-bits per pixel - full color images that allow near 16 million possible values, e.g. consumer photographs, web and email applications
• 12- to 14-bit RGB, 36-42 bits per pixel - raw camera images for professional photography
• 16-bit half precision RGB, 48 bits per pixel - HDR images used in real time rendering
• 32-bit floating-point RGB, 96 bits per pixel - HDR images for software rendering