Overview of Transforming Between Image Domains

Some processes performed on an image in the spatial domain may be very computationally expensive. These same processes may be significantly easier to perform after transforming an image to a different domain. These transformations are the basis for many image filters, applied to remove noise, to sharpen, or extract features. Domain transformations also provide additional information about an image and can offer compression benefits.

The most common representation of a pixel's value and location is spatial, where it appears in three dimensions (x, y, and z). Pixel value and location in this space is usually referred to by column (x), row (y), and value (z), and is known as the spatial domain. However, a pixel's value and location can be represented in other domains.

In the frequency or Fourier domain, the value and location are represented by sinusoidal relationships that depend upon the frequency of a pixel occurring within an image. In this domain, pixel location is represented by its x- and y-frequencies and its value is represented by an amplitude. Images can be transformed into the frequency domain to determine which pixels contain more important information and whether repeating patterns occur. See Transforming Between Domains with FFT for more information on the frequency domain.

In the time-frequency or wavelet domain, the value and location are represented by sinusoidal relationships that only partially transform the image into the frequency domain. Like the transformation to the full frequency domain, the transformation to the time-frequency domain helps to determine the important information in an image. See Transforming Between Domains with Wavelets for more information on the time-frequency domain.

In the Hough domain, pixels are presented by sinusoidal lines. Since straight lines within an image are transformed into the Hough domain as intersecting sinusoidal lines, these intersections can be used to determine if and where straight lines occur within an image. See Transforming to and from the Hough and Radon Domains for more information on the Hough domain.

In the Radon domain, a line of pixels occurring in an image is represented by a single point. This transformation is useful for detecting specific features and image compression. Since transforming images to and from the Hough and Radon domains use similar methods, the Radon image representation is described in the same section as the Hough representation. See Transforming to and from the Hough and Radon Domains for more information on the Radon domain.

Note
In this book, Direct Graphics examples are provided by default. Object Graphics examples are provided in cases where significantly different methods are required.

The following list introduces the image domain transformations and associated IDL image transformation routines covered in this chapter.

Table 7-1: Image Transformation Tasks and Related Routines
Task	Routine(s)	Description
Transforming Between Domains with FFT	FFT	Transform images into the frequency domain and back into the spatial domain with the Fast Fourier Transform. Then show how to use this process to remove noise from an image.
Transforming Between Domains with Wavelets	WTN	Transform images into the time-frequency domain and back into the spatial domain with the Wavelet transform. Then show how to use this process to remove noise from an image.
Transforming to and from the Hough and Radon Domains	HOUGH RADON	Transform images into the Hough and the Radon domains and back into the spatial domain with the Hough and Radon transforms. Then show how to use these processes to detect straight lines and improve contrast within an image.

Note
This chapter uses data files from the IDL examples/data directory. Two files, data.txt and index.txt, contain descriptions of the files, including array sizes.