Image transformation

Geometric transformations of images in the spatial domain

Translation transformation

Construct a translation transformation matrix and then apply this matrix to (matrix left-multiply) each pixel of the original image to achieve the effect of image translation.

Rotation transformation

In general, the rotation operation will have a center of rotation, which is usually the center of the image, and the size of the image will usually change after the rotation.

Scaling transformation

The position of each pixel in an image can be thought of as a point or as a vector on a two-dimensional plane. Image scaling is, essentially, scaling the per-pixel vector

Image transformation in the frequency domain

Fourier transform

The Fourier transform is a linear integral transform used to transform signals between the time and frequency domains and has many applications in physics and engineering. It is named in honor of the French scholar Joseph Fourier because its basic idea was first systematically proposed by him. In fact the Fourier transform is like determining the basic composition of a substance by chemical analysis; signals come from nature and can also be analyzed to determine their basic composition.

Similar to a dichroic prism in optics that divides white light into different colors by frequency, the Fourier transform divides a signal into different frequency components and is known as a mathematical prism. The effect of the Fourier transform is shown in Figure 2-20. Corresponding to the digital image, the high-frequency signals are often the edge signals and noise signals in the image, while the low-frequency signals contain signals such as the image contour and background.

Discrete cosine transform

The Discrete Cosine Transformation (DCT) is a transform related to the Fourier Transform, which is similar to the Discrete Fourier Transform but uses only real numbers. The discrete cosine transform is equivalent to a discrete Fourier transform that is roughly twice as long as a real even function (since the Fourier transform of a real even function is still a real even function), and in some transformations it is necessary to shift the position of the inputs or outputs by a half-unit (there are eight standard types of DCTs, four of which are common).

The discrete cosine transform, especially its second type, is often used by signal processing and image processing for lossy data compression of signals and images, both still and moving images. This is due to the strong “energy concentration” property of the discrete cosine transform: most of the energy of natural signals (including sound and images) is concentrated in the low-frequency part of the discrete cosine transform, and the decorrelation of the discrete cosine transform is close to that of the K-L transform when the signal has statistical properties close to those of a Markov process -It has the optimal decorrelation performance.

Your task

Choose your favorite image transformation method and try to transform an image from your phone.

Share your thoughts and ideas in the comments below.