Comparison of Image Transforms

Lecture

Signal transforms can be ranked by the coding gain they achieve (Fig. 1). Coding gain is the degree to which the variances of the transform coefficients are redistributed. The unit impulse basis expansion gives the smallest gain, and the Karhunen-Loève transform (KLT) gives the largest. Transforms with high coding gain (the discrete cosine transform, DCT, and the wavelet transform) are characterized by a sharply uneven distribution of the variances of the subband coefficients. Low-frequency subbands are unsuitable for a steganographic coding algorithm because most of the information load is concentrated in them, and high-frequency subbands are unsuitable because of the high processing noise. Therefore, mid-frequency bands must be used, in which the influence of these two factors is approximately equal. Since there are few such bands, the capacity of the steganographic channel is low.

Fig. 1

The Hadamard transform is based on a square Hadamard matrix, whose elements are equal to plus or minus one and whose rows and columns form orthogonal vectors. A normalized Hadamard matrix of order Comparison of Image Transforms satisfies the relation . Among the orthonormal Hadamard matrices, the smallest is the second-order matrix .

The wavelet transform is an integral transform that consists of the convolution of a wavelet function with a signal. The wavelet transform converts a signal from a time representation into a time-frequency one. It is a way of transforming a function (or signal) into a form that either makes certain quantities of the original signal easier to study or allows the original data set to be compressed. The wavelet transform of signals is a generalization of spectral analysis. The term "wavelet" means a "small wave". Wavelets is the general name for mathematical functions of a certain shape that are localized in both time and frequency and in which all the functions are obtained from a single base function by modifying it (shifting and stretching it).

The Haar wavelet is one of the first and simplest wavelets. It is based on an orthogonal system of functions proposed by the Hungarian mathematician Alfréd Haar in 1909. Haar wavelets are orthogonal and have compact support and good spatial localization, but they are not smooth. Later, Ingrid Daubechies developed the theory of orthogonal wavelets and proposed using functions computed iteratively, which came to be called Daubechies wavelets. Without loss of quality, it compresses the size of an image file that has long runs of identical brightness values. It preserves the main features of the image by lowering its resolution and substantially reducing the data when the detail matrices are discarded.

The discrete cosine transform (DCT) is one of the orthogonal transforms. It is a variant of the cosine transform for a vector of real numbers. It is used in lossy information compression algorithms, for example MPEG and JPEG. This transform is closely related to the discrete Fourier transform and is a homomorphism of its vector space. Mathematically, the transform can be carried out by multiplying a vector by a transform matrix. The inverse transform matrix is equal, up to a factor, to the transposed matrix. In mathematics, the matrices are chosen so that the transform is orthonormal and the constant factor equals one. In computer applications this is not always the case. Different periodic extensions of the signal lead to different types of DCT.

The Karhunen-Loève transform (KLT). The method of transforming continuous signals into a set of uncorrelated coefficients was developed by Karhunen and Loève. As stated in article [30], Hotelling [29] was the first to propose a method for transforming discrete signals into a set of uncorrelated coefficients. However, in most works on digital signal processing, both the discrete and the continuous transforms are called the Karhunen-Loève transform or the eigenvector expansion.

When transforms with a low coding gain (Hadamard or Fourier) are used, there are more mid-frequency subbands. Consequently, the capacity is also higher, so it is better to use transforms with smaller coding gains, which are poorly suited to signal compression. One of the transforms that allows compression-resistant embedding of multimedia data is the fast wavelet transform (FWT). Using wavelet transforms in steganographic coding methods can solve the main tasks that were set: minimizing the distortions introduced and resisting attacks by a passive adversary.

Comparison of Image Transforms

All these transforms give a more compact representation in terms of energy than the original image.

"Energy compaction" means that most of the information content is placed in a small part of the representation.

Representation	Compaction	and / but ...
Image	Poor	Easy to interpret
Fourier	Good	Convolution theorem
Cosine	Better	Fast
Hotelling	Best	Basis functions depend on the signal
Wavelets	Good	Some spatial representation

Comparison of Image Transforms

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See also

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Lectures and tutorial on "Digital image processing"

Terms: Digital image processing