The Discrete Cosine Transform

Abstract

Abstract. Each discrete cosine transform (DCT) uses N real basis vectors whose components are cosines. In the DCT-4, for example, the jth component of v k is cos(j + 1 2. These basis vectors are orthogonal and the transform is extremely useful in image processing. If the vector x gives the intensities along a row of pixels, its cosine series c k v k has the coefficients c k = (x, v k )/N . They are quickly computed from a Fast Fourier Transform. But a direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are.We prove orthogonality in a different way. Each DCT basis contains the eigenvectors of a symmetric "second difference" matrix. By varying the boundary conditions we get the established transforms DCT-1 through DCT-4. Other combinations lead to four additional cosine transforms. The type of boundary condition (Dirichlet or Neumann, centered at a meshpoint or a midpoint) determines the applications that are appropriate for each transform. The centering also determines the period: N − 1 or N in the established transforms, N − in the other four. The key point is that all these "eigenvectors of cosines" come from simple and familiar matrices.

Interpreting principal component analyses of spatial population genetic variation

“…Mathematically, the explanation for the highly structured patterns is that covariance matrices from spatial data have eigenvectors related to sine waves of increasing frequency [17,18,19], and PC-maps are direct visual representations of such eigenvectors (see Supplemental Material). To give three specific examples, consider a situation where the covariance between two populations depends only on the geographic distance between them, and assume that sufficient genetic data (loci/alleles) are available to accurately estimate this covariance structure.…”

“…S3B), and the eigenvectors of any (large) Toeplitz matrix are known to be closely approximated by sinusoidal functions [19]. A wellstudied special case occurs when the covariance between populations decays exponentially with distance 2 , where the eigenvectors are approximately the columns of the discrete cosine transform (DCT) matrix [17,18].…”

“…fig. S3B), with eigenvectors that are approximated by the twodimensional DCT commonly used in image compression 4 [18]. The first two eigenvectors are commonly two orthogonal gradients, and the next two have a "saddle" and a "mound" shape ( fig.…”

Native Development Kit and Software Development Kit Comparison for Android Applications

“…The second application consisted in performing the discrete cosine transform of a 16 x 16 matrix. These matrices were loaded with random numbers of double type, commonly used in the compression of data and images (Lam 1998 andStrang 1999).…”

Histopathological Breast-Image Classification Using Local and Frequency Domains by Convolutional Neural Network

“…Strang first introduced the DCT method in 1974 [37]. A few DCT methods are available, and among them DCT-II methods have been largely utilized for image analysis.…”