Orthogonalized Adaptive Gabor Basis for Energy Compaction in Two-Dimensional Block Transforms

Author :

DR. NANIA Francesco Antonio, Italy, Terme V.re (ME)

mail : miadebora@hotmai.lit

I. Academic Preface and Transform Rationale

This manuscript formally derives the mathematical foundation of an optimized N×N block transform, designated the Orthogonalized Gabor Block Transform (OGBT). The use of Gabor functions in signal processing is motivated by their optimal joint spatial-frequency localization [1]. The central objective is the synthesis of an orthonormal basis that exploits these characteristics, overcoming the limitations of the naturally non-orthogonal Gabor family [2]. By systematically enforcing orthogonality via the Gram-Schmidt procedure, we maximize the energy compaction ratio of the transform coefficients, a critical requirement for high-efficiency data compression schemes [3].

The analysis is based on a block size N=8, leveraging the parameters σ=4.0 (Gabor window variance) and fmax=1.0 (maximum normalized frequency).

II. Mathematical Synthesis of the Transform Basis

Let B be the image block (B∈RN×N), and let T be the resulting one-dimensional (1D) orthonormal transform matrix (T∈RN×N).

A. The Gaussian Localization Window

The normalized Gaussian window samples w[k] are given by:

(Formula 1 - Finestra Gaussiana)

where k∈{0,…,N−1} and c=(N−1)/2.

B. Initial Gabor Basis Generation

The AC basis vectors, for n∈{1,…,N−1}, are formally defined as:

(Formula 2 - Componenti AC della Base)

where the DC component is T[0,k]=1/sqrt(N).

C. Orthogonalization via Gram-Schmidt

The normalized vector tiortho is derived iteratively for i=1,…,N−1, using the Modified Gram-Schmidt procedure [4].

(Formula 3 - Gram-Schmidt)

III. Two-Dimensional Transform Formulation

The 2D OGBT operates on the block B using the separable approach.

A. Forward Two-Dimensional OGBT

The coefficient matrix C is obtained through the double application of the 1D transform:

(Formula 4 - Trasformata Diretta Matrice)

The generic coefficient C[i,j] is expressed via the double summation:

(Formula 5 - Trasformata Diretta Somma)

B. Inverse Two-Dimensional OGBT

Due to the orthonormality property (T−1=TT), the inverse transform, which reconstructs the block R, is:

(Formula 6 - Trasformata Inversa Matrice)

The generic reconstructed element R[i,j] is formally written as:

(Formula 7 - Trasformata Inversa Somma)

This property ensures perfect reconstruction in the absence of quantization, adhering to Parseval's theorem for energy conservation [3].

IV. References

[1] D. Gabor, "Theory of communication," Journal of the Institution of Electrical Engineers - Part III: Radio and Communication Engineering, vol. 93, no. 26, pp. 429-457, Nov. 1946.

[2] H. G. Feichtinger and T. Strohmer, Gabor Analysis and Algorithms: Theory and Applications. Boston, MA, USA: Birkhäuser, 1998.

[3] A. K. Jain, Fundamentals of Digital Image Processing. Englewood Cliffs, NJ, USA: Prentice-Hall, 1989.

[4] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed. Baltimore, MD, USA: Johns Hopkins University Press, 2013.