We consider Gabor systems generated by a Gaussian function and prove certain classical results of Paley and Wiener on nonharmonic Fourier series of complex exponentials for the Gabor expansion. In particular, we prove a version of Plancherel-Po ́ lya theorem for entire functions with finite order of growth and use the Hadamard factorization theorem to study regularity, exactness and deficiency of Gabor systems.

A single reference image is used for detecting the face area in gray scale images. The algorithm uses Gabor transform as a powerful tool for feature extraction. Zero-crossing of Gabor transform is used for constructing the binary feature vectors and Hamming distance is used for measuring the similarity between feature vectors. Our developed method uses a pyramidal structure for finding the face images with different sizes. The algorithm has considerable robustness to the cluttered backgrounds and non-uniform illumination. More than 11,000 face images have been tested. The results show a very high detection rate for non occluded face images with low facial expression. The algorithm can also be used for correcting the head rotation angle in input images. The algorithm can also be used for correcting the head rotation angle in input images. The proposed method is suitable for the cases that contain one face image.

Hyperspectral anomaly detection is one of the main challenging topics in both military and civilian fields. The spectral information contained in a hyperspectral cube provides a high ability for anomaly detection. In addition, the costly spatial information of adjacent pixels such as texture can also improve the discrimination between anomalous targets and background. Most studies miss the worthful spatial characteristics. Moreover, some works that include the spatial features in the anomaly detection process extract features from each hyperspectral band that is a two dimensional image while the original structure of hyperspectral cube contains three dimensions. Ignoring the nature of hyperspectral image leads to lose the contained spectral-spatial correlations in the hyperspectral cube. To deal with this difficulty, in this work, the fused spectral and spatial features obtained by applying 3D Gabor filters are proposed for hyperspectral anomaly detection. Exploiting the 3D structure of hyperspectral cube by capturing multiple scales, orientations and its spectral-dependent characteristics provides an appropriate spectral-spatial feature space for anomalous targets detection. The extracted features are applied to the regularized RX detector to provide the detection map. The experiments show the superior performance of the proposed Gabor 3D based detector in terms of detection accuracy and computation time

The duals of Gabor frames have an essential role in reconstruction of signals. In this paper we find a necessary and sufficient  condition for two Gabor systems $\left(\chi_{\left[c_1,d_1\right)},a,b\right)$ and $\left(\chi_{\left[c_2,d_2\right)},a,b\right)$ to form dual frames for $L_2\left(\mathbb{R}\right)$, where $a$ and $b$ are positive numbers and $c_1,c_2,d_1$ and $d_2$ are real numbers such that $c_1