This correspondence discusses an extension of the well-known phase correlation technique to cover translation, rotation, and scaling. Fourier scaling properties and Fourier rotational properties are used to find scale and rotational movement. The phase correlation technique determines the translational movement. This method shows excellent robustness against random noise.

An FFT-based technique for translation, rotation, and scale-invariant image registration

Signals and Systems, Alexander D. Poularikas, University of Alabama in Huntsville Fourier Transforms, Kenneth B. Howell, University of Alabama in Huntsville Sine and Cosine Transforms, Pat Yip, McMaster University, Ontario The Hartley Transform, Kraig J. Olejniczak, University of Arkansas, Fayetteville Laplace Transforms, Samuel Seely (deceased), Westbrook, Connecticut The Z-Transform, Alexander D. Poularikas, University of Alabama in Huntsville Hilbert Transforms, Stefan L. Hahn, Warsaw University of Technology Radon and Abel Transforms, Stanley Deans, University of South Florida The Hankel Transform, Robert Piessens , Katholieke Universieit Leuven, Belgium Wavelet Transform, Yunlong Sheng, Laval University, Quebec The Mellin Transform, Jacqueline Bertrand, Pierre Bertrand, Universite de Paris VII, and Jean-Philippe Ovarlez, ONERA/DES, France Mixed Time-Frequency Signal Transformations, G. Faye Boudreaux-Bartels, University of Rhode Island Fractional Fourier Transforms, Dr. Mustafa Abushagur, University of Alabama in Huntsvill, Dr. Ahmed M. Almanasrah, Photronix, Malaysia The Lapped Transforms, Ricardo L. de Queiroz, Xerox Corporation The Discrete Time and the Discrete Transforms, Alexander D. Poularikas, University of Alabama in Huntsville Discrete Time and the Discrete Transforms, Alexander D. Poularikas, University of Alabama in Huntsville Appendices, Alexander D. Poularikas, University of Alabama in Huntsville Index

The transforms and applications handbook

We propose a new algorithm for the accurate detection and localization of copy–move forgeries, based on rotation-invariant features computed densely on the image. Dense-field techniques proposed in the literature guarantee a superior performance with respect to their keypoint-based counterparts, at the price of a much higher processing time, mostly due to the feature matching phase. To overcome this limitation, we resort here to a fast approximate nearest-neighbor search algorithm, PatchMatch, especially suited for the computation of dense fields over images. We adapt the matching algorithm to deal efficiently with invariant features, so as to achieve higher robustness with respect to rotations and scale changes. Moreover, leveraging on the smoothness of the output field, we implement a simplified and reliable postprocessing procedure. The experimental analysis, conducted on databases available online, proves the proposed technique to be at least as accurate, generally more robust, and typically much faster than the state-of-the-art dense-field references.

Efficient Dense-Field Copy–Move Forgery Detection

https://hal.archives-ouvertes.fr/hal-00325095/document

Robust and Efficient Fourier-Mellin Transform Approximations for Gray-Level Image Reconstruction and Complete Invariant Description

This work presents a new method for estimating planar roto-translations that operates in the frequency domain and as such, is not based on features. Since the proposed technique uses all the image information, it is very robust against noise, it can be very accurate; estimation errors on the rotational angle varies from a few hundredths to a few tenths of a degree, depending on the noise level. Experimental evidence of this performance is presented, and the mathematical reasons behind these characteristics are explained in depth. Another remarkable feature of the algorithm consists in that it works in Cartesian coordinates, bypassing the need to transform from the Cartesian to the polar domain, which, typically, is a numerically delicate and computationally onerous task. The proposed technique can become an effective tool for unsupervised estimation of roto-translations by means of implementations based on FFT algorithms.

A noise-robust frequency domain technique for estimating planar roto-translations

A modified Mellin transform for digital implementation is developed and applied to range radar profiles of naval vessels. The scale invariance property of the Mellin transform provides a means for extracting features from the profiles which are insensitive to the aspect angle of the radar. Past implementations of the Mellin transform based on the FFT have required exponential sampling, interpolation, and the computation of a correction term, all of which introduce errors into the transform. In addition, exponential sampling results in a factor of ln N increase in the number of data points. An alternate implementation, developed in the paper, utilizes a direct expansion of the Mellin integral definition. This direct Mellin transform (DMT) eliminates the implementation problems associated with the FFT approach, and does not increase the number of samples. A scale and translation invariant transform is developed from a modification of the DMT. The MDMT applied to the FFT of the radar profiles results in the desired insensitivity without having the low-pass filtering characteristic that exists in other Fourier-Mellin implementations.

Imre Kiss

Papers

A New Implementation of the Mellin Transform and its Application to Radar Classification of Ships