Two-dimensional S-transform Research Articles

A region of interest based approach for texture analysis of medical images in space–frequency domain

Abstract The two-dimensional S-transform (ST-2D) is a space–frequency representation (SFR) useful in image processing especially for texture analysis. The high computation time and inefficiency in computing the local spectrum at each pixel in an image motivated the authors to develop the two-dimensional fast time–frequency transform (FTFT-2D). However, in practice, the SFRs at individual pixels are seldom used for analysis. In most medical images, the spectral distribution over a region of interest (ROI) is of higher significance than that at individual pixels. This paper proposes a two-dimensional ROI-based fast time–frequency transform (FTFT-2D-ROI) for computing the average local spectrum and spectral statistics for an ROI faster and more efficiently using three skipping strategies. The spectral statistics for an ROI have been computed using mean and standard deviation of SFR magnitude in both Cartesian and polar coordinate systems. Simulations were performed with rectangular and non-rectangular ROIs while employing the three skipping strategies of FTFT-2D-ROI for a brain MRI image. Moving from skipping strategy I toward III, the computation time of FTFT-2D-ROI decreases significantly but at a cost of some degradation in accuracy of the SFR. The computational performance of the proposed FTFT-2D-ROI is also studied for different sizes of images, and found to be superior than that of the ST-2D. Further, sensitivity analysis was performed to study the impact of different parameters on the performance of the FTFT-2D-ROI. The end user can choose the skipping strategy based on the trade-off between accuracy and computation time.

Two-dimensional S-transform profilometry with an asymmetry Gaussian window

Abstract Two-dimensional S-Transform is an extension of one-dimensional S-Transform. It analyzes the fringe pattern to obtain the useful information from the instantaneous local information in two directions. Due to the application of Gauss Window, time resolution in the time–frequency spectrum is degraded because of the long front taper. In this paper, an asymmetry Gaussian window is introduced in the two-dimensional S-Transform method. With a decreased front taper and an increased back taper for the Gauss window, the proposed two-dimensional S-Transform method can reconstruct the measured object more accurately.

A frequency-based window width optimized two-dimensional S-Transform profilometry

Abstract A new scheme is proposed to as a frequency-based window width optimized two-dimensional S-Transform profilometry, in which parameters p u and p v are introduced to control the width of a two-dimensional Gaussian window. Unlike the standard two-dimensional S-transform using the Gaussian window with window width proportional to the reciprocal local frequency of the tested signal, the size of window width for the optimized two-dimensional S-Transform varies with the p u th ( p v th ) power of the reciprocal local frequency f x ( f y ) in x ( y ) direction. The paper gives a detailed theoretical analysis of optimized two-dimensional S-Transform in fringe analysis as well as the characteristics of the modified Gauss window. Simulations are applied to evaluate the proposed scheme, the results show that the new scheme has better noise reduction ability and can extract phase distribution more precise in comparison with the standard two-dimensional S-transform even though the surface of the measured object varies sharply. Finally, the proposed scheme is demonstrated on three-dimensional surface reconstruction for a complex plastic cat mask to show its effectiveness.

二维S变换在光学三维面形测量中的应用

二维S变换在光学三维面形测量中的应用

Incorporating approximate rotational invariance into two‐dimensional S‐transform

The discrete two-dimensional (2D) S-transform (ST-2D) is a space-frequency representation that provides the local spectrum at each pixel in an image. The inherent localisation properties of the ST-2D make it a preferred candidate for space-frequency analysis. However, the ST-2D is sensitive to the image rotation. This limits its use in many image processing applications such as texture analysis. This study describes the rotational sensitivity of the ST-2D, and explains how this limits its application. To overcome these problems, a novel rotationally invariant S-transform (RIST) is formulated, and its rotational invariance is investigated. The RIST can provide approximate rotational invariance for magnitudes and complex values over a wide range of rotation angles, but with a trade-off of higher computation time. The RIST may be useful for many medical, scientific and industrial applications.

Texture analysis of images using a two-dimensional fast time-frequency transform.

The two-dimensional S-transform (ST-2D) is a time-frequency representation that is widely used in medical image processing but prohibitive in both storage and computation time. The high computation time required for generating local spectrum discourages the use of ST-2D for analyzing textures in medical images. A two-dimensional fast time-frequency transform (FTFT-2D) for computing the local spectrum instantaneously and accurately is proposed. It can also be used to compute the complete redundant discrete ST-2D output, if needed. It reduces the storage requirement by generating a compressed form of the ST-2D. In addition, the memory efficient and adaptive nature of the FTFT-2D make it suitable for user-specific requirements.

A comparison of one and two-dimensional S-Transform in fringe pattern demodulation

Abstarct S-Transform, as a time-frequency analysis approach, has been successfully introduced into fringe pattern demodulation in recent years. The paper presents a thorough discussion on the application of this technique in analyzing fringe pattern. A brief review of the S transform profilometry is followed by the comparison between the one-dimensional S-Transform method and the two-dimensional S-Transform method, including one-dimensional S-Transform ridge method, one-dimensional S-Transform filtering method, two-dimensional S-Transform ridge method and two-dimensional S-Transform filtering method in the application of the phase retrieval from the fringe patterns with different noise, contrast, nonlinearity and fringe breaking. Numerical computer simulations and experiments were carried out to show the 1DST method is sensitive to the noise but suitable to detect detail of the measured object, while 2DST method with capability of smooth processing for the surface of the measured object is adapt to demodulate the fringe pattern with serious noise.

Error caused by sampling in S-Transform Profilometry

S-Transform, combining the advantages of both the windowed Fourier transform and the wavelet transform, is a suitable approach for the phase demodulation of the deformed fringe pattern to obtain the reconstruction of the tested object in optical three-dimensional (3D) surface measurement. In the S-Transform algorithm, the translated spectra are needed to calculate the S-Transform coefficients. When the signal is discrete, its Fourier spectrum is periodically repeated in the frequency domain. That is to say, not only the large height variation rate of the tested object can cause spectrum aliasing in the same frequency “island”, but also the spectrum repetition caused by sampling operation may lead to the spectrum overlapping between adjacent “islands” if the sampling rate is not high enough to satisfy the sampling theory. The paper researches the sampling problem of the S-Transform for guaranteeing its accuracy in fringe analysis. The frequency-domain description of the one-dimensional S-Transform and the two-dimensional S-Transform coefficients of the discrete fringe pattern are deduced. In addition, the criteria for the selection of sampling frequency in the S-Transform Profilometry are derived. Computer simulations and processing results of practical experiments demonstrate our analysis.

Application of two-dimensional S-Transform in fringe pattern analysis

Abstract As a novel approach for extracting phase of the fringe patterns in optical 3D shape measurement, two-dimensional S-Transform was presented. It can be regarded as the extension of one-dimensional S-Transform but possesses better localized time-frequency analysis ability and more effective de-noising capacity because it analyzes the image to obtain the needed parameters by using the instantaneous local information in two directions. In this paper, the theoretical derivation was given and the performance of this approach was verified in both simulation and experimental validation by comparing with the method of one-dimensional S-Transform.