Tai-Chiu Hsung

The discrete periodic Radon transform

In this correspondence, a discrete periodic Radon transform and its inversion are developed. The new discrete periodic Radon transform possesses many properties similar to the continuous Radon transform such as the Fourier slice theorem and the convolution property, etc. With the convolution property, a 2-D circular convolution can be decomposed into 1-D circular convolutions, hence improving the computational efficiency. Based on the proposed discrete periodic Radon transform, we further develop the inversion formula using the discrete Fourier slice theorem. It is interesting to note that the inverse transform is multiplication free. This important characteristic not only enables fast inversion but also eliminates the finite-word-length error that may be generated in performing the multiplications.

Optimizing the multiwavelet shrinkage denoising

Denoising methods based on wavelet domain thresholding or shrinkage have been found to be effective. Recent studies reveal that multivariate shrinkage on multiwavelet transform coefficients further improves the traditional wavelet methods. It is because multiwavelet transform, with appropriate initialization, provides better representation of signals so that their difference from noise can be clearly identified. We consider the multiwavelet denoising by using multivariate shrinkage function. We first suggest a simple second-order orthogonal prefilter design method for applying multiwavelet of higher multiplicities. We then study the corresponding thresholds selection using Steins unbiased risk estimator (SURE) for each resolution level provided that we know the noise structure. Simulation results show that higher multiplicity wavelets usually give better denoising results and the proposed threshold estimator suggests good indication for optimal thresholds.

Orthogonal discrete periodic Radon transform: part I: Theory and realization

The discrete periodic Radon transform (DPRT) was proposed recently. It was shown that DPRT possesses many useful properties that are similar to the conventional continuous Radon transform. Using these properties, a 2-D signal can be processed by some 1-D approaches to reduce the computational complexity. However, the non-orthogonal structure of DPRT projections introduces redundant operations that often lower the efficiency of the technique in applications. In this paper, we propose the orthogonal discrete periodic Radon transform (ODPRT) in which a new decomposition approach is introduced. All ODPRT projections are modified to be orthogonal such that redundancy is eliminated. Furthermore, we consider the efficient realization for computing ODPRT and its inverse that make the proposed ODPRT more feasible in practical applications.

Efficient blind image restoration using discrete periodic Radon transform

Restoring an image from its convolution with an unknown blur function is a well-known ill-posed problem in image processing. Many approaches have been proposed to solve the problem and they have shown to have good performance in identifying the blur function and restoring the original image. However, in actual implementation, various problems incurred due to the large data size and long computational time of these approaches are undesirable even with the current computing machines. In this paper, an efficient algorithm is proposed for blind image restoration based on the discrete periodic Radon transform (DPRT). With DPRT, the original two-dimensional blind image restoration problem is converted into one-dimensional ones, which greatly reduces the memory size and computational time required. Experimental results show that the resulting approach is faster in almost an order of magnitude as compared with the traditional approach, while the quality of the restored image is similar.

Orthogonal discrete periodic Radon transform: part II: applications

In this paper, we study the properties and possible applications of the newly proposed orthogonal discrete periodic Radon transform (ODPRT). Similar to its previous version, the new ODPRT also possesses the useful properties such as the discrete Fourier slice theorem and the circular convolution property. They enable us to convert a 2-D application into some 1-D ones such that the computational complexity is greatly reduced. Two examples of using ODPRT in the realization of 2-D circular convolution and blind image resolution are illustrated. With the fast ODPRT algorithm, efficient realization of 2-D circular convolution is achieved. For the realization of blind image restoration, we convert the 2-D problem into some 1-D ones that reduces the computation time and memory requirement. Besides, ODPRT adds more constraints to the restoration problem in the transform domain that makes the restoration solution better. Significant improvement is obtained in each case when comparing with the traditional approaches in terms of quality and computation complexity. They illustrate the potentially widespread applications of the proposed technique.

Wavelet based speech presence probability estimator for speech enhancement

A reliable speech presence probability (SPP) estimator is important to many frequency domain speech enhancement algorithms. It is known that a good estimate of SPP can be obtained by having a smooth a-posteriori signal to noise ratio (SNR) function, which can be achieved by reducing the noise variance when estimating the speech power spectrum. Recently, the wavelet denoising with multitaper spectrum (MTS) estimation technique was suggested for such purpose. However, traditional approaches directly make use of the wavelet shrinkage denoiser which has not been fully optimized for denoising the MTS of noisy speech signals. In this paper, we firstly propose a two-stage wavelet denoising algorithm for estimating the speech power spectrum. First, we apply the wavelet transform to the periodogram of a noisy speech signal. Using the resulting wavelet coefficients, an oracle is developed to indicate the approximate locations of the noise floor in the periodogram. Second, we make use of the oracle developed in stage 1 to selectively remove the wavelet coefficients of the noise floor in the log MTS of the noisy speech. The wavelet coefficients that remained are then used to reconstruct a denoised MTS and in turn generate a smooth a-posteriori SNR function. To adapt to the enhanced a-posteriori SNR function, we further propose a new method to estimate the generalized likelihood ratio (GLR), which is an essential parameter for SPP estimation. Simulation results show that the new SPP estimator outperforms the traditional approaches and enables an improvement in both the quality and intelligibility of the enhanced speeches.

Efficient fringe image enhancement based on dual-tree complex wavelet transform

In optical phase shift profilometry (PSP), parallel fringe patterns are projected onto an object and the deformed fringes are captured using a digital camera. It is of particular interest in real time three-dimensional (3D) modeling applications because it enables 3D reconstruction using just a few image captures. When using this approach in a real life environment, however, the noise in the captured images can greatly affect the quality of the reconstructed 3D model. In this paper, a new image enhancement algorithm based on the oriented two-dimenional dual-tree complex wavelet transform (DT-CWT) is proposed for denoising the captured fringe images. The proposed algorithm makes use of the special analytic property of DT-CWT to obtain a sparse representation of the fringe image. Based on the sparse representation, a new iterative regularization procedure is applied for enhancing the noisy fringe image. The new approach introduces an additional preprocessing step to improve the initial guess of the iterative algorithm. Compared with the traditional image enhancement techniques, the proposed algorithm achieves a further improvement of 7.2 dB on average in the signal-to-noise ratio (SNR). When applying the proposed algorithm to optical PSP, the new approach enables the reconstruction of 3D models with improved accuracy from 6 to 20 dB in the SNR over the traditional approaches if the fringe images are noisy.

On the convolution property of a new discrete Radon transform and its efficient inversion algorithm

In this paper, a new discrete Radon transform (DRT) and the inverse transform algorithm are proposed. The proposed DRT preserves most of the important properties of the continuous Radon transform, for instance, the Fourier Slice theorem, convolution property, etc. With the convolution property, the computation of a two-dimensional (2-D) cyclic convolution can be decomposed as a number of one-dimensional (1-D) ones, hence greatly reduces the computational complexity. Based on the proposed DRT, we further derive the inverse transform algorithm. It is interesting to note that it is a multiplication free algorithm that only additions are required to perform the inversion. This important characteristic not only reduces the complexity in computing the inverse transform, but also eliminates the finite word length error that may be generated in performing the multiplications.

Generalized Discrete Multiwavelet Transform With Embedded Orthogonal Symmetric Prefilter Bank

Prefilters are generally applied to the discrete multiwavelet transform (DMWT) for processing scalar signals. To fully utilize the benefit offered by DMWT, it is important to have the prefilter designed appropriately so as to preserve the important properties of multiwavelets. To this end, we have recently shown that it is possible to have the prefilter designed to be maximally decimated, yet preserve the linear phase and orthogonal properties as well as the approximation power of multiwavelets. However, such design requires the point of symmetry of each channel of the prefilter to match with the scaling functions of the target multiwavelet system. It can be very difficult to find a compatible filter bank structure; and in some cases, such filter structure simply does not exist, e.g., for multiwavelets of multiplicity 2. In this paper, we suggest a new DMWT structure in which the prefilter is combined with the first stage of DMWT. The advantage of the new structure is twofold. First, since the prefiltering stage is embedded into DMWT, the computational complexity can be greatly reduced. Experimental results show that an over 20% saving in arithmetic operations can be achieved comparing with the traditional DMWT realizations. Second, the new structure provides additional design freedom that allows the resulting prefilters to be maximally decimated, orthogonal and symmetric even for multiwavelets of low multiplicity. We evaluated the new DMWT structure in terms of computational complexity, energy compaction ratio as well as the compression performance when applying to a VQ based image coding system. Satisfactory results are obtained in all cases comparing with the traditional approaches.

New sampling scheme for region-of-interest tomography

The wavelet localization technique was previously applied to the study of region-of-interest (ROI) tomography. It achieves a significant saving in the required projections if only a small region of a tomographic image is of interest. In this paper, we first show that with the same sampling scheme, a simple interpolation applied to the samples can give a result at least as good as that using the original wavelet localization approach. It implies that the use of the wavelet transform is not the key to the reduction of the sampling requirement. In fact, the quality of the reconstructed ROI is largely determined by the structure of the sampling scheme. Rather than directly reducing the projection number, the use of the wavelet theory permits a clear understanding of how to achieve a good sampling pattern. Based on an error analysis using the wavelet theory, we further suggest a new sampling scheme such that the number of required projections in each angle is reduced in a multiresolution form. A new multiresolution interpolation algorithm is then used to interpolate the missing samples to obtain the full projections. As a result, more than 84% of projections are saved, as compared with the traditional approach, in reconstructing an ROI of 32/spl times/32 pixels in an image of 256/spl times/256 pixels. A series of simulations was performed to reconstruct different sizes of the ROI. All results show that the signal-to-error ratios of the reconstructed ROI are comparable with that using full projection data set.