Complex Cepstrum Research Articles

Noise robust one-dimensional blind deconvolution of medical ultrasound images

Recently, several blind cepstral deconvolution methods for medical ultrasound images were compared experimentally. The results indicated that the generalized cepstrum or the complex cepstrum with phase unwrapping give the blind homomorphic deconvolution algorithms with the best performance. However, the frequency domain phase unwrapping for pulse estimation, which is an essential part of both methods, is sensitive to the sensor noise when the values of the spectrum are small due to the randomness of the tissue response. The noise introduces abrupt changes in the phase. The phase degradation due to the noise causes variable spatial and gray scale resolution in image sequences following deconvolution. This paper introduces a noise robust Bayesian phase unwrapping method using a noncausal Markov random chain model. The prior regularizing term accounts for the noise and smoothes the phase. The phase unwrapping is formulated as a least mean square optimization problem. The optimization is done noniteratively by solving a difference equation using the cosine transform. The resulting improvement in radial and lateral blind deconvolution is demonstrated on six short ultrasound image sequences recorded in vitro or in vivo.

Complex cepstral filtering of images and echo removal in the radon domain

Homomorphic filtering using the complex cepstrum has been used in many applications of one-dimensional (1D) signal processing, notably speech and seismic signal analysis. Some work has been done in extending cepstral filtering to the 2D domain. An application that we are particularly interested in is 2D echo removal. We show in this paper that 1D cepstral filtering techniques may be applied to the projections of images in the Radon domain. The filtered projections may then be used to reconstruct an image without the echoes. Results with synthetic images are presented to illustrate the strengths of the method. The technique could find applications in analysis of textured images and images obtained via reconstruction from projections.

Hybrid higher-order cepstrum and functional link network-based blind equaliser (HOCFLN)

A new hybrid higher-order cepstrum (HOC) and functional link network (FLN)-based blind equaliser (HOCFLN) is presented. The system initially uses the complex cepstrum of the 1-D slice of the fourth-order cumulants of the unknown received signal to partially estimate the equaliser coefficients, then it switches to an FLN adaptive equaliser operating in the decision directed mode (DDM) to further improve the mean-squared error (MSE) convergence. In this system two nonlinearities are used; one on the input data where the HOC is used and the other one in the FLN equaliser filter. It is shown that in the new HOCFLN system multiple nonlinearities give significant performance improvement with less computational complexity, particularly in severe channel distortion compared to the conventional equalisation algorithms. This method can accommodate both nonminimum phase MA and ARMA channels. Performance results for channels exhibiting abrupt characteristic changes are also shown.

Pseudo Complex Cepstrum Using Discrete Cosine Transform

Two new algorithms are proposed, which obtain pseudo complex cepstrum using Discrete Cosine Transform (DCT). We call this as the Discrete Cosine Transformed Cepstrum (DCTC). In the first algorithm, we apply the relation between Discrete Fourier Transform (DFT) and DCT. Computing the complex cepstrum using Fourier transform needs the unwrapped phase. The calculation of the unwrapped phase is difficult whenever multiple zeros and poles occur near or on the unit circle. Since DCT is a real function, its phase can only be 0 or π and the phase is unwrapped by representing the negative sign by exp (−jπ) and the positive sign by exp (j0) . The second algorithm obviates the need for DFT and obtains DCTC by representing the DCT sequence itself by magnitude and phase components. Phase is unwrapped in the same way as the first algorithm. We have tested DCTC on a simulated system that has multiple poles and zeros near or on the unit circle. The results show that DCTC matches the theoretical complex cepstrum more closely than the DFT based complex cepstrum. We have explored possible uses for DCTC in obtaining the pitch contour of syllables, words and sentences. It is shown that the spectral envelope obtained from the first few coefficients matches reasonably with the envelope of the signal spectrum under consideration, and thus can be used in applications, where faithful reproduction of the spectral envelope is not critical. We also examine the utility of DCTC as feature set for speaker identification. The identification rate with DCTC as feature vector was higher than that with linear prediction-derived cepstral coefficients.

Comparison of cepstrum-based methods for radial blind deconvolution of ultrasound images

This paper compares the performance of seven different cepstrum-based methods for radial blind deconvolution of medical ultrasound images. The first is the generalized cepstrum method. The second is the spectral root cepstrum method. These methods have received little attention so far. The last five methods are all based on the complex cepstrum, but different computational techniques in the spatial and frequency domain are employed. Using in vivo radio frequency data from a clinical scanner, the generalized cepstrum method gave the best images closely followed by the complex cepstrum using phase unwrapping or polynomial rooting. The complex cepstrum method using higher-order statistics was ranked as low as number five. These results are an important guideline for selecting a specific cepstrum-based radial deconvolution method for implementation in ultrasound scanners.

Performance of blind equalization with higher order statistics in indoor radio environments

This paper analyzes the performance of blind equalization using the complex cepstrum of third-order moments applied to 4-QAM time division multiple access (TDMA) indoor radio communication systems. In particular, we have modeled a dispersive indoor channel with Rice statistics. We used the blind algorithms to estimate the channel-impulse response, and from this, we computed the equalizer coefficients using a classical minimum mean square error (MMSE) algorithm. In order to evaluate the system performance, we calculated the bit error rate (BER) of a decision feedback equalizer (DFE) that uses a tricepstrum algorithm to estimate the channel-impulse response. The results are compared with those obtained using a least sum of square errors (LSSE) algorithm as a channel estimator and considering the exact channel response. The results obtained show that this kind of blind equalizer performs better than the more classically trained equalizer when Rice channels with a strong direct path and signal-to-noise ratios (SNRs) lower than 20 dB are taken into account. However, some problems relating to the length of time needed for convergence must be solved.

Synthesis of IIR digital filters exhibiting simultaneous amplitude and phase responses for VLSI implementations

A method is described for the design and implementation, in a double-lattice form, of selective linear phase digital filters. It is shown that these filters are optimum in the sense that all available degrees of freedom are used in adjusting the filter parameters to meet the specified characteristics. The designed filters, apart from satisfying the linear phase requirement, exhibit equiripple loss response over both the pass- and stopbands. A novel computationally efficient method based on computing the complex cepstrum is proposed for the construction of orthogonal double-lattice structures that realize the designed filters. It is based on the important property of cepstrum functions, namely decomposing mixed sequences into both minimum and maximum phase sequences. Illustrative examples are given to support the theoretical analysis. © 1997 John Wiley & Sons, Ltd.

Restoration of medical ultrasound images using two-dimensional homomorphic deconvolution

Describes how two-dimensional (2D) homomorphic deconvolution can be used to improve the lateral and radial resolution of medical ultrasound images recorded by a sector scanner. The recorded radio frequency ultrasound image in polar coordinates is considered as a 2D sequence of angle and depth convolved with a 2D space invariant point-spread function (PSF). Each polar coordinate sequence is transformed into the 2D complex cepstrum domain using the fast Fourier transform for Cartesian coordinates. The low-angle and low-depth portion of this sequence is taken as an estimate of the complex cepstrum representation of the PSF. It is transformed back to the Fourier frequency domain and is used to compute the deconvolved angle and depth sequence by 2D Wiener filtering. Two-dimensional homomorphic deconvolution produced substantial improvement in the resolution of B-mode images of a tissue-mimicking phantom in vitro and of several human tissues in vivo. It was better than lateral or radial homomorphic deconvolution alone, and better than 2D Wiener filtering with a PSF recorded in vitro. >

A multidimensional isomorphic operator and its properties-a proposal of finite-extent multidimensional cepstrum

We propose a new multidimensional homomorphic operator that replaces the conventional complex cepstrum transformation. We treat multidimensional signals of finite support since any signal we can actually observe and deal with is of finite support. We first show that for any sequence of finite support there exists a coordinate transformation that transforms the support of a given sequence into the first quadrant in the multidimensional signal space. We then propose a new multidimensional homomorphic operator /spl Psi/ which transforms a sequence of finite support into another sequence of finite support in the first quadrant. It is proved that the operator /spl Psi/ is an isomorphism between two multidimensional signal spaces of finite support where finite convolution and usual addition, respectively, are defined as binomial operations. It is also shown that unlike the conventional complex cepstrum, the proposed operator /spl Psi/ is quite simple to compute and requires no complicated procedure like phase unwrapping, while it maintains the special features of the conventional complex cepstrum transformation that are useful in homomorphic signal processing. Moreover we clarify some algebraic structure of the multidimensional signal space with the finite convolution as a binomial operation. Finally it is shown by a numerical example that the deconvolution system using the proposed operator /spl Psi/ gives a much better result than the conventional complex cepstrum method. >

Nonminimum phase channel deconvolution using the complex cepstrum of the cyclic autocorrelation

New discrete time blind deconvolution methods are proposed for nonminimum phase linear channels driven by cyclo-stationary inputs. The methods rely exclusively on second-order statistics and do not impose any constraints on the distribution of the channel input as in the case of methods based on higher-order statistics. The output of the channel is fractionally sampled and then the complex cepstrum of the cyclic autocorrelation is obtained. It is shown that this complex cepstrum preserves nonminimum phase information and thus the identification of nonminimum phase channels is possible. Practical constraints in the implementation of the methods and channel identifiability conditions are discussed. The applicability of the methods to both channel identification and fractionally spaced linear and DFE equalization is described and verified by means of computer simulations. >

The complex cepstrum applied to two-dimensional images

The purpose of this paper is to provide an analytical understanding of the two-dimensional (2D) complex cepstrum and to apply the cepstrum in image processing. The analytical cepstrum is first developed in one dimension for the single echo case. This is followed by a more valuable analytical derivation for multiple echoes in one and two dimensions. The 2D, digital implementation of the forward and inverse cepstrum is then described. Three key applications are presented for 2D images: echo removal, automatic intensity control, and enhancement of repetitive features. A significant recommendation for future research is to investigate optimal filtering techniques based on the described analytical results.

The cross-bicepstrum: definition, properties, and application for simultaneous reconstruction of three nonminimum phase signals

The recovery of three signals from their cross-bispectrum (or the identification of the impulse responses of three parallel linear time-invariant (LTI) systems from the cross-bispectrum of their system outputs) by computing the complex cepstrum of the cross-bispectrum, as long as the signals (or impulse responses) have no zeros on the unit circle, is discussed. It is shown that the three signals can be separated completely and (approximately) recovered in the cross-bicepstrum domain, except for their magnitude and linear phase factors. The computation of the cross-bicepstrum can be seen as a method for the simultaneous computation of the ordinary complex cepstra of three nonminimum-phase signals without the need for phase unwrapping. Both least-squares and fast Fourier-transform (FFT)-based methods for computing the bicepstral coefficients are presented. Simulation examples of signal reconstruction in Gaussian white and nonGaussian colored noise and of system identification are included. The results are extended to nth-order cross-spectra, and the factorization problem for these spectra is discussed. >

Time-domain cepstral transformations

The realization of homomorphic systems for convolution is addressed, motivated by the limitations of the Fourier-transform-based method commonly used in homomorphic (complex cepstral) filtering. The time-domain cepstral transformation (TDCT) method, which is entirely based on time-domain calculations, thus avoiding or minimizing the problems associated with the FT method, is introduced. Explicit transformations of an ordinary mixed-phase time sequence (belonging to convolution space) into its complex cepstrum time sequence (belonging to additive cepstrum vector space) and vice versa are derived. The method does not require unwrapped phase calculations, and no specific windows are used to precondition the signal in order to produce a more accurate representation of the complex cepstrum. It trades reduced computational efficiency for improved performance. Examples comparing Fourier-based and TD cepstrum transformations are presented. >

System identification for the ECG using CZT

A new approach for extraction of clinically useful parameters from the ECG signal is presented using the system identification technique of CZT on the DCT-transformed signal. A one to one relationship between the model singularities and the significant points in the time signal is arrived at. The method allows the determination of the R-R interval needed in rhythm analysis. The complex cepstrum is used for identifying and removing the effect of zeros outside the unit circle. A significant data compression of 1 in 10 is achieved. A large number of continuous strips of ECG data are analyzed and the results are presented.

Removal of spurious reflections from computational fluid dynamic solutions with the complex cepstrum

The complex cepstrum is shown to remove spurious reflections from artificial boundaries in computational fluid dynamic (CFD) solutions. First, the complex cepstrum theory is presented. A time sequence consisting of a direct signal and reflections is analyzed theoretically with the complex cepstrum, and it is shown that the direct signal uncontaminated by reflections may be recovered in the time domain. Next, the complex cepstrum is applied to one- and three-dimensional CFD solutions, and spurious reflections from the boundary conditions are removed. By eliminating spurious reflections introduced by artificial boundary conditions, the applicability of CFD methods to aeroacoustic problems is enhanced greatly.

Identification of nonminimum phase MA systems using cepstral operations on slices of higher order spectra

Recursive and least-squares methods for reconstructing the impulse response of a nonminimum phase, linear time-invariant (NMP-LTI) MA system from its (k+1)st-order spectrum (k>1) are presented. The methods are nonparametric. They are based on computing the complex cepstrum of the impulse response sequences of the unknown system from the diagonal slice of the (k+1)st-order spectrum. The computational complexity of the system identification problem can be considerably reduced by using slices of higher-order spectra. >

A new method of pitch modification by complex cepstrum analysis-synthesis

This paper proposes a new method of high-quality pitch modification by complex cepstrum analysis-synthesis algorithm (hereafter referred to as the CCAS algorithm). The algorithm of the new conversion method presented here is as follows: the complex cepstra obtained within each pitch-synchronized frame are divided into pitch components and other spectral components by using comb filters. The pitch pulses obtained by inverse transformation of the pitch components are equalized adaptively to zerophase, while the spectral components also are transformed inversely into unit responses in time-domain. This equalization makes the pitch components more impulsive, resulting in clear separation of the voice source from the vocal tract characteristics. Speech synthesis is performed by convoluting the phase-adjusted pitch pulses with the nit responses. The results of listening tests showed that the pitchmodified speech by the CCAS method subjectively retains as high quality as the original speech.

Design of zero-phase spherically symmetric N-dimensional IIR digital filters

The paper presents a new method for designing zero-phase N-dimensional IIR digital filters satisfying given amplitude design specifications. The method is based on applying the N-dimensional extension of McClellan transformation to the square of a one-dimensional IIR digital filter design recently proposed by Charalambous. The resulting N-dimensional digital filter is of zero-phase and hyperspherically symmetric but unstable. Stabilisation of the filter is achieved by dividing it into a product of 2N filters, each of which is stable if realised through a recursion in the suitable quadrant. The stabilisation procedure uses the properties of the complex cepstrum. Three examples are introduced to show the performance and validity of the design method presented. The designs obtained are also tested in image processing applications.

Combined linear and homomorphic deconvolutions for processing bandpass measurements (US flaw detection)

The deconvolution presented is a combination of available techniques. It is effective for extracting the impulse response of individual flaws from ultrasonic data generated by flaw clusters. The technique requires output signals from both the flawed and unflawed systems. By combining linear deconvolution and homomorphic deconvolution, using the complex cepstrum, the desired impulse response can be extracted in cases where the signals contain overlapping wavelets. For the important case of bandpass signals, for which the complex cepstrum does not exist, the technique can still be applied by including a bandpass mapping in conjunction with homomorphic processing. Application of this combined approach to ultrasonic signals reflected from epoxy specimens produced the impulse response of a single cavity from measurements which were bandpass and contained overlapping signals from adjacent cavities. The recovered response, which compared well with the theoretical response of a single cavity, was sufficiently resolved to yield the radius of the spherical cavity.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

CAS-CAS transform for 2-D signals: A few applications

A few applications of a separable Hartley-like (CAS-CAS) transform in two-dimensional (2-D) signal processing is presented. The applications discussed include (i) the interpolation of signals, (ii) the computation of Hilbert transform, and (iii) the complex cepstrum computation. The computational advantage of the proposed methods over the algorithms using 2-D FFT are discussed.