Quadratic-constrained Least-squares Research Articles

Optimal apodization design for medical ultrasound using constrained least squares part II simulation results

In the first part of this work, we introduced a novel general ultrasound apodization design method using constrained least squares (CLS). The technique allows for the design of system spatial impulse responses with narrow mainlobes and low sidelobes. In the linear constrained least squares (LCLS) formulation, the energy of the point spread function (PSF) outside a certain mainlobe boundary was minimized while maintaining a peak gain at the focus. In the quadratic constrained least squares (QCLS) formulation, the energy of the PSF outside a certain boundary was minimized, and the energy of the PSF inside the boundary was held constant. In this paper, we present simulation results that demonstrate the application of the CLS methods to obtain optimal system responses. We investigate the stability of the CLS apodization design methods with respect to errors in the assumed wave propagation speed. We also present simulation results that implement the CLS design techniques to improve cystic resolution. According to novel performance metrics, our apodization profiles improve cystic resolution by 3 dB to 10 dB over conventional apodizations such as the flat, Hamming, and Nuttall windows. We also show results using the CLS techniques to improve conventional depth of field (DOF).

Efficient biorthogonal cosine-modulated filter banks

Biorthogonal modulated filter banks, when compared to paraunitary ones, provide the advantage that the overall system delay can be chosen independently of the filter length, thus allowing to design low delay filter banks. They have recently been studied by several authors. In this paper, we connect two different design methods, namely the quadratic constrained least-squares optimization and the principle of cascading sparse self-inverse matrices. Moreover, we show how factorizations into zero-delay and maximum-delay matrices can be utilized in order to achieve desirable features such as structure-inherent perfect reconstruction, no DC leakage of the filter bank, and a low implementation cost.

Efficient design method of cosine-modulated QMF banks satisfying PR property

In this paper the design problem of perfect-reconstruction cosine-modulated QMF banks has been formulated as a quadratic-constrained least-squares (QCLS) minimization problem in which all constrained matrices of the QCLS optimization problem are symmetric and positive definite. A cost function which is a convex function of desired prototype filter coefficients is constructed so that this kind of QCLS optimization problems can be efficiently solved. So a global minimizer of this problem can be easily obtained. Results of two design examples are presented to support the derivations and analyses.

An efficient design method of cosine-modulated QMF banks satisfying PR property

In this paper the design problem of perfect-reconstruction cosine-modulated QMF banks has been formulated as a quadratic-constrained least-squares (QCLS) minimization problem in which all constrained matrices of the QCLS optimization problem are symmetric and positive definite. A cost function which is a convex function of desired prototype filter coefficients is constructed so that this kind of QCLS optimization problems can be efficiently solved. So a global minimizer of this problem can be easily obtained. Results of two design examples are presented to support the derivations and analyses.

Neural networks design approach for cosine-modulated FIR filter banks and compactly supported wavelets with almost PR property

The design problem of paraunitary cosine-modulated FIR filter banks satisfying the perfect-reconstruction (PR) property has been formulated as a quadratic-constrained least-squares (QCLS) minimization problem in which all constraint matrices of the QCLS optimization problem are symmetric and positive definite. In this paper, we propose a neural network to efficiently solve this QCLS optimization problem in real time, whose energy function coincides with the objective function of the above QCLS minimization problem. A detailed analysis of the proposed network shows that the coefficient set of the designed prototype filter is just the output of the network at the minimizer of energy function of the network and, furthermore, that all its minimizers are global minimizers. Additional regularity conditions, which are imposed on the filter bank to obtain the cosine-modulated orthonormal bases of compactly supported wavelets, are transformed into the same formulation as PR condition, i.e. the constraint matrices for regular conditions are also symmetric and positive definite. So a similar method can be used to construct cosine-modulated wavelets. Compared to other design methods, the proposed network has many desirable features such as fast convergence, random initialization, global minimizer to be obtained, high stopband attenuation and so on. Results of computer simulation are presented to support our derivations and analyses.

The theory and design of arbitrary-length cosine-modulated filter banks and wavelets, satisfying perfect reconstruction

It is well known that FIR filter banks that satisfy the perfect-reconstruction (PR) property can be obtained by cosine modulation of a linear-phase prototype filter of length N=2mM, where M is the number of channels. In this paper, we present a PR cosine-modulated filter bank where the length of the prototype filter is arbitrary. The design is formulated as a quadratic-constrained least-squares optimization problem, where the optimized parameters are the prototype filter coefficients. Additional regularity conditions are imposed on the filter bank to obtain the cosine-modulated orthonormal bases of compactly supported wavelets. Design examples are given.

Design of low-delay two-channel FIR filter banks using constrained optimization

Two approaches for the design of two-channel perfect reconstruction FIR filter banks with short reconstruction delays are presented. The approaches are based on constrained optimization. In the first approach, a low-order filter is first designed and the objective function of the filter bank is formulated as a quadratic programming problem with linear constraints. Then the Lagrange-multiplier method is used to design a higher-order filter. The method is simple, efficient, flexible, and an exact solution is obtained by solving a set of linear equations. The second approach can be used to design filters of equal as well as unequal lengths. In this approach, the design problem is formulated as a quadratic-constrained least-squares minimization problem which can be solved using standard minimization algorithms. Design examples are given to illustrate the advantages of the proposed approaches. The quality of reconstruction is considered very good and superior to those of existing methods.

Digital filter bank design quadratic-constrained formulation

Formulate the filter bank design problem as an quadratic-constrained least-squares minimization problem. The solution of the minimization problem converges very quickly since the cost function as well as the constraints are quadratic functions with respect to the unknown parameters. The formulations of the perfect-reconstruction cosine-modulated filter bank, of the near-perfect-reconstruction pseudo-QMF bank, and of the two-channel biorthogonal linear-phase filter bank are derived using the proposed approach. Compared with other design methods, the proposed technique yields PR filter banks with much higher stopband attenuation. The proposed technique can also be extended to design multidimensional filter banks. >