Michailovich Oleg, Adam Dan
Department of Bio-Medical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel.
IEEE Trans Med Imaging. 2004 Jan;23(1):7-25. doi: 10.1109/TMI.2003.819932.
In most approaches to the problem of two-dimensional homomorphic deconvolution of ultrasound images, the estimation of a corresponding point-spread function (PSF) is necessarily the first stage in the process of image restoration. This estimation is usually performed in the Fourier domain by either successive or simultaneous estimation of the amplitude and phase of the Fourier transform (FT) of the PSE This paper addresses the problem of recovering the FT-phase of the PSF, which is an important reconstruction problem by itself. The purpose of this paper is twofold. First, it provides a theoretical framework, establishing that the FT-phase of the PSF can be effectively estimated by a proper smoothing of the FT-phase of the appropriate radio-frequency (RF) image. Second, it presents a novel approach to the estimation of the FT-phase of the PSF, by solving a continuous Poisson equation over a predefined smooth subspace, in contrast to the discrete Poisson equation solver used for the classical least mean squares phase unwrapping algorithms, followed by a smoothing procedure. The proposed approach is possible due to the distinct properties of the FT-phases, among which the most important property is the availability of precise values of their partial derivatives. This property overcomes the main disadvantage of the discrete schemes, which routinely use wrapped (principal) values of the phase in order to approximate its partial derivatives. Since such an approximation is feasible subject to the restriction that the partial phase differences do not exceed pi in absolute value, the discrete schemes perform satisfactory only for few practical situations. The proposed approach is shown to be independent of this restriction and, thus, it performs for a wider class of the phases with significantly lower errors. The main advantages of the novel method over the algorithms based on discrete schemes are demonstrated in a series of computer simulations and for in vivo measurements.
在大多数解决超声图像二维同态反卷积问题的方法中，估计相应的点扩散函数（PSF）必然是图像恢复过程的第一阶段。这种估计通常在傅里叶域中通过对PSF的傅里叶变换（FT）的幅度和相位进行逐次或同时估计来执行。本文解决了恢复PSF的FT相位的问题，这本身就是一个重要的重建问题。本文的目的有两个。首先，它提供了一个理论框架，证明通过对适当的射频（RF）图像的FT相位进行适当的平滑处理，可以有效地估计PSF的FT相位。其次，它提出了一种估计PSF的FT相位的新方法，即通过在预定义的平滑子空间上求解连续泊松方程，这与用于经典最小均方相位解缠算法的离散泊松方程求解器不同，随后进行平滑处理。由于FT相位的独特性质，所提出的方法是可行的，其中最重要的性质是其偏导数精确值的可用性。该性质克服了离散方案的主要缺点，离散方案通常使用相位的包裹（主）值来近似其偏导数。由于这种近似在偏相位差绝对值不超过π的限制下是可行的，所以离散方案仅在少数实际情况下表现令人满意。所提出的方法被证明不受此限制，因此，它适用于更广泛的相位类别，误差显著更低。在一系列计算机模拟和体内测量中证明了该新方法相对于基于离散方案的算法的主要优点。
