Matrix Inverse -- from Wolfram MathWorld
in which case X is unique and called the generalized inverse with prescribed range T and null space T,S. It guarantees that there is a relationship between the. The pseudoinverse defined by Rao did not satisfy all the restrictions imposed by . matrices and * denotes the conjugate transpose of a matrix, then relation (). In case of an inexact field the relation should hold up to a threshold. In case of two chain complexes, these relation should hold for each pair of pseudo inverse .
An improvement of the SR1 procedure, which is derived on the basis of the specific structure of the underlying Toeplitz matrix, is presented in the same section. An application of introduced methods in image restoration is presented in Section 5. Preliminaries and Motivation Toeplitz matrices or diagonally constant matrices are matrices having constant diagonal entries.
Toeplitz matrices which are applicable in the image restoration process contain nonzero main diagonal parallels above the main diagonal, where defines the blurring process. In what follows, let us consider the Toeplitz matrix of such form: The assumption is active.
To clarify notation, Toeplitz matrices of the general form 3 will be denoted shortly by We investigate the use of the SR1 update method, as described in [ 15Algorithm 2], during the numerical computation of the Moore-Penrose inverse of Toeplitz matrices satisfying the pattern.
Also, we examine different improvements of the original method. The improvements are based on appropriate adaptations of the SR1 method and the S-M formula to the characteristic structure of underlying matrices of type. The method of SR1 updates is based on the expression which computes the Moore-Penrose inverse of the first columns of the initial matrix using the Moore-Penrose inverse of its first columns.
LAPACK/ScaLAPACK Development • View topic - pseudo inverse
In detail, the SR1 method from [ 15 ] starts from the well-known representation of the Moore-Penrose. If the th row of. This leads to a computationally efficient implementation of the regularized pseudoinverse filtering approach using the 2-D fast Fourier transform FFT.
The derivation of the filtering equation is shown in detail and the regularization procedure is fully described. Using FIELD, we present simulation data to show the 2-D point-spread functions PSFs for imaging systems employing linear arrays with fine and coarse sampling of the imaging aperture.
PSFs are also computed for a coarsely sampled array with different levels of regularization to demonstrate the tradeoff between contrast and spatial resolution. These results demonstrate the well-behaved nature of the PSF with the variation in a single regularization parameter.
- Moore–Penrose inverse
- Matrix Inverse
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Specifically, the 6 dB axial and lateral dimensions of the PSF increase gradually with increasing value of the regularization parameter. On the other hand, the peak grating lobe level decreases gradually with increasing value of the regularization parameter. The results are supported by image reconstructions from a simulated cyst phantom obtained using finely and coarsely sampled apertures with and without the application of the regularized 2-D PIO.
Introduction High-frequency ultrasound HFUS has been used for noninvasive visualization of living tissues at or near microscopic levels in many clinical or biological applications, such as dermatology [ 1 ], [ 2 ], ophthalmology [ 3 ]—[ 5 ], intravascular intracardiac imaging [ 6 ]—[ 10 ], nonvascular endoluminal imaging [ 11 ], cartilage imaging [ 12 ], [ 13 ], and small-animal studies [ 14 ], [ 15 ].
Moore-Penrose pseudoinverse is not an adjoint
Researchers have recently achieved some success in extending the resolution to microscopic levels [ 3 ], [ 16 ], [ 17 ] using high-frequency systems with single-element transducers. At least one high-frequency ultrasound imaging system of this kind has been commercialized for small-animal research applications. Various array transducers for HFUS imaging applications have been proposed recently [ 16 ]—[ 23 ].
These efforts are motivated by the success of array transducers in clinical applications in the frequency range of 2 to 16 MHz. The benefits of arrays in ultrasonic imaging are well understood and amply demonstrated by their wide use in clinical scanners.
Linear probes with center frequencies up to 15 MHz are available on some commercial scanners. A typical linear array is implemented with element-to-element spacing in the 1 to 1. Several transducer materials and fabrication methodologies have been developed in recent years [ 17 ], [ 21 ], [ 23 ], [ 24 ].
The benefits of these advances in materials and fabrication techniques have not been fully realized for high-frequency arrays above 20 MHz. One of the primary challenges for current transducer technologies is the element size requirement in this frequency range.
Even if the fabrication problems are solved, additional problems due to increased cross coupling, increased element impedance, and increased variability in element sensitivity may arise.
Therefore, based on the existing knowledge of transducer materials and fabrication methodologies, it is expected that high-frequency arrays in the range of 25 to 35 MHz will be realized with 1.
Moore-Penrose pseudoinverse is not an adjoint
Conventional beamforming with these coarsely sampled arrays results in increased grating lobe levels leading to reduced dynamic range and loss of contrast. Pseudoinverse and matched filtering has been successfully used in ultrasonic imaging to restore axial resolution, primarily in conjunction with coded excitation [ 25 ]—[ 29 ].
The algorithm in [ 25 ], [ 30 ] employs a filter bank for parallel processing of echo data from multiple directions from a single beamforming operation. The coefficients of the filters were computed based on a regularized inverse of a discretized 2-D axial-lateral pulse-echo propagation operator from the array to the region of interest ROI.
For a single A-line, the sampled beamformer output is related to the scatter distribution within the ROI by a discretized propagation operator matrix as described in [ 25 ]. The propagation operator was represented as a block row matrix with each block representing the array response in a given lateral direction. Using the range-shift invariance assumption [ 25 ], the matrix elements of the propagation operator were represented as Toeplitz.
This allowed for the use of the computationally efficient discrete Fourier transform DFT in finding a pseudoinverse operator resulting in the filter bank implementation for the parallel imaging in multiple directions from a single beamforming operation. In principle, the filter bank could be designed to remove some of the beamforming artifacts, but this is limited to the removal of uncorrelated grating-lobe components from the desired beam direction s.