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### Abstract

A number of imaging systems exhibit speckle, which is caused by the interaction of a coherent pulse reflecting off of random reflectors. The limitations of these systems are quite serious because the speckle phenomenon creates a pattern of nulls and peaks from subresolvable scatterers or targets that are difficult to interpret. Another limitation of these pulse-echo imaging systems is that their resolution is dependent on the full spatial extent of the propagating pulse, usually several wavelengths in the axial or propagating dimension and typically longer in the transverse direction. This limits the spatial resolution to many multiples of the wavelength. This paper focuses on the particular case of ultrasound B-scan imaging and develops an inverse filter solution that eliminates both the speckle phenomenon and the poor resolution dependency on the pulse length and width to produce super-resolution ultrasound (SURUS) images. The key to the inverse filter is the creation of pulse shapes that have stable inverses. This is derived by use of the standard Z-transform and related properties. Although the focus of this paper is on examples from ultrasound imaging systems, the results are applicable to other pulse-echo imaging systems that also can exhibit speckle statistics.

This work was supported by the Department of Electrical and Computer Engineering in the Hajim School of Engineering of the University of Rochester. The author is grateful to Zhilin (Aaron) Cong who produced superb programs for solving and incorporating inverse Z-transforms.

I. INTRODUCTION

II. THEORY

III. RESULTS

IV. DISCUSSION AND CONCLUSIONS

### Key Topics

- Ultrasonography
- 22.0
- Medical imaging
- 16.0
- Speckle
- 15.0
- Speckle patterns
- 9.0
- Fourier transforms
- 6.0

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## Figures

(Color online) A conventional pulse with a Gaussian envelope, sampled at twice the center frequency in (a). The zeros of the Z-transform of the sampled pulse is shown in (b). Because some zeros lie in and around the unit circle, the inverse is unstable and unbounded.

(Color online) A conventional pulse with a Gaussian envelope, sampled at twice the center frequency in (a). The zeros of the Z-transform of the sampled pulse is shown in (b). Because some zeros lie in and around the unit circle, the inverse is unstable and unbounded.

(Color online) An asymmetric pulse formed by multiplying the Gaussian envelope with a geometric series is shown in (a). The zeros of the Z-transform are retracted into the unit circle as shown in (b). This leads to a stable inverse filter.

(Color online) An asymmetric pulse formed by multiplying the Gaussian envelope with a geometric series is shown in (a). The zeros of the Z-transform are retracted into the unit circle as shown in (b). This leads to a stable inverse filter.

(Color online) Another asymmetric pulse with the form of a Gaussian multiplied by the square root of *t* is shown in (a). The zeros of its Z-transform are shown in (b) indicating the availability of a stable inverse filter shown in (c).

(Color online) Another asymmetric pulse with the form of a Gaussian multiplied by the square root of *t* is shown in (a). The zeros of its Z-transform are shown in (b) indicating the availability of a stable inverse filter shown in (c).

(Color online) An unmodulated asymmetric envelope proposed for the transverse beampattern is shown in (a). Its Z-transform zeros are shown in (b) and corresponding stable inverse filter in (c).

(Color online) An unmodulated asymmetric envelope proposed for the transverse beampattern is shown in (a). Its Z-transform zeros are shown in (b) and corresponding stable inverse filter in (c).

A gray-scale plot of the two dimensional (2D) function that is separable and asymmetric in both vertical and horizontal directions. The vertical (axial) function is given by the function shown in Fig. 3(a), and the horizontal (transverse) function is the function shown in Fig. 4(a).

A gray-scale plot of the two dimensional (2D) function that is separable and asymmetric in both vertical and horizontal directions. The vertical (axial) function is given by the function shown in Fig. 3(a), and the horizontal (transverse) function is the function shown in Fig. 4(a).

A set of random reflectors with patches of zeros in the shape of letters (a) undergoes 2D convolution with the pulse of Fig. 5. The resulting speckle envelope is shown in (b), and the letters “UR” are not visible due to the distribution of the speckle statistics. After inverse filtering in the vertical direction, the results are improved in (c), and after horizontal inverse filtering the final result is given in (d); 5% rms white noise was added to the convolution result before inverse filtering, so the operations are well conditioned in the presence of modest additive noise.

A set of random reflectors with patches of zeros in the shape of letters (a) undergoes 2D convolution with the pulse of Fig. 5. The resulting speckle envelope is shown in (b), and the letters “UR” are not visible due to the distribution of the speckle statistics. After inverse filtering in the vertical direction, the results are improved in (c), and after horizontal inverse filtering the final result is given in (d); 5% rms white noise was added to the convolution result before inverse filtering, so the operations are well conditioned in the presence of modest additive noise.

(Color online) Histograms showing the probability density functions for the original scatterers (a), the result of convolution with a pulse (b), and after inverse filtering (c). The standard Rayleigh speckle statistics are demonstrated in (b), whereas the original Gaussian statistics are restored in (c).

(Color online) Histograms showing the probability density functions for the original scatterers (a), the result of convolution with a pulse (b), and after inverse filtering (c). The standard Rayleigh speckle statistics are demonstrated in (b), whereas the original Gaussian statistics are restored in (c).

(Color online) A sampled pulse (a) and its Z-transform zeros (b) indicating the availability of a stable inverse filter.

(Color online) A sampled pulse (a) and its Z-transform zeros (b) indicating the availability of a stable inverse filter.

(Color online) The same pulse sampled at twice the sampling rate of Fig. 8, shown in (a), and its Z-transform zeros in (b). A movement toward the unit circle is evident. Another doubling of the sampling rate results in (c) and (d). The trend is toward the unit circle and instability.

(Color online) The same pulse sampled at twice the sampling rate of Fig. 8, shown in (a), and its Z-transform zeros in (b). A movement toward the unit circle is evident. Another doubling of the sampling rate results in (c) and (d). The trend is toward the unit circle and instability.

The effect of inexact reconstruction parameters. A region of reflectors is shown in (a). Imaging with a pulse using a geometric parameter of 0.7 yields a conventional speckle image in (b). Reconstruction with horizontal (exact) and vertical (inexact, geometric parameter 0.6 used) are shown in (c) and (d), respectively. The resulting inverse filtered image is a reasonable reconstruction of the original scatterers, despite the use of an erroneous model.

The effect of inexact reconstruction parameters. A region of reflectors is shown in (a). Imaging with a pulse using a geometric parameter of 0.7 yields a conventional speckle image in (b). Reconstruction with horizontal (exact) and vertical (inexact, geometric parameter 0.6 used) are shown in (c) and (d), respectively. The resulting inverse filtered image is a reasonable reconstruction of the original scatterers, despite the use of an erroneous model.

(Color online) Two functions in the spatial domain, a Rayleigh and a Gaussian, representing asymmetric and symmetric pulse envelopes, are given in (a). Their Fourier Transform magnitudes are given in (b). A somewhat greater distribution of the higher spatial frequencies is required to generate an asymmetric beam envelope.

(Color online) Two functions in the spatial domain, a Rayleigh and a Gaussian, representing asymmetric and symmetric pulse envelopes, are given in (a). Their Fourier Transform magnitudes are given in (b). A somewhat greater distribution of the higher spatial frequencies is required to generate an asymmetric beam envelope.

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