This work presents an ASIC designed to be integrated on the tip of an intra-vascular ultrasound catheter, capable of interfacing 64 ultrasonic transducer elements to an imaging system using only one micro-coaxial cable. A prototype chip has been taped out that can be used to transmit acoustic waves using a small-area transmit switch. The received echo signal is amplified by a phantom powered LNA and read-out at the other end of the micro-coaxial cable by a transimpedance amplifier.

In this thesis, we investigate a sparse basis for ultrasound images, so that we can use sparse regularization in imaging. Actually, there are few previous researches explicitly demonstrating that medical ultrasound images can be sparsified for some dictionary. We consider various orthogonal transforms such as wavelet transforms, cosine transforms and wave atom transforms. Then, we perform those transforms on various ultrasound images and analyzes their sparsity. These ultrasound images include the images of two computer ultrasound phantoms and beamformed ultrasound images with good quality from real people. We looked at sparsity of the true pre-beamformed images, as well as beamformed images. We also consider constructing a specific ultrasound image dictionary using the K-SVD algorithm. We observed that, the pre-beamformed images hardly haVe no sparse basis, and the sparsity of beamformed images will only increase slightly if we use different 1D-DWT in each direction. We also found that the wide overdetermined dictionary generated by K-SVD significantly increases sparsity. After this, we simulate the ultrasound image reconstruction from the ultrasound RF measurements, and we analyze the effects of the different sparse spaces on the reconstruction performance. We observed that, the L1-regularization can work for ultrasound imaging better than L2-regularization, but the orthogonal transforms as well as the dictionary do not improve the reconstruction image quality much.

Current research on osteoporosis detection and transcranial imaging suggests that it is necessary to find a method to calculate the speed of sound in curved bones. A popular method to find the speed of sound in bones is by the bidirectional axial transmission technique. Currently, this method assumes that the arrival time of the head wave depends linearly on the source-receiver distance. For curved bones this assumption is not valid. The arrival time of the head wave as a function of the source-receiver distance follows a curved trajectory. In this report, we attempt to find an extension to the current method that also works for curved bones, and evaluate how accurate the results of this extended method are. This extension is based on the semblance method, that quantifies how well a certain trajectory agrees with the received data. In the first part of our project, theoretical trajectories are calculated for flat and semicircular bones. Furthermore, a numerical method is discussed to find these trajectories for arbitrary geometries. Comparing the trajectories corresponding to different velocities using the semblance methods allows for the determination of the speed of sound. The accuracy of this extended method is evaluated by computer simulations. In these simulations, a Philips P41 Cardiac Sector Probe consisting of 96 elements was used to perform measurements on three different geometries. These geometries consisted of three different single interfaces between water (simulating soft tissue) and bone. The first interface consists of a straight interface between a 2 mm thick layer of water and a 5 mm thick layer of bone (that continues until the bottom edge of the simulation). The second simulation is the same, but the interface is rotated 2 degrees around its midpoint. In the third and fourth simulation, the interface is a semicircle with a 50 mm and 40 mm radius, respectively. The speed of sound in the bone was taken to be 3 mm/𝜇s. The received data was interpolated by a cubic interpolation with a refinement of 4. Both the bidirectional and the extended, trajectory-based method were applied on the raw and interpolated data. For the flat interface, the bidirectional method performed slightly better than the trajectory-based method (0.13% and 0.30% error, respectively). The same is true for the tilted interface (0.16% and 0.23% error). Interpolation did not affect the results here. For the 50 mm semicircular interface, the trajectory-based method performed much better than the bidirectional method: it had a 0.10% error compared to a 2% one. This was also true when the interpolation was applied: then the errors were 0.3% and 1.97%. Similar improvements were seen for the 40 mm radius semicircle. The higher error of the trajectory-based method for the flat and tilted interfaces can be explained by small errors in the additional assumptions that this method makes. One drawback of the trajectory-based method is that there can be problems interpreting the results. This can be seen in the semblance plots, which plot the semblance coefficient (how well the trajectory fits the data) against the test velocity. For the bidirectional method, these plots clearly have a single biggest peak. For the trajectory-based methods the plot contains multiple large peaks. Furthermore, for the semicircular bone this plot contains a flat peak, which can cause multiple test velocities to have approximately the same semblance. This is a problem, since it means that small changes to the experimental setup can cause big changes in the calculated velocity. Indeed, this can be seen for the trajectory-based method on the 40 mm radius semicircular bone, which has a 1.6% error without interpolation (0.43% with interpolation). Further research could examine solutions to this problem, such as increasing the window size of the semblance method, taking the average velocity over flat peaks, or using an alternative to the semblance method. Finally, we conclude that the trajectory-based method represents an improvement over the bidirectional method when examining curved bones.

In this research, a Shive wave machine is used to study (a) the velocity of waves throughout different media and (b) the transition of waves between two different media. The Shive wave machine used in this research consists of 32 parallel aluminum bars attached perpendicularly to three parallel central wires. When a perturbation is applied to one of the bars, a torsional wave is initiated in the Shive wave machine, which is mapped to a transverse wave at the extremities of the bars. The velocity of a wave in the Shive wave machine is theoretically determined by four variables: (i) The distance between the two outside wires and the central wire; (ii) the tension in the two outside wires; (iii) the distance between the bars; (iv) the moment of inertia of the bars. The transition of waves between two different media is theoretically determined by the wave velocities in the two media. The theory states that a part of the wave is reflected and a part is transmitted at the intersection between the two media. The results for (a) show that the measured velocity is higher than theoretically expected, which may be caused by an incorrect measurement of (ii). The results for (b) show that a change in (i) throughout the system does not comply with the theoretically expected ratios for reflection and transmission, whereas a change in (iv) does comply with the theoretically expected ratios.

In medical ultrasound imaging phased array transducers are used for non-invasive imaging. For 3D intra-cardiac-echography technical borders are reached since blood vessel size limits the dimensions of the array. This dimensional limitation lowers lateral resolution. To solve part of the problem, non conventional ways of lateral resolution improvements are needed. For linear arrays R. Reeg [1] proposed null subtraction imaging (NSI), which resulted at best in 30 dB lower side lobes and a width reduction of up to 25 times for the main lobe. This is a non-linear image processing technique based on implementing multiple apodizations in post-processing. Three images are made from which one has a sharp drop to zero in the middle which can be exploited by subtracting it from the other images. This results in a improvement over a regular apodization scheme. To verify the technique for 2D phased array imaging, simulations are done using Field-II [2, 3] while taking array dimension into account. Measurements take place on an artificial phantom with evenly spaced line scatter targets. For the experiment a P4-l phased array transducer was used in combination with a Verasonics Vantage 256TMwhere a sub-aperture of 30 elements is used. The experiment is done on a CIRS 040GSE Phantom, where evenly spaced line scatter targets and lateral resolution targets are looked at. To form an image a simple delay-and-sum (DAS) algorithm was used. The effect of NSI on the speckle in an image is also measured, where for normal developed speckle a signal to noise ratio (SNR) of 1.91 is expected. Results for simulation were as expected with an average beamwidth reduction of 22.5 times and on average higher main lobe to side lobe ratio (MSR) of 33.4 dB for NSI imaging opposed to using a rectangular apodization. Experimentally an average beamwidth reduction of 5 times and an on average higher MSR of 20.5 dB are realised. The beamwidth reduction is lower then expected, this is due to rounding errors in the DAS algorithm, where data can only be delayed by integer elements. For the lateral scatter targets no extra targets can be distinguished when using NSI imaging opposed to rectangular apodization imaging, only the targets that where already visible show reduction in beamwidth and side lobe levels. When using NSI speckle SNR is reduced from 1.78 to on average 0.50 and seems to scale with energy present in the image. An hybrid image between the NSI image and rectangular apodization image is made to solve this. It results in reduced beamwidth, restored speckle SNR but the MSR is reduced. It can be concluded that the method works for phased array imaging under perfect conditions, the simulation. To get the full potential of the technique more research is needed towards the effects of improper sampling. Which in this case resulted in less then ideal ’sharp’ drops, lowering the reduction in beamwidth. Also the reduced speckle needs further research to understand it well. Once this is all achieved the next step to 3D imaging can be made.

In this Bachelor project we researched the applicability of finite difference methods to solve the multi dimensional Westervelt equation. The Westervelt is an equation describing the propagation of a non-linear wave. The goal of this project was to solve the full Westervelt equation in higher dimensions. We used Julia for our implementations of the finite difference method. The choice of Julia allowed us to easily write high performing code. We were unsuccessful in modelling the full Westervelt equation since we were unable to implement attenuation into our models. We successfully implemented the non-linear wave equation without attenuation for a number of models, including a three dimensional model. The inability to implement attenuation into our models hampered the flexibility of our models. When the non-linearity created a shock front, the top of the wave would experience numerical errors which quickly made the whole solution unphysical and ultimately unstable. An added implementation of attenuation would dampen the leading edge of the wave preventing it from becoming too steep. Additionally we made two comparisons with the linear wave equation to verify the correctness of our implementations. We used a uniform grid to discretize the spatial dimensions. After which we used non-stiff FDM solvers with adaptive time stepping algorithms to solve for the remaining time dimension. We analyzed the use of stiff FDM solvers but these proved too computationally expensive to be viable, especially for our 3D model with 64.000 points.

In this thesis two topics are discussed. The covariant formulation ofMaxwell’s equations of electromagnetism and the formulation of said equations in the context of a rotating frame of reference. Through the development of the necessary theories of Differential Geometry and Special Relativity we show how to formulateMaxwell’s equations in terms of the covariant derivative and in terms of the hodge operator and differential operator. Using this formulation we study the transformation properties of these equations under Lorentz transformations and we conclude that they remain invariant under said transformations. Secondly,we discuss literature concerning the problem of electromagnetism in the context of a rotating frame of reference. We show that the method of direct transformation to a rotating frame results in an unobservable frame of reference and we show that it is impossible to reconstruct Coriolis like forces in an observable frame of reference.

This essay shows a two dimensional implementation of the finite element method for the Westervelt equation. To do this the finite element method is first applied to the linear wave equation, then to non-linear diffusion and finally to the Westervelt equation. Both an element by element and a faster vectorized implementation are given for the finite element method. To verify the numerical solution two analytical solutions are used. The first is a one dimensional wave and the second a circularly symmetric wave. We found that the two-dimensional implementation was successful in computing the Westervelt equation. The error of the solution scales with the mesh size with a power of around 1.7. It was also found that the time step used to compute the solution needs to be small enough for the implementation to converge.