A Pseudo-Dynamic Delay Calculation Using Optimal Zone Segmentation for UltraCompact Ultrasound Imaging Systems

The implementation of dynamic delay calculations (DDCs) is challenging for ultra-compact ultrasound imaging due to the enormous computation and power consumption requirements. Here, we present an efficient pseudo-DDC method based on optimal zone segmentation (PDC-Optimal), which significantly decreases these requirements relative to an unconstrained DDC method: reductions in flip-flops of 84.35% and in look-up tables of 94.19%, respectively. The reductions lead to an up to 94.53% lower dynamic power consumption and provide image quality comparable to the unconstrained DDC method. The proposed PDC-Optimal method also provides adaptive flexibility between beamforming accuracy and battery life using the delay error allowance, a user-definable parameter. A conventional pseudo-DDC method using uniform zone segmentation (PDC-Conv) presented substantial image degradation in the near imaging field when the same number of zone segments was used. Therefore, the PDC-Optimal method provides an efficient yet flexible DDC solution to improve the experiences for ultra-compact ultrasound imaging system users.


Introduction
In point-of-care ultrasound (US), demands for advanced imaging algorithms have been emphasized [1][2][3][4].However, the implementation of such advanced features in an ultra-compact US system remains challenging because complex computations must be performed with limited hardware resources, circuit volume, and battery capacity, decreasing clinical efficacy.
Several efficient beamforming and/or compressive sensing algorithms have been proposed to address such demands [5][6][7][8][9][10].The dynamic receive beamformer is one of the most complex processing components in a US imaging system; our previous research found that it contains 46.5% of the total hardware resources and 25.4% of the total power consumption of a system-on-chip solution for portable US imaging [11].Here, we particularly focus in an efficient implementation of dynamic delay calculations (DDC) during dynamic receive beamforming, which allocates complex square roots and multipliers at each channel [12].Although a time-sharing pseudo-DDC method (PDC-TS) has been proposed to alleviate this burden, it does not provide any flexibility with the complex multiplexing hardware structure [11,13].A look-up-table (LUT)-based DDC method was proposed to address this problem and provide both high flexibility and reduced hardware usage by adopting a pseudo-dynamic delay interpolation concept, which is referred to as the "PDC-Conv" method in this paper.The PDC-Conv method interpolates between the representative delays of adjacent "zone segments" divided with a uniform distance within an entire imaging depth.With this method, both the use of computational units and the LUT size can be reduced [2,14].However, its practical use in a US system has been difficult, as there are no clear design guidelines to regulate beamforming quality and hardware usage (e.g., the number and length of a zone segment, bit quantization in delay calculation, etc.).The lack of guidelines leads to irregular imaging performance, especially at near imaging depth, due to the fixed number and length of zone segments regardless of the dynamic changing in delay increments over the imaging depth.
In this paper, we propose a pseudo-DDC method using an optimal zone segmentation with non-uniform intervals (PDC-Optimal) for an efficient implementation of the ultra-compact US imaging system with a clear design methodology.This would provide more precise beamforming over an imaging field of view compared with the PDC-Conv method when using the same amount of hardware resources.In addition, the proposed PDC-Optimal method will empower adaptive management of dynamic power consumption as a function of user-definable delay error allowance, which enables an optimized user experience on various diagnostic circumstances requiring different levels of image quality and amounts of battery life.

Dynamic Delay Calculation for Diagnostic U2.3ltrasound
Figure 1 shows the conceptual diagram of the dynamic receive beamforming block for the diagnostic US, which consists of the delay compensation for the digitized RF signals at each channel and its summation at each imaging depth.Each DDC event for a dynamic delay can be equated by where (x ch , z ch ) denotes the Cartesian geometry of the ch-th channel of the US array transducer ranging from 1 to the total number of channels (e.g., 128-256 in general); R is the distance between the center of the US transducer and a focusing point (x, z); t is the elapsed time since an acoustic transmission; c is the mean speed of sound in soft biological tissue, i.e., 1540 m/s.
method in this paper.The PDC-Conv method interpolates between the representative delays of adjacent "zone segments" divided with a uniform distance within an entire imaging depth.With this method, both the use of computational units and the LUT size can be reduced [2,14].However, its practical use in a US system has been difficult, as there are no clear design guidelines to regulate beamforming quality and hardware usage (e.g., the number and length of a zone segment, bit quantization in delay calculation, etc.).The lack of guidelines leads to irregular imaging performance, especially at near imaging depth, due to the fixed number and length of zone segments regardless of the dynamic changing in delay increments over the imaging depth.
In this paper, we propose a pseudo-DDC method using an optimal zone segmentation with non-uniform intervals (PDC-Optimal) for an efficient implementation of the ultra-compact US imaging system with a clear design methodology.This would provide more precise beamforming over an imaging field of view compared with the PDC-Conv method when using the same amount of hardware resources.In addition, the proposed PDC-Optimal method will empower adaptive management of dynamic power consumption as a function of user-definable delay error allowance, which enables an optimized user experience on various diagnostic circumstances requiring different levels of image quality and amounts of battery life.

Dynamic Delay Calculation for Diagnostic U2.3ltrasound
Figure 1 shows the conceptual diagram of the dynamic receive beamforming block for the diagnostic US, which consists of the delay compensation for the digitized RF signals at each channel and its summation at each imaging depth.Each DDC event for a dynamic delay can be equated by where (x ch , z ch ) denotes the Cartesian geometry of the ch-th channel of the US array transducer ranging from 1 to the total number of channels (e.g., 128-256 in general); R is the distance between the center of the US transducer and a focusing point (, ); t is the elapsed time since an acoustic transmission; c is the mean speed of sound in soft biological tissue, i.e., 1540 m/s.In general, an electronic circuit for the DDC operates at an analog-to-digital conversion (ADC) frequency, usually >20 MHz, and its real-time calculation should be repeated at each beamforming point of multiple vertical scanlines (e.g., 128-512) comprising an image frame.For example, when assuming a DDC sequence for 128 scanlines and an 8 cm imaging depth at a 40 MHz ADC frequency

Dynamic receive beamformer
Beamformed output In general, an electronic circuit for the DDC operates at an analog-to-digital conversion (ADC) frequency, usually >20 MHz, and its real-time calculation should be repeated at each beamforming point of multiple vertical scanlines (e.g., 128-512) comprising an image frame.For example, when assuming a DDC sequence for 128 scanlines and an 8 cm imaging depth at a 40 MHz ADC frequency in a 32-channel US system, 510.20 million of DDC events per second would be needed for a 30 Hz frame rate.Therefore, an efficient electronic DDC implementation would be crucial to secure the efficacy of an ultra-compact US imaging system in terms of hardware resource utilization and power consumption.

Pseudo-Dynamic Delay Calculation (PDC) Using Optimal Zone Segmentation
The PDC-Optimal method consists of a delay accumulator and a dedicated LUT module at each channel (Figure 2a).The LUT at each channel stores a preset for each zone segment, and it is fed to a dedicated delay accumulator for interpolating a dynamic delay at each focusing point during receive beamforming.Figure 2b shows how one can determine the delay increment and length of the N-th zone segment, i.e., ∆d N (ch) and L N (ch), as a function of the user-definable delay error allowance, τ. τ is defined by half of the target delay accuracy (e.g., 0.125, 0.25, 0.5, and 1 of τ for 0.25, 0.5, 1, and 2 of the target time delay resolution in the sample domain, respectively).This definition yields f s /2τ of receive beamforming accuracy, in which f s is the data sampling frequency.
Electronics 2019, 8, x FOR PEER REVIEW 3 of 10 in a 32-channel US system, 510.20 million of DDC events per second would be needed for a 30 Hz frame rate.Therefore, an efficient electronic DDC implementation would be crucial to secure the efficacy of an ultra-compact US imaging system in terms of hardware resource utilization and power consumption.

Pseudo-Dynamic Delay Calculation (PDC) Using Optimal Zone Segmentation
The PDC-Optimal method consists of a delay accumulator and a dedicated LUT module at each channel (Figure 2a).The LUT at each channel stores a preset for each zone segment, and it is fed to a dedicated delay accumulator for interpolating a dynamic delay at each focusing point during receive beamforming.Figure 2b shows how one can determine the delay increment and length of the N-th zone segment, i.e., ∆d N ch and LN(ch), as a function of the user-definable delay error allowance, τ. τ is defined by half of the target delay accuracy (e.g., 0.125, 0.25, 0.5, and 1 of τ for 0.25, 0.5, 1, and 2 of the target time delay resolution in the sample domain, respectively).This definition yields fs / 2τ of receive beamforming accuracy, in which fs is the data sampling frequency.In Figure 2b, the ideal receive delay calculated by the unconstrained DDC method (i.e., d ideal (z, ch) ) is represented as a dashed line.Other solid lines present the receive delay of the PDC-Optimal method corresponding to τ.As one can see, as τ increases, the PDC-Optimal method has a larger N-th zone length although its delay error from the ideal delay is worsening.Note that modern US imaging systems are generally designed for 0.25 of target time delay resolution with a 4-f0 data sampling rate, in which f0 is the center frequency of an employed US transducer, leading to 16-f0 beamforming resolution [15,16].The zone segment is defined by Δd test ch ranges from 0 to 1 in a bit quantization resolution, and index i ≥ 1 (integer number).The initial bit quantization resolution of Δd test ch (i.e., Q ) is in 18 bit, determined for the highest-possible delay calculation accuracy from the default bit width of basic IPs in a modern field programmable gated array (FPGA).The Q would be further optimized in the following design procedures.d N-1 ch is the final delay at the previous N-1 th zone segment; z N ch is the starting depth index of the N-th zone segment, given by The Δd test ch defining L N (ch) is the optimal delay increment in the N-th zone segment, denoted by ∆d N ch in this paper.The pre-defined preset for each zone segment is stored in an LUT In Figure 2b, the ideal receive delay calculated by the unconstrained DDC method (i.e., d ideal (z, ch)) is represented as a dashed line.Other solid lines present the receive delay of the PDC-Optimal method corresponding to τ.As one can see, as τ increases, the PDC-Optimal method has a larger N-th zone length although its delay error from the ideal delay is worsening.Note that modern US imaging systems are generally designed for 0.25 of target time delay resolution with a 4-f 0 data sampling rate, in which f 0 is the center frequency of an employed US transducer, leading to 16-f 0 beamforming resolution [15,16].The zone segment is defined by ∆d test (ch) ranges from 0 to 1 in a bit quantization resolution, and index i ≥ 1 (integer number).The initial bit quantization resolution of ∆d test (ch) (i.e., Q) is in 18 bit, determined for the highest-possible delay calculation accuracy from the default bit width of basic IPs in a modern field programmable gated array (FPGA).The Q would be further optimized in the following design procedures.d N−1 (ch) is the final delay at the previous N-1 th zone segment; z N (ch) is the starting depth index of the N-th zone segment, given by The ∆d test (ch) defining L N (ch) is the optimal delay increment in the N-th zone segment, denoted by ∆d N (ch) in this paper.The pre-defined preset for each zone segment is stored in an LUT allocated at each channel: ∆d N (ch) and L N (ch).Therefore, total LUT size is dependent to Q so that an optimization would be needed.If the bit resolution for ∆d N (ch) is too low, the quantization error would deteriorate the designed beamforming accuracy in the N-th zone.However, a redundant bit resolution is either suboptimal, as the total data width of the preset would be increased.Therefore, we used a heuristic optimization between the number of zone segments and the LUT size for an individual US transducer, which is calculated by which is subject to an unsigned fixed number Q, given u1.Q − 1 as a fractional bit resolution.The pseudo-dynamic delay is calculated in our PDC-Optimal method as follows.In the first zone segment, the system writes d ideal (1, ch) Q = ∆d 1 [ch] on a delay latch ("D" in Figure 2a) to be used as d N−1 [ch].When operating, a delay accumulator generates a pseudo-dynamic delay using the calculated preset for each zone segment, which is given by in which i indicates each depth sample point within a zone segment, i.e., 1, 2, . . ., L N (ch).The preset is updated when translating to the next zone segment.

Simulation Setup
The delay calculation errors of the PDC-Conv and PDC-Optimal methods were evaluated compared to the unconstrained DDC method.The evaluation was performed with specific transducer specifications: 128-channel linear US array transducer at 15 MHz f 0 with 100 µm of element pitch.The -6 dB fractional bandwidth was 48% from 11.4 MHz to 18.6 MHz.A 32-channel US system was assumed.The preset for the LUT (i.e., L N (ch), and ∆d N (ch)) was pre-calculated for 20 mm of imaging depth, which is typical in US diagnostics with the given f 0 .The f s was 60 MHz, i.e., 4 f 0 .Moreover, we conducted a comparative study for the ultrasound brightness mode (B-mode) images reconstructed by PDC-Conv and PDC-Optimal methods.The imaging quality expected from the aforementioned system configuration and imaging specifications was compared in a Field-II simulation [17].The target geometry was comprised of the wire targets aligned in an axial direction at a center field of view from 2 mm to 17 mm in 1 mm intervals.The hardware resource utilization and dynamic power consumption were also estimated respectively in 32-channel unconstrained DDC, PDC-TS, and PDC-Optimal methods synthesized and implemented by a Vivado 14.4 on an Artix-7 FPGA (xc7A200T, Xilinx Inc., San Jose, CA, USA).Note that the estimated dynamic power consumption will project computational complexity in each method from the heterogeneous resources available in the FPGA (e.g., multipliers, adders, multiplexers, memories, etc.).The input voltages and operating frequency were equally applied for each method: f s , 40 MHz; 2.5 V for low-voltage differential signaling (LVDS) input; 3.3 V and 1.8 V for low-voltage complementary metal oxide semiconductor (LVCMOS) input.

Results
Figure 3a shows the mean number of zone segments across all channels as a function of Q for 0.125, 0.25, 0.5, and 1 of τ.As shown in Figure 3b, the total LUT size decreases with decreasing Q, although the mean number of zone segments remains fixed.However, once Q becomes lower than the optimal, the number of zone segments increases because a delay error would more easily exceed τ with quantization error in Equation (2).Conversely, each τ yielded a different optimal bit quantization resolution: 11, 9, 9, and 8 for 0.125, 0.25, 0.5, and 1 of τ, respectively (Figure 3b), which leads to differing mean numbers of zone segments: 12, 10, 7, and 6, respectively (Figure 3a).As a result, the total LUT sizes per channel were 982, 700, 502, and 380 bytes for 0.125, 0.25, 0.5, and 1 of τ, respectively.
Electronics 2019, 8, x FOR PEER REVIEW 5 of 10 quantization resolution: 11, 9, 9, and 8 for 0.125, 0.25, 0.5, and 1 of τ, respectively (Figure 3b), which leads to differing mean numbers of zone segments: 12, 10, 7, and 6, respectively (Figure 3a).As a result, the total LUT sizes per channel were 982, 700, 502, and 380 bytes for 0.125, 0.25, 0.5, and 1 of τ, respectively.Figure 4 presents the mean delay error across all delay approximation channels.An identical number of zone segments was used for the PDC-Conv and PDC-Optimal methods.For both methods, the errors were proportional to the number of zone segments, defined by τ in the proposed PDC-Optimal method.The delay error was well delimited in the proposed PDC-Optimal method at 0.125, 0.25, 0.5, and 1 over the entire imaging depth (Figure 4b).However, the PDC-Conv method exhibited unregulated performance especially in the near imaging depth, as shown in Figure 4a.In quantitative evaluation, the means and standard deviations of the dynamic delay errors throughout the imaging depths (0-20 mm) were 6.25 ± 1.82, 7.50 ± 2.39, 10.16 ± 3.94, and 11.66 ± 4.73 for the PDC-Conv method and 0.12 ± 0.02, 0.24 ± 0.04, 0.48 ± 0.09, and 0.99 ± 0.01 for the PDC-Optimal method, respectively at 0.125, 0.25, 0.5, and 1 of τ (p < 0.0001).Figure 4 presents the mean delay error across all delay approximation channels.An identical number of zone segments was used for the PDC-Conv and PDC-Optimal methods.For both methods, the errors were proportional to the number of zone segments, defined by τ in the proposed PDC-Optimal method.The delay error was well delimited in the proposed PDC-Optimal method at 0.125, 0.25, 0.5, and 1 over the entire imaging depth (Figure 4b).However, the PDC-Conv method exhibited unregulated performance especially in the near imaging depth, as shown in Figure 4a.In quantitative evaluation, the means and standard deviations of the dynamic delay errors throughout the imaging depths (0-20 mm) were 6.25 ± 1.82, 7.50 ± 2.39, 10.16 ± 3.94, and 11.66 ± 4.73 for the PDC-Conv method and 0.12 ± 0.02, 0.24 ± 0.04, 0.48 ± 0.09, and 0.99 ± 0.01 for the PDC-Optimal method, respectively at 0.125, 0.25, 0.5, and 1 of τ (p < 0.0001).
Figure 5a shows the point spread functions (PSFs) at different imaging depth for PDC-Conv and PDC-Optimal methods.In this figure, PSFs in the left and right subcolumns are reconstructed by PDC-Conv and PDC-Optimal methods corresponding to τ, respectively.Figure 5b,c shows a quantitative evaluation of the correlation coefficients (CCs) of the PSFs and those by the unconstrained DDC images at each depth.According to the average CCs in the superficial imaging depth (i.e., 0-5 mm), the PDC-Optimal method consistently yielded improved image quality compared to the PDC-Conv method: 1.00 ± 0.00, 0.99 ± 0.00, 0.97 ± 0.01, and 0.84 ± 0.05 for the PDC-Optimal method and 0.99 ± 0.01, 0.98 ± 0.01, 0.82 ± 0.27, and 0.60 ± 0.28 for the PDC-Conv method for 0.125, 0.25, 0.5, and 1 of τ, respectively.Otherwise, there was no significant difference in the deeper imaging depth (6-20 mm) between the methods.Both methods yielded > (0.98 ± 0.03), even when τ is set to 1.These results confirm the efficacy of the PDC-Optimal method: a well-defined design methodology with user-definable parameter τ, which is not available with the PDC-Conv method, and increasing uniformity in image quality compared to the PDC-Conv method.
Electronics 2019, 8, x FOR PEER REVIEW 6 of 10 unconstrained DDC images at each depth.According to the average CCs in the superficial imaging depth (i.e., 0-5 mm), the PDC-Optimal method consistently yielded improved image quality compared to the PDC-Conv method: 1.00 ± 0.00, 0.99 ± 0.00, 0.97 ± 0.01, and 0.84 ± 0.05 for the PDC-Optimal method and 0.99 ± 0.01, 0.98 ± 0.01, 0.82 ± 0.27, and 0.60 ± 0.28 for the PDC-Conv method for 0.125, 0.25, 0.5, and 1 of τ, respectively.Otherwise, there was no significant difference in the deeper imaging depth (6-20 mm) between the methods.Both methods yielded > (0.98 ± 0.03), even when τ is set to 1.These results confirm the efficacy of the PDC-Optimal method: a well-defined design methodology with user-definable parameter τ, which is not available with the PDC-Conv method, and increasing uniformity in image quality compared to the PDC-Conv method.Table 1 shows the hardware utilization and power consumption in the unconstrained DDC, PDC-TS, and PDC-Optimal methods.Active hardware resources for the PDC-Optimal method were significantly lower than those required by the unconstrained DDC method: the use of FF by 84.35-87.23%and the use of LUT by 94.19-95.78%for τ ranging from 0.125 to 1. Compared to the PDC-TS method, the hardware complexity of the PDC-Optimal method was comparable when τ is 0.125, although 13.83% and 49.54% of FF and LUT requirements could be further reduced when τ is 1.These savings in hardware resources reduces power consumption.The PDC-Optimal method decreased the dynamic power usage by 92.04-94.53%compared to the unconstrained DDC method and saved 20.00-45.00% of the dynamic power compared to the PDC-TS method for τ values ranging from 0.125 to 1.  Table 1 shows the hardware utilization and power consumption in the unconstrained DDC, PDC-TS, and PDC-Optimal methods.Active hardware resources for the PDC-Optimal method were significantly lower than those required by the unconstrained DDC method: the use of FF by 84.35-87.23%and the use of LUT by 94.19-95.78%for τ ranging from 0.125 to 1. Compared to the PDC-TS method, the hardware complexity of the PDC-Optimal method was comparable when τ is 0.125, although 13.83% and 49.54% of FF and LUT requirements could be further reduced when τ is 1.These savings in hardware resources reduces power consumption.The PDC-Optimal method decreased the dynamic power usage by 92.04-94.53%compared to the unconstrained DDC method and saved 20.00-45.00% of the dynamic power compared to the PDC-TS method for τ values ranging from 0.125 to 1.

Discussion and Conclusion
In the last few decades, there have been extensive accomplishments in point-of-care US imaging with several electronic systems in dedicated units [2,11], tablet PCs [18], and smartphone-based form factors [19].This technological evolution inevitably necessitates more efficient receive beamforming for better clinical efficacy in terms of beamforming precision, system volume, and battery life.In this paper, we present the PDC-Optimal method for efficient implementation of ultra-compact US imaging systems as well as flexible user experience in various clinical settings.Using the L N (ch) and ∆d N (ch) optimized for each imaging zone (Figures 2 and 3), the PDC-Optimal method presented the flexible yet regulated beamforming accuracy by a user-definable delay error allowance, τ.The beamforming accuracy of the PDC-Optimal method could be comparable to the unconstrained DDC method, whereas the PDC-Conv method suffered from substantial image degradation in the near imaging field when the same number of zone segments was used (Figure 4).Moreover, the PDC-Optimal method could significantly reduce hardware complexity and dynamic power consumption compared to the unconstrained DDC and PDC-TS methods (Table 1), which will enable a more compact system volume and a lighter weight with a longer battery life.We already embodied the proposed PDC-Optimal method in 16-channel tablet PC-based and smartphone-based prototypes using a low-cost FPGA (Spartan-6 LX150, Xilinx Inc., USA) [18][19][20].The prototype was 180 × 55 × 35 mm 3 and ~180 g, supporting B-mode and color Doppler mode at 1 of τ.The optimization of the PDC-Optimal method in these prototypes is currently being conducted and will be presented in a future publication.
These advantages of the PDC-Optimal method may lead to the improvement in its clinical efficacy in point-of-care diagnostics.For example, in prehospital settings in disaster and battlefield medicine, requiring as many diagnostic tasks as possible with a single charging, the PDC-Optimal method may be configured for the longest-possible operation time while moderating beamforming accuracy (e.g., τ ≥ 1).Moreover, if there is any need for high-precision imaging during the imaging sessions, an update of the LUT (e.g., τ < 1) would shortly take effect and provide a higher image quality (up to the unconstrained DDC accuracy) over the entire imaging depth.Note that the transition of τ may necessitate a short FPGA reconfiguration time, but it could be minimized by the partial reconfiguration method [21].On the other hand, in a daily diagnostic setting at patients' bedside with accessible power supplies nearby, the maximal performance setting may be continuously used (i.e., τ = 0.125 for 16-f 0 temporal resolution in delay compensation).In summary, the proposed PDC-Optimal method with a clear design methodology will support dynamic performance calibration for the best user experiences in the widest possible clinical settings with a single US imaging platform.
Our further work will be first focused on a translational investigation to establish the standards in beamforming accuracy and battery life in various clinical settings with considerations on different US transducer specifications (e.g., FAST examination for internal bleeding or pneumothorax using low-f 0 linear/convex arrays with wide aperture, bedside triage of skin or musculoskeletal diseases using high-f 0 linear arrays with narrow aperture, and extreme settings for disaster, battlefield, and/or outdoor diagnostics).This study will help us to find the detailed parameters for each application to secure an effective range of beamforming accuracy (e.g., relative mainlobe width to DDC), the minimal battery life, and an ergonomically applicable system volume.From the quantitative design specifications established, a prototype will be further optimized to support the clinical diagnosis in the most effective form factor. Afterward, the prototype system will be validated by a comprehensive performance evaluation and human clinical trials in practical circumstances.In addition, we also envision the extended use of the proposed PDC-Optimal method beyond the conventional US diagnostics.The photoacoustic (PA) imaging is an emerging biomedical imaging modality, in which a rich optical absorptive contrast at acoustic imaging depth (several centimeters) can be obtained [22][23][24].Recently there have also been arising demands for, and investigations on, compact PA imaging systems [25][26][27], where an efficient beamforming solution is essential [28][29][30].This may be needed for space-limiting clinical applications such as surgical and interventional guidance [24,[31][32][33].Moreover, a compact system volume in PA imaging may extend its scientific efficacy.For example, it may enable an implantable and/or wearable PA neurosensing system for continuous monitoring, leading to successful translations of current neuroscientific findings and neuro-engineering innovations into practice [24][25][26][27][28][29][30][31][32][33][34][35][36].

Figure 1 .
Figure 1.Illustration for dynamic receive beamforming in a medical ultrasound system.

Figure 1 .
Figure 1.Illustration for dynamic receive beamforming in a medical ultrasound system.

Figure 2 .
Figure 2. The proposed PDC-Optimal method.(a) Schematic diagram; (b) determination of the N th zone length, LN(ch), which provides flexibility between beamforming accuracy and the number of calculations needed.Dotted line presents an ideal delay for each beamforming depth, z, at a channel, ch.LUT: look-up table; RTC: real-time controller; D: delay latch; RF: radio-frequency signal.

5 LFigure 2 .
Figure 2. The proposed PDC-Optimal method.(a) Schematic diagram; (b) determination of the N th zone length, L N (ch), which provides flexibility between beamforming accuracy and the number of calculations needed.Dotted line presents an ideal delay for each beamforming depth, z, at a channel, ch.LUT: look-up table; RTC: real-time controller; D: delay latch; RF: radio-frequency signal.

Figure 3 .
Figure 3.The optimization results for bit resolution of ∆d N [ch] in the PDC-Optimal method subject to allowance error.(a) The mean zone size averaged over all channels; (b) total LUT memory size.

Figure
Figure5ashows the point spread functions (PSFs) at different imaging depth for PDC-Conv and PDC-Optimal methods.In this figure, PSFs in the left and right subcolumns are reconstructed by PDC-Conv and PDC-Optimal methods corresponding to τ , respectively.Figure5b,cshows a quantitative evaluation of the correlation coefficients (CCs) of the PSFs and those by the

Figure 3 .
Figure 3.The optimization results for bit resolution of ∆d N [ch] in the PDC-Optimal method subject to allowance error.(a) The mean zone size averaged over all channels; (b) total LUT memory size.

Figure 3 .
Figure 3.The optimization results for bit resolution of ∆d N [ch] in the PDC-Optimal method subject to allowance error.(a) The mean zone size averaged over all channels; (b) total LUT memory size.

Figure
Figure5ashows the point spread functions (PSFs) at different imaging depth for PDC-Conv and PDC-Optimal methods.In this figure, PSFs in the left and right subcolumns are reconstructed by PDC-Conv and PDC-Optimal methods corresponding to τ , respectively.Figure5b,cshows a quantitative evaluation of the correlation coefficients (CCs) of the PSFs and those by the
• Q and LUT Q are the corresponding quantization operator and LUT memory size, respectively.The optimization of Q is conducted downward from an 18 bit resolution to lower resolutions(18, 17,  16, etc.)The LUT size L N (ch) is evaluated for each quantization resolution, and ∆d N (ch) Q giving the minimal LUT Q is set as ∆d N [ch].Note that the bit resolution for L N (ch) is defined by the maximal zone length in sample (e.g., 10 bits for 1,024 samples = a 2 cm imaging depth at 40 MHz of f s and 1540 m/s of mean sound propagation speed in a biological tissue).

Table 1 .
Hardware resource utilization and on-chip power consumption of the DDC, PDC-TS, and PDC-Optimal methods in a target FPGA.FF: flip-flop; LUT: look-up table.

Table 1 .
Hardware resource utilization and on-chip power consumption of the DDC, PDC-TS, and PDC-Optimal methods in a target FPGA.FF: flip-flop; LUT: look-up table.