1. FPGA implementation of undecimated wavelet transform denoising and
Fourier deconvolution for photoacoustic microscopy
Ryan T. Maxson, Scott P. Mattison, Brian E. Applegate
Laboratory for Optical and Molecular Imaging
Department of Biomedical Engineering, Texas A&M University
Wiener
Deconvolution
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Wavelet Transform
(undecimated)
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Wavelet Thresholding
Hilbert
Transform
Amplitude
Envelope
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FIFO to Host
for Display
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How can we recover x(n) from y(n) ?
References
• Mattison, Scott P., Ryan T. Maxson, and Brian E. Applegate. "Continuous real-time
photoacoustic demodulation via field programmable gate array for dynamic imaging of
zebrafish cardiac cycle."Biomedical optics express 4.8 (2013): 1451-1463.
• Neelamani, Ramesh, Hyeokho Choi, and Richard Baraniuk. "ForWaRD: Fourier-
wavelet regularized deconvolution for ill-conditioned systems." Signal Processing, IEEE
Transactions on 52.2 (2004): 418-433.
• Holan, Scott H., and John A. Viator. "Automated wavelet denoising of photoacoustic
signals for circulating melanoma cell detection and burn image reconstruction."Physics
in medicine and biology 53.12 (2008): N227.
Supplementary Information
• UWT is used for shift invariance
• Filters are derived from Haar wavelet
• Wavelet design decomposes signal into a “partition of unity” frequency responses
• Reconstruction is simplified to a summation of decomposition scales.
• Wiener filter coefficients can be recalculated on the fly requiring several clock
cycles of offline time.
• Hilbert Transform is approximated as FIR filter to save resources
PAM FPGA Processor (Proposed Method)
vDeconvolution
Wiener Filter
• Performed on RF signal
• Peaks are resolved but there are
numerous inversion artifacts
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wDenoising
Undecimated Wavelet Transform (UWT)
xEnvelope – Analytic Signal
• Noise is mostly suppressed
• Peak resolution is maintained
• Further SNR enhancement and resolution
maintenance
• Non-parametric, independent of detection
configuration
mz  2556.
FPGA Benchmarks
Component
Latency
(clock cycles)
Delay (ns)
@ 80 MHz
Deconvolution 9 112.5
Wavelet Decomposition 15 187.5
Thresholding/Reconstruction 5 62.5
Hilbert Transform 19 237.5
Enveloping 24 300
Routing/Miscellaneous 5 62.5
TOTAL 77 962.5 ns
Acquisition System
• Off-axis configuration
• 532 nm Nd:YAG laser pulsed @ 50 kHz
•10x Fluor objective lens
• Galvanometer mirror raster scanning for volumetric data
• 27 MHz Panametrics ultrasound transducer
• National Instruments PXIe-7965R, Xilinx Virtex-5 SX95T
)(
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ndl )(
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ndL )(~ naL
Generalized linear system x(n): input (axial absorption profile)
h(n): transducer impulse response
γ(n): random additive noise
y(n): measured output (RF signal)
)()()()( nnhnxny  
Results
Photoacoustic microscopy (PAM) is an optical imaging modality which combines
optical excitation with acoustic detection pathways to achieve optical resolution
deep in tissues. Real-time PAM is currently hindered by the need for post-
processing in the form of signal enveloping to achieve morphologically accurate
axial information. Additionally, axial resolution is worse than lateral resolution
resulting in an asymmetric voxel. Resolution improvement can be achieved by
inverting the ultrasound transducer’s frequency response using a deconvolution
scheme. To ensure numerical stability and limit inversion artifacts, a significant
computational cost is required beyond that of enveloping, further contributing to
processing bottlenecks. We demonstrate the use of a field programmable gate
array (FPGA) to perform Fourier-regularized deconvolution and wavelet denoising
enabling real-time processing in parallel with single detector off-axis PAM (OA-
PAM) signal acquisition. Simulations and real data acquired from the FPGA
processor show an improvement in axial resolution along with increased SNR.
Abstract:
Implementation
)(~ nx
The mapping of the
axial optical
absorption profile to
the detected RF
signal can be modeled
as a linear system
with convolution and
additive noise.
System Inversion: We model an inverse system that obtains the best estimate )(~ nx
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Introduction
Chick
Embryo
RF signal Proposed MethodComparison Method
Cross Sections
(B-scan)
Sample A-lines
142.3--
Depth
Depth
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Wavelet Filters
Frequency Response (Magnitude) Impulse Response
: Collective frequency response (unity) : Collective impulse response (delayed Dirac delta)
Individual responses of 4th level decomposition (4 detail + 1 approximation set of coefficients)
Rhodamine Capillaries – Volume Rendering
Two 50 μm inner diameter capillary tubes filled with rhodamine 6G and arranged in a crosshair fashion.
Dimensions are 128 pixels in the lateral directions and 64 in the axial dimension.
Z: Axial direction (depth)
X&Y : Lateral dimensions
Calculated Average Diameter : 65 μm
uRF input
Comparison Method
Envelope of analytic signal
mz  7593.
Conclusions
•Continuous throughput after 962.5 ns initial latency
•Axial resolution enhancement of 40% over RF envelope
•Wavelet thresholding removed most noise without
noticeable signal loss.
•Except for a priori knowledge of transducer response,
system is generalized and should be widely applicable
Total acquisition and processing time: 328 ms
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f
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z ltheoretica
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 : transducer angle with respect to optical axis (40º)
: speed of sound in water (1482 m/s)
:transducer bandwidth (27 MHz)
v
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Objective:
Create a PAM FPGA processor which accomplishes the following:
• Inversion of the ultrasound receiver’s blurring operator via deconvolution
• Optimally balances the restoration of axial resolution and the suppression of noise and artifacts
• High adaptability to changes in detection configurations
Considerations:
• Additive noise makes deconvolution highly unstable (“ill-posed”)
• Deconvolution is best modeled in Fourier domain for ideal conditions, but PAM signal lacks sparsity
in its Fourier coefficients
• Continuous throughput requirement  Iterative techniques are not practical
•“ill-posed” problem requires significant computation spanning multiple transformation domains
• Fourier-wavelet approach is a widely applicable solution that achieves optimal results due to its
balance of time/frequency uncertainty
•Two point absorbers
•Shift and sum (PSF)
•Added random Gaussian noise (σ=0.15)
mz 2556.
Axial Resolution z
Simulated Data
69.25Thickness (μm)
biomed.tamu.edu/lomi
)(ny
Peak SNR (Simulated Data)
Signal
PSNR
(dB)
RF 10.35
RF Envelope 17.12
Deconvolved/Denoised
Envelope
18.83