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The proposed Doppler data acquisition performed by using a real Doppler ultrasound data (heart data). The data downloaded from the H. Torp group website. The length of the Doppler data was 2032 and the numbers of measurement used for reconstruction are, 5%, 20%, 40%, 60% and 80%. Software programs written in MATLAB (MathWorks, Inc., Natick, MA) were developed and used to generate the Doppler ultrasound spectrogram, before and after applying the CS theory. Also all the reconstruction algorithm developed in MATLAB program.
4.2 Doppler Signal Reconstruction
Different reconstruction algorithms propose for signal and image reconstruction via CS theory. In the work four different algorithms were used to reconstruct the Doppler ultrasound signal, the algorithms are
Reconstruction via â„“1 Minimization (â„“1-norm)
Reconstruction via Orthogonal Matching Pursuit (OMP)
Reconstruction via Compressive sampling Matching Pursuit (CoSaMP)
Reconstruction via Regularized Orthogonal Matching Pursuit (ROMP)
4.2.1 Reconstruction via â„“1 Minimization
The Doppler ultrasound data were loaded into Matlab, The coefficient matrix A in term of M x N have been selected non uniformly at random, which is done with normalized vectors sampled independently and uniformly using the sparse model, the Doppler signal represented linearly as follows:
The basis A selected to suit the discrete cosine transform and assume that most of the coefficients x are zero, so x is sparse. In real signal it's not possible to collect signal without noise, thus the noise z added to the signal, the linear signal described as:
Where the vector f is M x 1, z in an M-dimensional measurement noise vector and M << M
For solving the Doppler signal (4.2), another linear operator is needed. The linear operator chosen to be as follows:
Where, is a random sample from and is the subset of the rows of the identity operator.
To recover the signal we need to recover the coefficient by solving where .
By using â„“1-norm
Recover the sparse signal. The recovered signal applied into the Matlab code to generate the recovered Doppler spectrum.
4.2.2 Orthogonal Matching Pursuit
Orthogonal matching pursuit algorithm from sparse approximation used to reconstruct the Doppler ultrasound signal. Doppler ultrasound signal was sampled randomly and constructed by using CS via OMP algorithm to create a reconstructed Doppler signal, which is used to generate a Doppler ultrasound spectrogram using a much fewer number of measurements M. The data constructed by using OMP begins by finding the column of A most related to the measurements. The algorithm then repeats this step by correlating the columns with the signal residual, which is obtained by subtracting the contribution of a partial estimate of the signal from the original measurement vector. The measurement model is:
Where A is a measurement matrix in N x M, y is an M-dimensional and x is a sparse signal with k nonzero.
The signal x reconstructs by solving the relation (4.5) with the OMP algorithm as follows:
Input: Loaded Doppler signal vector A
Output: sparse signal vector x
Initialize the residual r0 = y. At each iteration, the observation vector is set, y = A*r, and add the index to the coordinate of its the largest coefficient in the magnitude. By solving the least square problem, the residual is updated r = u - Ay. Repeating this m = 2*k times give the recovered Doppler signal x. The recovered signal used to generate Doppler spectrogram by using MATLAB.
4.2.3 Compressive Sampling Matching Pursuit
To reconstruct the Doppler signal using CoSaMP algorithms we first need to create the measurement matrix A, later we create the sparse coefficient, which have a problem specific structure. We intend to reconstruct a vector x, the Doppler ultrasound signal in our case, with a few numbers of non-zero components, that is, with a CoSaMP recovery algorithm. Many others algorithms exist for signal recovery proposed in .
The Doppler signal with a length of 2032 was sampled randomly and constructed by CoSaMP using a few numbers of measurements M. To reconstruct the signal as mentioned before the measurement matrix A was selected randomly and then reconstruct the signal by solving the measurement vector y. The measurement vector is given in equation (4.5).
Applying CoSaMP to reconstruct the Doppler data by solving the measurement vector (4.5), lead to a good approximation of Doppler signal x. By using the largest coordinates, an approximation of the signal is found at each iteration. After each new residual is formed, reflecting the missing portion of the signal, the measurements are updated. This is repeated until all the recoverable portion of the signal is found. The whole CoSaMP algorithm for reconstructing the signal described below:
Counter t = 1
Find the 2s column most correlated with
Add them to the index set
Re-evaluate the solution by least square
Prune: Ω = the k largest coefficients of
Until stopping creation
4.2.4 Regularized Orthogonal Matching Pursuit
ROMP algorithm used to reconstruct Doppler ultrasound signal using a few numbers of points. The reconstruction performed using the Doppler signal of length 2032 and fife different numbers of measurements.
Doppler ultrasound signal with a length of 2032 was sampled randomly and constructed by ROMP using a few numbers of measurements M and sparsity level n. To reconstruct the signal, an N x M Gaussian measurement matrix A was selected randomly and then reconstructs the signal with ROMP by solving the measurement vector y. The measurement model was given in relation (4.5).
The reconstructed signal used later to generate the reconstructed Doppler spectrogram.
4.3 Reconstruction Time
The reconstruction relative time was calculated for each number of measurements in all the recovery algorithms used to reconstruct the Doppler signal. We have computed the relative process time by using Matlab program v. 7.0.1, which allow us to run the CS recovery algorithms. The recovery algorithms run on a TOSHIBA laptop model 2008 with Intel® Celeron @ 2.6 GHz, 3.0 GB of main memory and 512 MB RAM. The operating system of the laptop was Windows XP Home Edition Service Pack 2.
Each algorithm with specific numbers of measurements runs several times, the average relative time was calculated and compared for each.
The number of iterations for â„“1-norm algorithm only was evaluated at each numbers of measurements used. The process was repeated several times and the average was calculated at each measurements. The result shows that there is no significant difference in the number of iterations by using different numbers of measurements.
4.4 Reconstructed Image Evaluation
Root mean Square Error (RMSE) and Peak Signal-to-Noise Ratio (PSNR) expressed in dB were used to evaluate the quality, accuracy of the reconstructed images and compared between the resulting images. Those methods are widely used for evaluating the recovered images using random sampling.
4.4.1 Root Mean Square Error (RMSE)
The efficient of reconstructed images evaluated by using the root mean square error (RMSE) which is widely-used quantitative measurement. The RMSE calculated for two images I and II with dimension of (m-by-n), where I is the original image and II is the reconstructed image. The RMSE measurement is easily computed by the square root of (mean square error MSE) the average squared difference between every pixel in recovering image and the original image. The RMSE calculated as follows:
Where, I(i, j) and II(i, j) are the pixel values of the original and recovered image respectively and m, n are the size of an image.
4.4.2 Peak Signal-to-Noise Ratio (PSNR)
PSNR reflects the differences of the information contained between an original and recovered image. The PSNR numbers are reported in Decibels (dB) as a measure of the relative weight between two images. A higher number in dB indicates a higher correlation. The PSNR is directly proportional to the image quality. When PSNR is higher this indicates that the reconstruction is of higher quality. PSNR calculated as follows:
4.5 Applying Parallel Computing to the Doppler Signal
Parallel computing is an effective method proposed for process time reduction or analysis large set of data. During reconstruction the Doppler spectrogram it's very important to keep the time of reconstruction very low so as to display the image in real-time. The use of parallel computing techniques can enable us to utilize the number of processors to run comprehensive analysis in a reasonable amount of time.
The Matlab program for the reconstruction was run on serial implantation first, and then we run the same file in parallel implementation using Matlab parallel computing toolbox package in duo-core CPU. Parallelization techniques applied to the Doppler data after prepare the data for compressed sensing, before solving the CS algorithm the parallel algorithm was started as shown in figure 4-2.
Figure 4-2 Serial and parallel methods for CS reconstruction
The parallel initialized using the Matlab command as stated in , which allows multiple Matlab processes run on parallel computer clusters or multicores. In our case duo-core CPU, two Matlab started at the beginning of the initialization. The data separated between the channels, then the data on each channel reconstructed and then sum up to get the final recovered Doppler signal.