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The reconstruction of 3D ultrasound (US) images from mechanically registered, but otherwise irregularly positioned, B-scan slices is of great interest in image guided therapy procedures. Conventional 3D ultrasound algorithms have low computational complexity, but the reconstructed volume suffers from severe speckle contamination. Furthermore, the current method cannot reconstruct uniform high-resolution data from several low-resolution B-scans. In this paper, the minimum mean-squared error (MMSE) method is applied to 3D ultrasound reconstruction. Data redundancies due to overlapping samples as well as correlation of the target and speckle are naturally accounted for in the MMSE reconstruction algorithm. Thus, the reconstruction process unifies the interpolation and spatial compounding. Simulation results for synthetic US images are presented to demonstrate the excellent reconstruction.

Medical ultrasound is a widely used imaging modality because of its real-time, nonradioactive, low-cost, and portable nature. While currently the majority of clinical ultrasound is based on 2D cross-sectional slices, a lot of recent researchers have shown their interest in 3D ultrasound, which is anticipated to have a lot of advantages over conventional 2D ultrasound by increasing spatial anatomical detail, facilitating accurate measurement of
organ volumes and improving diagnostic comprehensibility [

Fenster and Downey described various methods that have been used to perform 3D ultrasound imaging [

In freehand system, information about the third dimension is achieved by the transducer’s
movement. In clinical diagnostics, physician usually moves the transducer along
one direction to acquire a series of slices. Those slices are nearly but not
exactly parallel, some of which may intersect. These 2D images are usually
interpolated into a Cartesian volume for visualization and data analysis. Each
pixel in the 2D B-scans is placed in the corresponding position in the 3D
volume. If B-scans do not intersect a particular voxel in the Cartesian volume, the voxel’s value is estimated by interpolation. This process is depicted in Figure

Illustration of 3D volume reconstruction from 2D freehand B-scans.

Many 3D interpolation algorithms have been proposed, such as voxel nearest neighbor
(VNN) [

Besides the performance of interpolator, the 3D volume’s image quality also depends on
other variables such as transducer geometry, frequency, focal zone position, and time gain compensation. Within a slice, the resolution (in-plane) is determined by the pulse bandwidth and transducer aperture. In the direction perpendicular to the slice (elevation),
the resolution is determined by the thickness of the slice and the inter-slice
distance. In general, the in-plane resolution is much higher than the elevation
resolution due to the transducer thickness and large elevation sampling
intervals. The volume interpolated from a single sweep data set therefore has
nonuniform spatial resolution. This effect is shown in Figure

Nonuniform resolution feature in bilinear reconstruction volume from freehand 2D slices: (a) high resolution in cross-sectional view, (b) low resolution in sagittal view.

In clinical imaging, the operator usually acquires several sets of B-scans of the same
target region from different interrogation angles and different sweep directions to increase the details in each look. Another benefit of this technique is that redundant data are acquired with statistically independent speckle patterns. Compounding those data, the underlying image information will sum constructively while the speckle artifact will be averaged out, resulting in a reduced speckle image. This technique, known as “spatial compounding,” has been proven effective [

However, there is a tradeoff between speckle reduction and
spatial resolution [

Compounded 3D volume for two orthogonal datasets: (a) cross-sectional view, (b) sagittal view.

The conventional compounding method does not address spatial resolution disparity and just simply averagesredundant data. Each sweep of data is first interpolated into a 3D volume. Then those 3D volumes are spatially registered and averaged. The final compounded volume has low-spatial resolution due to averaging of nonuniform resolution data. From a statistical signal processing perspective, both interpolation and spatial compounding are estimations of an underlying signal in the presence of speckle noise. Thus, the two processes can be unified with a single objective of minimizing the estimation error. In this paper, we propose using the minimum mean-squared error (MMSE) principle to optimally combine the acquired 2D data, achieving interpolation and compounding simultaneously.

The remainder of this paper is organized as follows. Section

In this section, we derive the MMSE reconstruction algorithm based on statistical
models of the image and speckle noise. Let

Let

The inversion of measurement covariance matrix

Appling the matrix inversion approximation to (

For multiple datasets, each sweep of B-scans has poor resolution in the elevation direction due to transducer’s thickness and large sampling interval. In the frequency domain, each sweep only occupies a narrow strip. The direction of the strip is different for each data set,
reflecting the different direction and look angle of each sweep. Reconstruction
of the original 3D high-resolution image from a single sweep of B-scan is an
ill-posed problem. However, when multiple images of the same source are blurred
by different blurring kernels, a well-posed reconstruction problem can be formulated
if the data sets are acquired with well-distributed angles in the frequency
domain. In image processing techniques, this is also called “super-resolution (SR) image reconstruction”, which refers to the process of
reconstructing a high-resolution image from multiple low-resolution images. SR
image reconstruction is an active research field, which is introduced in the
two review papers [

A lot of approaches for SR image reconstruction have been proposed. These include: iterated backprojection methods [

The cause of resolution degradation using simple averaging methods becomes clear in the
frequency domain. When multiple data sets are averaged, the nonoverlapping high-frequency components are weighted down relative to the overlapping low-frequency components, thereby
lowering the resolution. Our approach does not weight the frequency components
of the data set equally. Rather, we weight them according to the SNR at the
current frequency, in a MMSE sense similar to the Wiener filter. In more
detail, let

According to standard statistical signal processing theory, the MMSE estimate of

Note that if there is only one dataset, (

In this section, we validate the MMSE reconstruction method using both synthetic and
experimental ultrasound data. The synthetic simulation allows the computation
of ground truth information and thus quantification of algorithm performance.
The method to synthesize ultrasound data is described in [

The 3D ultrasound signal

In the first simulation, we reconstruct a spherical object from one sweep of synthetic ultrasound data. We synthesize 71 parallel slices. In each slice, the target is a dark circular disk with varying radii. From these 71 slices, 20 slices are randomly selected. We then try to reconstruct the original 3D data from these 20 slices.

To test the performance of MMSE reconstruction method, we compared its result with the most
frequently used interpolation method—bilinear interpolation method. In the bilinear
method, each reconstruction point is estimated by the nearest two samples in
the B-scans that fall on either side of it. The point is then set to the
inverse distance-weighted average of the two contributing samples. In our
algorithm, not only two nearest samples, but also all samples within a window are also used to estimate the reconstruction point. Different window sizes may be used according to the image content. Here, we select a

Reconstruction result for a spherical target from synthetic ultrasound data: (a) a slice of simulation data, (b)-(c) a slice of reconstruction result from bilinear and MMSE methods, (d) sagittal view of simulation data, (e)-(f) sagittal view from bilinear and MMSE reconstruction results.

In the second experiment, we reconstruct a 3D image from clinical B-scans. Prostate
cancer is reported to be the second most frequently diagnosed cancer in the United States
male population and is the third most frequent cause of cancer death. Radiation
oncology treatment is one of the primary methods for killing cancer cells in
the prostate. Accurate location of the prostate is important in delineating
prostate boundary and improving dose delivery accuracy of radiation therapy.
Figure

(a) A slice of clinical US B-scan of prostate, (b) a slice of bilinear reconstruction, and (c) a slice of MMSE reconstruction.

In the final simulation, we validate the multiset MMSE reconstruction algorithm. In the reconstruction, we consider date sets that are synthesized but yet with realistic speckle patterns and noise statistics. The benefits of using synthetic data are that they are easier to obtain, and there is no data registration error and no truncation of the field of view. This allows us to focus on evaluating the resolution degradation problem due to compounding disparate resolution data.

The simulation process is as follows. First, we synthesize two ideal 3D ultrasound images

Two synthetic 3D ultrasound image sets

The acquired image is the average of several adjacent slices due to transducer
thickness. So, we convolve

Two reconstructed 3D ultrasound image sets: (a)

We then generated 3D compounded ultrasound images

Comparison of compounded 3D ultrasound images by two methods: (a) cross-sectional view of the dataset by conventional method, (b) sagittal view of the dataset by conventional method, (c) cross-sectional view of the dataset by MMSE method, (d) sagittal view of the dataset by MMSE method.

From the simulation above, we have demonstrated the algorithm with compounding of 2 data sets, and the advantages are clear. It is expected that performance difference to be even bigger, if more data sets are compounded. This technique could increase small-lesion detectability and give more accurate measurement of organ volume.

In this paper, we develop a novel reconstruction method in medical ultrasound. Both the signal and noise’s statistics are incorporated in the MMSE reconstruction formulation. The MMSE reconstruction outperforms bilinear reconstruction in terms of speckle signal-to-noise ratio and contrasts between the target and homogeneous regions.

Another advantage of MMSE reconstruction method is that it can be applied in 3D compounding. Conventional ultrasound compounding only considers the data spatial redundancy and simply averages all data sets. Our method takes into account the different degrees of redundancy for different frequency components. The frequency components are weighted not equally but according to the Wiener filter/MMSE principle. The result is that high-frequency components are better preserved.

We also observe that this reconstruction algorithm is very computationally affordable, since each frequency component of the reconstruction is estimated separately, and the transformation between spatial and frequency domains can be done via FFT. The limitation of the proposed algorithm is that the 2D freehand B-slices must be acquired approximately parallel and equally-spaced, because only in this case, the sampling and interpolation process can be modeled as a low-pass filter.

It should be noted that our reconstruction formula is derived under the assumption of additive noise. But in ultrasound data, the speckle is better modeled as multiplicative noise. Deriving a MMSE reconstruction formula applicable to multiplicative noise is a difficult task since nonlinear estimation problems are involved. In this paper, we apply MMSE reconstruction to ultrasound data with multiplicative noise, even though additive noise is assumed in the formulation. Both synthetic and experimental results have demonstrated MMSE reconstruction method’s good performance.

This work was supported in part by NSF Grant no. 0635288.