Introduction

    Ultrasound images have numerous medical applications:  measuring blood flow through vessels, estimating the extent of prostatic cancers, assessing the health of fetuses.  These images are most useful when the structures of interest are clearly defined.  Unfortunately, ultrasound systems are inherently very noisy, rendering it difficult to interpret the images and to successfully auto-detect features in the images.
    Simoncelli and Adelson (1996) have demonstrated the effectiveness of Gaussian white noise removal using a Bayesian wavelet coring technique.  Their technique requires prior knowledge of the probability density functions of both the noise and the original, clean signal, neither of which are typically known for an ultrasound image.  The focus of this project is to apply the Bayesian wavelet coring technique to a set of ultrasound images of the carotid artery and determine the effectiveness of this denoising method for this type of image.

Method

Statistics of Ultrasound Images

In order to apply the Bayesian wavelet coring technique to ultrasound images, the probability density functions for the noise and the original signal needed to be determined.  A series of 25 ultrasound images were obtained of the carotid artery, from the same perspective and at approximately the same point in the cardiac cycle.  By averaging these 25 images together, I obtained a "clean" signal.  The noise was then calculated by subtracting one of the ultrasound images from the "clean" image.

Histograms of the wavelet subbands of the "clean" image and the noise were constructed to determine the probability density function.  The highest subbands were ignored, because there were not enough coefficients to construct a histogram that could be modeled.  However, the other subbands clearly had Laplacian distributions, of the form:

                            P_x(x) = exp (-| x / s | ^ p).

The model parameters, p and s, depended on the histogram shape, which varied depending on the range of x values and the number of bins used in the histogram.  For this implementation, the full range for the subband coefficients were used, and the number of bins used was the square root of the number of pixels in the subband.  The parameters were determined by using Matlab's constr function to minimize the relative entropy (the Kullback-Leibler divergence):

                            relative entropy = P_x(x) * log10 ( P_x(x) / hist_freq(x))

                                    P_x(x) is as defined above for a particular p and s

                                    hist_freq (x) is the histogram values divided by the total
                                    number of subband pixels

Figure 1 shows the histogram frequencies for one subband of the clean ultrasound image and the noise associated with it.
 

Bayesian Wavelet Coring

Assuming that the image to be denoised can be written as y = x + n, where x is the clean image and n is the noise, we can use Bayes' rule to estimate y, given the probability densities of the noise and signal:

                            estimate = integral [ P_x|y(x | y) * x * dx ]
                                           = integral [ P_n(y - x) * P_x(x) * x * dx ] / integral [ P_n(y - x) * P_x(x) * dx ]
                                           = convolution (P_n, P_x * x) / convolution (P_n, P_x)

Figure 2 shows the Bayesian estimator determined from the models of the histograms of Figure 1. 

Results

Figure 3 shows a comparison between the "clean" ultrasound image, a test image, and the results from denoising using a Wiener filter and using the Bayesian coring algorithm. Table 1 below compares the SNR for 5 different ultrasound images with their corresponding denoised images. Improvement in SNR was seen for the images filtered with the Wiener filter. Bayesian coring produced variable changes in SNR, depending on how the noise was modeled. If the noise was assumed to have a Gaussian distribution, the resulting image had a higher SNR than the original test image and the Wiener filtered image. On the other hand, if the noise was assumed to have a Laplacian distribution, the resulting image had a lower SNR than either the original test image or the Wiener filtered image.

Discussion

The Bayesian wavelet coring method is able to denoise an image better than the Wiener filter, provided that the noise is modeled correctly. Since the relationship between the probability density function of the noise and of the clean image determines the coring function, large variances in the noise, as compared to that of the clean image, and/or poorly modeled noise will result in distorted coring functions. This will result in distorted estimated images and poor SNRs, similar to those produced by assuming a Laplacian distribution for the noise. Further investigation is needed to determine the reason for the distorted coring functions seen under this assumption: Matlab may not be producing an accurate fit, since the fit heavily depends on the bounds and starting values for the parameters s and p; the noise may actually have a Gaussian distribution; the images may be too noisy, so that the probability distribution functions of the noise have a larger spread than that of the images. Despite these issues with the noise model, the Bayesian wavelet coring has proven that it can effectively denoise ultrasound images.

Acknowledgments

Many thanks go to Dr. David Heeger for his assistance with the implementation of the wavelet coring.  I'd also like to thank Hewlett-Packard for the loan of the ultrasound machine, which was used to collect the test images for this project.

References

Simoncelli, Eero and Adelson, Edward, 1996, "Noise Removal Via Bayesian Wavelet Coring," Proceedings of the 3rd IEEE International Conference on Image Processing, Vol. I, pp. 379-382.