N

which is equal to the adaptive filter we wanted to construct (Eq. (39)). The under-braced terms sums to 1 because of the dual basis relation between the two bases Mk and Nk.

5.4 Estimation of Multidimensional Local Anisotropy Bias

Knutsson [27] has described how to combine quadrature filter responses into a description of local image structure using tensors. His use of tensors was primarily driven by the urge to find a continuous representation of local orientation. The underlying issue here is that orientation is a feature that maps back to itself modulo n under rotation, and direction is a feature that maps back to itself modulo 2n. That is why vector representations work well in the latter case (e.g., representing velocity) but not in the former case.

Knutsson used spherically separable quadrature filters [28],

where u is the vector valued frequency variable, p = |w|, u = 1^, and R(p) and Dk (U) are the radial and the directional functions, respectively,

where the vectors ek describe the principal axes of the local signal spectrum (the locality is defined by the radial frequency functions of the quadrature filters).

The distribution of the eigenvalues, A1 < X2 < ... < , describes the anisotropy of the local spectrum. If the tensor is close to rank 1, i.e., there is only one large eigenvalue (A1 >>1K, ke {2,..., N}), the spectrum is concentrated to the line in the Fourier domain defined by e1. Further, if the tensor is close to rank 2 the spectrum is concentrated to the plane in the Fourier domain spanned by e1 and e2.

5.5 Tensor Mapping

The control tensor C used in the adaptive scheme is based on a normalization of the tensor described in the previous section (Eq. 54):

where nk is the filter direction, i.e., D(m) varies as cos2(^), where รง is the angle between u and the filter direction, and where a is a term defining the trade-off between resolution and stability similar to Eq. (33). However, here the resolution trade-off is adaptive in the same way as the stationary Wiener filter a = 0 gives maximum resolution, and the larger the value of a, the smoother the Wiener solution. Maximum resolution is given when 11 is large compared to a, and the smaller 11 the smoother the solution. If the "resolution parameter" a = 0, the control tensor will be is the radial frequency function. Such functions are Gaussian functions on a logarithmic scale and are therefore termed lognormal functions. B is the relative bandwidth in octaves and p0 is the center frequency of the filter. R(p) defines the frequency characteristics of the quadrature filters.

With this normalization, the largest eigenvalue of C is yl = l,

and the resulting adaptive filter Hc becomes an allpass filter along signal direction

Ifit is desired to increase the image contrast, this can be done by increasing the high frequency content in dim parts of the image data adaptively. Assuming a normalization of the tensors T so that the largest eigenvalue is 1 globally, max(2:) = 1, this can be achieved by increasing the exponent of the ^ in the denominator.

This will increase the relative weights for low and medium-low signal components compared to the largest = 1). More elaborate functions for remapping of the eigenvalues can be found in [4,29,30].

5.6 Examples of Anisotropic Filtering in 2D and 3D

The filters used in the examples below have in the 2D examples size 15 x 15, and 15 x 15 x 15 in the 3D. The center frequencies of the quadrature filters differ in the examples, but the relative bandwidth is the same, B = 2. We have for simplicity approximated the stationary Wiener filter with the following lowpass filter:

Here p is the radial frequency variable, and pp is the cutoff

frequency. The more noise in the image, the lower the cutoff frequency plp that should be used.

Figure 7 shows the result of 2D adaptive filtering of MR data from breast imaging. The original image is shown to the left. To the right, the result from anisotropic adaptive filtering is shown. The quadrature filters used for estimating the local structure had center frequency ffl0 = re/3 and the lowpass filter H had a cutoff frequency plp = re/4. The control tensor C was defined by Eq. (56) with a = 1% of the largest ^ globally.

Figure 8 shows the result of 3D adaptive filtering of MR data through the skull. The original image is shown to the left. The result from adaptive filtering is shown to the right. The quadrature filters used had center frequency ffl0 = re/2 and the lowpass filter H had a cutoff frequency plp = re/3. The control tensor C was defined by Eq. (60) with a = 1% of the largest ^ globally. Note that details with low contrast in the original image have higher contrast in the enhanced image.

Figure 9 shows the result of 3D (2D + time) adaptive filtering of ultrasound data of a beating heart. The left row shows images from the original time sequence. The right row shows the result after 3D filtering. The quadrature filters used had center frequency ffl0 = re/6 and the cutoff frequency of the lowpass filter H was plp = re/4. The control tensor C is defined by Eq. (60) with a = 5% of the largest 11 globally.

More details on implementing filters for anisotropic adaptive filtering can be found in [4] where the estimation of local structure using quadrature filters is also described in detail. An important issue that we have not discussed in this chapter is that in medical imaging we often face data with a center-to-center spacing between slices that is larger than the in-plane pixel size. Westin et al. [30] introduced an affine model of the frequency characteristic of the filters to compensate for the data sampling anisotropy. In addition

FIGURE 7 2D adaptive filtering of data from MR breast imaging. (Left) Original image. (Right) Adaptively filtered image. The adaptive filtering reduces the unstructured component of the motion related artifacts.

FIGURE 8 3D adaptive filtering of a coronal MR data set of the head with dynamic compression of the signal. (Left) Original image. (Right) Adaptively filtered image. Note the improved contrast between brain and cerebrospinal fluid (CSF).

FIGURE 9 Spatio-temporal adaptive filtering of ultrasound data of the heart. (Left) The images for the original image sequence. (Right) Result after 3D adaptive filtering. Note the reduction of the specular noise when comparing between filtered and unfiltered image sets.

to changing the frequency distribution of the filters, they also show how this affine model can provide subvoxel shifted filters that can be used for interpolation of medical data.

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