The filters described in the previous sections are usually optimized for å specific type of noise and sometimes a specific type of signal. However, this is not usu-ally the case, and in many applications the signal characteristics vary considerably within the signal. In images, for example, neighborhoods of edges are character-ized by large signal variance as opposed to neighborhoods located in flat regions.
Space-varying filters are trying to incorporate information about the local signal neighborhood into the filtering process.
3.4.1 Varying filter sizes
One simple way of adapting the filtering process to the local signal characteristics is to vary the filter size based on estimation of local signal variance [41].
An example could be to start with a rather large window at each point. Then, if the estimated signal variance is above a certain threshold, the window size is reduced.
If the new local signal variance is still above a certain threshold, the window size is again reduced, and so on, until a window of size is reached.
EDGE
T
OUTPUT k(x,y)
1-k(x,y)
OR
+
Figure 3.5: Simple space-varying filter using a linear combination of two filters, from [40]
3.4.2 Linear combinations of multiple isotropic filters
Another way of obtaining local adaptability is by combining outputs from several different filters, each optimized for different signal characteristics. The weighting of the output from the different filters can be based on feature extractors tuned to the target signal characteristics of the specific filter [40].
Figure 3.5 shows a block diagram of a simple adaptive image filter differentiat-ing edges and flat neighborhoods. The feature extractor is an edge detector givdifferentiat-ing an output between 0 and 1 depending on how “edge-like” the local neighborhood is deemed to be.
3.4.3 Anisotropic window adaption
The filter window itself can be made adaptable to different signal structures. In images, for example, the filter window can be aligned with the edge, so as not to encompass pixels from both sides of the edge.
One way of estimating local structure is that of Granlund and Knutsson[14] which is explained in chapter 5. Other examples include fitting multi-variable polygons to the signal [31], and techniques based on binary morphological erosion and dilation [41]. More pixel-based filters, like the -trimmed mean, the sigma filter and the
-nearest neighbors, can be argued belong to another class of filters. Descriptions of those filters can be found in most introductory books on image processing.
Chapter 4
Discontinuity Filter and Speckle Tracking
This chapter presents two important topics referred to later in this thesis. The temporal discontinuity filter is used extensively throughout the experiments, and speckle tracking is used in addition to the energy-based method for estimating motion in the ultrasound image sequences.
4.1 Discontinuity filter
4.1.1 Introduction
This section describes the temporal filtering method for ultrasound image sequences suggested by Olstad[38]. The general idea is that the time evolution for each indi-vidual spatial coordinate is divided into homogeneous regions, whereupon the tem-poral filtering is performed with minimal interaction across the boundaries defining the regions.
A synthetic example of a time sequence divided into five homogeneous regions for a fixed spatial coordinate is shown in figure 4.1. The sequence could have been generated by a high-intensity object present in the time interval 5
and a rapidly moving structure present in the interval . Reconstruction based on piecewise constant and linear functions are superimposed on the intensity sequence.
4.1.2 Homogeneity measure requirements and discontinuity detection Let
1 be any sequence of temporal consecutive intensity val-ues. A homogeneity measure for is any nonnegative function:
satisfying
t0 t
Figure 4.1: Optimal discontinuity locations and piecewise reconstruction
whenever and .
Let
contain a set of ascending indices
M 5 M M M ! M 0
indicating discontinuities in a time window of0 samples. The discontinuities gen-erate the following subsequences:
5 5
An error function, , on the set of discontinuities,
, can then be defined using the homogeneity measure on each :
2
365
(4.1)
Since the set of all possible
is finite, a solution to minimizing (4.1) will always exist, but it is not necessarily unique. An arbitrary choice is made in the case of multiple minimums. The integer value is an algorithm parameter, and depends on a-priori information and computational complexity considerations.
4.1.3 Filtering and homogeneity measures 4.1.3.1 Piecewise constant functions
A natural choice of homogeneity measure is the square error,
8
where denotes the mean value of . Thus, measures how well the intensity function can be approximated by a function of constant segments.
An example of intensity reconstruction using optimal constant segments is shown in figure 4.1(a). This approach will have substantial impact on the noise level, and increase the sharpness of the intensity transitions. Efficient algorithms for the computation of the optimal discontinuity set utilizing dynamic programming can be found in [38]
4.1.3.2 Piecewise linear functions
Reconstruction using a piecewise linear function can be obtained by using the ho-mogeneity measure:
This makes A@ measure how well the intensity function can be approximated by a function of
linear segments, as illustrated in figure 4.1(b). Higher order parametric functions, such as splines, can also be utilized in the reconstruction [38].
4.1.3.3 Incorporating spatial persistence
Components modeling spatial persistence can be incorporated by weighting the intensity observations in the error measurement , i.e.,
$
where is a weight coefficient reflecting the spatial persistence of . could for example be based on the order statistic of in a local spatial neighborhood such that
if is the median and dropping towards zero as turns to the minimum or maximum value.
4.1.4 Shorter time windows
When using shorter time windows, ordering the window samples in a cyclic man-ner, as illustrated in figure 4.2, improves performance. The two discontinuities are computed such that the homogeneity measures of the two intervals in the cyclic ordering is minimized.
Allowing a small leakage across the boundaries, i.e., including weighted border pixels in the reconstruction, will improve the noise suppression at the cost of only a minor edge degradation.
0
Figure 4.2: Window samples ordered in a cyclic manner