Automatic detection of sparse landmarks

Increasing attention has been focused on the estimation of myocardium border deformation. In our proposed tracking approach we aim to select a group of landmarks on the myocardium border which can be best tracked across the cardiac cycle. Some points such as the mitral valve points are distinctive, but many other points on the myocardium boundary share similar

5.2. Methods 113

appearance and shape features, which leads to ambiguities when these points are being tracked. We propose an entropy-based landmark detector, which identifies landmarks that are recognisable in all frames throughout the cardiac cycle.

For the pixel x in frame f1, the detector defines a similarity measure based on sum of square

differences (SSD) between a patch centered at x and a patch centered at the pixel y_i in frame fn(n = 2..N ), which are denoted by Ωx and Ωyi accordingly in Figure 5.2. The size of the

patches are 5mm × 5mm and y_i is inside the search region Rn of the size of 40mm × 20mm

for x in frame fn. The search region is defined according to the analysis of the distribution

of the maximum displacement of the valve points in Chapter 4 (see Figure 4.10). Therefore a self-similarity map SMn in frame fn is generated for x as:

SMn= {SSDi, yi ∈ Rn}

x

f

R

_Ω

f

Ω

Figure 5.2 – This figures shows the search region Rn in frame fn for pixel x when calculate the

self-similarity map SMn. The similarity between x and yi is calculated based on patches Ωx and

Ωy_i.

Interestingly, when analysing different regions locally, we found that the corresponding distribution of the similarities were different. As shown in Figure 5.3a, the self-similarity map in the blood pool are more diffuse than those in the valve and apex regions. This is due to the fact that points in the valve and apex regions have more distinctive features which means that one is able to find good correspondences with high similarity. On the other hand, points in the blood pool have a more ambiguous appearance and thus less informative self-similarity maps will be found for those points.

Shannon entropy can quantify information content in a variable, so we then compute the entropy Hn of this self-similarity map SMn as follows:

Hn = −

X

k∈SMn

p(k) log p(k)

Here p(k) denotes the probability mass function (PMF) of k which gives the probability that a random similarity value in SMn is exactly equal to k. We then obtain the sum of the entropies

for pixel x:

H = X

n=2..N

Hn

Here n denotes the frames other than the ED phase. The entropy is maximized if all possible self similarities are equally likely to occur. On the other hand, a low value of Hn is usually

associated with less informative content (the value in the self-similarity map is either high or low), indicating a better alignment of the point x with only a small region in the searching region.

We hence assume that a low entropy of the similarity map corresponds to a more discriminative feature point. We choose four frames with an equal time interval from the whole image sequence to avoid unnecessary computations, and use these four frames to find a set of distinctive landmarks on the manually segmented contour of myocardium. This speeds up the landmark identification process substantially without losing generality.

However, we noticed that the entropy approach detects a number of landmarks that have distinctive appearance but are static and hence are less interesting. Points that undergo large deformation need to be selected too in order to well represent the deformation of the endocardium. In order to pick the landmarks with larger deformation, we analyze the dynamic information across the cardiac cycle by applying SVD to the cardiac cine MR images. SVD seeks to find a low rank approximation for a matrix and has a wide range of applications, such as compression and noise reduction in image processing [Andrews and Patterson III, 1976,S Wack

and D Badgaiyan, 2011].

5.2. Methods 115

corresponds to a frame in the sequence while each row of the matrix corresponds to a material pixel in the image. The SVD of X is defined as X = USVT, where U is a M × M matrix, and VT _{is a N × N matrix. The non-zero elements of the M × N diagonal matrix S are called}

singular values and can be sorted in descending order as S = diag(s1, ..., sN). Then, X can be

approximated by a matrix X(l) _{of rank l,}

X(l) =

l

X

k=1

ukskvkT (5.1)

The quality of the approximation can be measured by the ratio ω = Pl

j=1sj/

PN

i=1si. To

achieve a given quality of the approximation, we may only need a smaller number of singular values for a less complex image. In other words, if we specify a certain rank of decomposition l, with l < N , a higher approximation quality can be obtained for a less complex image compared to a more complex image. Instead of performing the SVD on the whole image, the same approach can be applied to patches of the image. For decomposition with rank l, different patches may have different approximation quality due to the fact that the complexity of patch is different from each other. In the case of a cine cardiac MR sequence, higher approximation ratios correspond to patches with no or little motion and lower approximation ratios corresponding to patches with larger motion, such as mitral valve point and apex (see Figure 5.3).

(a) (b) (c)

Figure 5.3 – This figure shows (a) self similarity maps at different locations, including valve points, apex, blood pool and a random point at the myocardium boarder. And a cine image overlaid with SVD image of approximation ratio set to (b)<= 80% (c) <= 70%.

Using the sum of the entropy and SVD approximation ratio, we select a set of sparse landmarks with the lowest values. A certain space is set between landmarks and edge conditions are checked in order to ensure that the selected landmarks can fully cover the endocardium and

epicardium.