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Geometric observation and oriented filters
As presented in Chapter 1, the difference between curvilinear, plane-like and blob-like structures is geometric. It is based on the size of their main dimensions. A curvilinear structure is longer in one of its dimensions, whereas plane-like structures are larger in two dimensions and blob-like structures have similar sizes in all dimensions. By counting the number of large sizes in all dimensions, one should be able to distinguish between the different structure types. To do so, we chose an approach based on oriented filters. We call oriented filter any filter which response depends on the orientation in which it is computed. Let F be an oriented filter and O be a set of chosen orientations in the 3D space such that Fo(I), o 2 O, is the response of filter F in the orientation o on image I. If a bright structure lies in orientation o, then Fo(I) will output a high value for this structure. Let us now consider a blob, a plane and a curvilinear structure, processed by such filter. The blob-like structure, as it lies in all orientations, presents a high response for all Fo(I). The plane-like structure will respond highly in almost all orientations except for those along its small size. In the same way, the curvilinear structure will only respond highly in a few orientations corresponding to those along its only large size (see Figure 12).
Our strategy to only preserve curvilinear structures relies on the geometric difference between structures. We propose to filter an image by an oriented filter and to classify each structures based on the number of high responses.
Alternative mathematical morphology based approach
In a preliminary work, partially presented in [59], we first explored an alternative approach to distinguish between curvilinear structures and plane-like structures, based on mathematical morphology tools. This approach also relies on path operators but involves, in addition, radial openings. Figure 13 illustrates this approach. In part due to the incoherence between the radial and path operators, we showed that this type of approach was not as effective as purely path-based approches to preserve curvilinear structures while removing other structures.
filtering curvilinear structures with line segments as structuring elements
A common approach to filter curvilinear structures in mathematical morphology is to use openings or closings with line segments as SE. For the sake of clarity, we only present this approach with openings as we consider bright structures of interest on a dark background.
However, it is easily transposable to dark structures on a bright background using closings instead of openings. Filtering an image by an opening with a line segment preserves curvilinear structures, but only those that everywhere include the SE. The SE length is a parameter that is chosen according to the application, but the orientation issue remains. In order to filter curvilinear structures with a arbitrary orientations, the classical approach is to use a bank of line segments with the same length, but with varying orientations. Openings with each of these SEs are applied and the final result is the union of all the openings (see Figure 19). This approach is sometimes called Radial Opening.
Radial opening requires the computation of many erosions/dilations with line segments. Van Herk et al. [96] proposed in 1992 an efficient algorithm to perform 2D erosions and dilations with a line segment of arbitrary size, but only in the main natural orientations of a 2D image (horizontal, vertical and the two diagonals). In 1996, Soille et al. [83] extended this work by proposing an algorithm to compute erosions and dilations along discrete lines at arbitrary angles. Nevertheless, this algorithm results in erosions and dilations which are not translation invariant. Finally, a translation invariant algorithm, with the same complexity as the latter, was proposed in 2001 [82]. Recently, Van Droogenbroeck and Buckley [95] proposed a novel algorithm to compute erosions but also openings with line segments. This is currently the fastest known algorithm.
Radial opening presents many advantages. First, as the opening is anti-extensive, this approach only decreases the signal of non-curvilinear structures. This means that it neither creates false positives nor moves edges of the sough structures. Moreover, this approach is non-local since it looks into a neighborhood of the size of the SE. This certainly avoids false detections, which may be caused by local analysis. Finally, the only parameters of a radial opening are the SE length and the number of different orientations. However, this approach assumes that curvilinear structures are locally straight. This is a restrictive hypothesis. Indeed, to filter tortuous parts of curvilinear structures, the SE length should be reduced to a length small enough for the straight SE to fit the tortuous parts of the curvilinear structure. Nevertheless, all the non-curvilinear structures into which the line SE can fit are also preserved by openings. Consequently, the smaller the SE length, more both tortuous curvilinear structures and non-curvilinear structures are preserved (see Figure 20).
Table of contents :
1 introduction
2 curvilinear structure features in the literature
2.1 Local approaches
2.1.1 Derivative-based
2.1.2 Filter banks
2.2 Non-local approaches
2.2.1 SE-based approaches
2.2.2 Connectivity-based approaches
2.2.3 Path-based approaches
i tubular features extraction
3 context
3.1 Motivation
3.2 General Strategy
3.2.1 Geometric observation and oriented filters
3.2.2 The oriented filter choice
3.2.3 Alternative mathematical morphology based approach
4 previous work on path operators
4.1 Mathematical morphology basic operators
4.1.1 Structuring elements
4.1.2 Dilation and erosion
4.1.3 Opening and closing
4.2 Filtering curvilinear structures with line segments as structuring elements
4.3 Path Operators
4.3.1 Paths
4.3.2 Binary and grey-level path opening
4.3.3 Space discretization
4.3.4 Path opening algorithms
4.4 Robust to noise path opening
4.4.1 Incomplete path opening (IPO)
4.4.2 Robust path opening (RPO)
5 the rorpo framework
5.1 Ranking orientations
5.1.1 Pointwise rank filter (PRF)
5.1.2 Counting high path opening responses
5.2 Intensity feature
5.2.1 The generic RORPO intensity operator
5.2.2 Setting the it threshold
5.2.3 Limit orientations in 3D
5.3 Directional feature
5.3.1 Finding the orientations of interest
5.3.2 Combining the orientations
5.4 Multiscale approach
5.4.1 Length based multiscale approach
5.4.2 Methodology
6 algorithmic considerations
6.1 RORPO algorithm
6.2 Simplified robust path opening
6.3 Parameters
6.3.1 Path length
6.3.2 Noise robustness parameter
6.3.3 Orientations
6.4 Computational cost
7 results and comparisons
7.1 Comparison framework
7.1.1 Compared methods
7.1.2 Evaluation criteria
7.1.3 Parameters Optimization
7.2 Synthetic images
7.2.1 Intensity feature comparison
7.2.2 Directional feature comparison
7.3 Real images
7.3.1 Intensity feature comparison
7.3.2 Directional feature comparison
ii variational regularization for curvilinear structure filtering and segmentation
8 introduction
9 previous work
9.1 Inverse problem and the need of a regularization term
9.2 Previous work on total variation in image analysis
9.2.1 Tikhonov regularization
9.2.2 Total variation
9.2.3 TV vs. Tikhonov regularization
9.2.4 Solving the total variation
9.3 Segmentation model
9.3.1 The Chan et al. model
9.3.2 Proximal splitting algorithms
10 directional total variation
10.1 Motivation
10.2 The directional total variation
10.2.1 Directional gradient
10.2.2 Embedding the curvilinear structure features
10.3 Solving the segmentation model
10.4 Results and Comparisons
10.4.1 Compared methods
10.4.2 Synthetic image segmentation
10.4.3 Segmentation of the Drive database
11 conclusion
Appendix
a notations
a.1 Image
a.2 Dot product of images
a.3 Norms of an image
a.4 Gradient of an image
a.5 Divergence of an image
a.6 Norms of the gradient of an image
b subgradient and proximity operator
b.1 Subgradient
b.2 Proximity operator
b.3 Link between the subgradient and the proximity operator
c proof of some statements
c.1 TV of a differentiable function
c.2 TV of the indicator function
list of publications and communications
acronyms
bibliography
