The research interests cover visualization of multidimensional data, optimization theory and applications, data mining in databases, multiple criteria decision support, neural networks, parallel optimization, image analysis. The author of more than 240 scientific publications, two monographs, five textbooks. Editor in chief of the international journals

Radiologists need to find a position of a slice of one computed tomography (CT) scan in another scan. The image registration is a technique used to transform several images into one coordinate system and to compare them. Such transversal plane images obtained by CT scans are considered, where ribs are visible, but it does not lessen the significance of our work because many important internal organs are located here: liver, heart, stomach, pancreas, lungs, etc. The new method is developed for registration based on the mathematical model describing the rib-bounded contour. Parameters of the mathematical model and of distribution of the bone tissue on the CT scan slice form a set of features describing a particular slice. The registration method applies translation, rotation, and scaling invariances. Several strategies of translation invariance and options of the unification of scales are proposed. The method is examined on real CT scans seeking for its best performance.

Image analysis becomes a top technology assisting to make decisions in medicine. Images come from various sources: radiology, echoscopy, magnetic resonance, thermovision, tomography, etc. Many diseases may be diagnosed and their treatment observed using the computed tomography (CT) that is a technology allowing the inside of objects to be spatially viewed, using computer-processed X-rays. CT scans are 3D images, i.e. a collection of 2D images (slices), representing human body cross-section with a transversal plane. Such collections of images require special methods and means to handle graphical data, e.g. image segmentation, medical modelling, image registration.

The image registration is a technique used to transform several images into one coordinate system. Although it has applications in many fields, the medical image registration is important among them for aligning and comparing different images (Treigys

In Graf

When analysing transversal plane images, obtained by computer tomography scans, the peculiarity of the problem is that parts of different ribs are visible on the same slice. It is the reason why the models in Kindig and Kent (

This research deals with CT scan slice registration, based on the mathematical model that describes the ribs-bounded contour developed in Bilinskas

The registration problem could be solved by using the meta data of the DICOM header of a CT scan. However, the available information is often error-prone. Güld

The goal of this paper is to develop a registration method where the model of the rib-bounded contour serves as the basis of the similarity criterion of images (slices). In this case, we have a method of the feature-based registration. The problem is to find the most relevant slice in one scan to the chosen slice from another scan of the same patient. Registration of slices must be done independently of the patient position on the bed and of the radiocontrast agent injection. Feature-based methods find a correspondence between image features, such as points (Bouguet,

In Bilinskas

The second part of the model is a line-segment bounded by two points

Mathematical model (1)–(4) is defined by an array of 8 parameters

An example of CT scan slice with a model curve (red line).

Radiologists need to find a position of a slice of one CT scan in another scan. Formally, having a slice

For image registration, we need discrete points of the curve of model (

Some registration methods need weights of model curve points. Weights are gathered by distributing bone tissue points among the curve points

Finally, for each slice, we have a set of bone tissue pixels (points)

The registration method below applies translation, rotation, and scaling invariance. These invariances are usual in the image registration (Szeliski,

The comparison of slices is based on:

the values of parameters describing the mathematical model (

the sequence of discrete points of the curve of model (

the weights of points of the curve.

Model (

Most often the compared CT scan slices have a different scale. The scale depends on the parameters of CT scanner. These parameters may vary in different scans. One of such parameters is the size of a pixel of the image.

In our model, we have scale parameters

The first option O1 is the usage of DICOM metadata tags, indicating the size of pixel. Let a pixel be quadratic. Denote the width of the source slice pixel by

The next two options take into account the specificity of the problem, the source and target slices are of the same patient. In these cases, the scaling is performed using specific features of the curve describing the bone tissue:

O2) the maximal width of the region, bounded by the curve,

O3) the area of the region, bounded by the curve.

If the maximal width is considered (O2), then the target slice model remains as it stands, and the parameters of the source slice model are revised as follows:

If the area of the region bounded by the curve is considered (O3), then the target slice model remains as it stands, and the parameters of the source slice model are revised as follows:

The efficiency of registration is examined experimentally in this paper, using different options O1, O2 and O3.

Without loss of generality and seeking for simplicity of notation, we redefine

There are several reasons generating the necessity to solve the problem of translation invariance. The patient lies in various positions on the bed during different scans, and models, corresponding to target and source slices, differ as usual.

The translation invariance can be realized in two steps: horizontal translation and the following vertical translation.

The models of bone tissue of the source and target slices have a vertical symmetry: the axis of symmetry crosses the abscissa axis at

Note that

The problem is more complicated to find optimal

Several strategies for finding

The simplest criterion in search of

Derivative of

The method was tested using two CT scans and searching for optimal positions of slices from the source scan with respect to target scan slices, i.e. applying Eq. (

A pointwise comparison has disadvantages. Figure

Red is the source slice model, blue is the correct target slice model, shifted using (

The same models as in Fig.

Figure

Let the comparison criterion in Eq. (

Some matching examples imply that the result could be improved even more. For example, the models of the source (Fig.

Source (a) and target (b) slices of the same position, respectively; models of these slices (c).

To solve the problem of the breastbone cave uncertainty, the model curve points, that do not have the bone tissue nearby, must be not included in the comparison of slices. It is done by introducing model point weights, as explained in Section

The comparison criterion of two slices is as follows:

The one dimensional search for finding

In this strategy, the problem for

Here

In the experiments, the values of functions

Division of the model curve into top (blue line) and bottom (red line) parts.

Equation (

Scans of the same patient are examined, where the relative position of one scan is known with respect to the other one. Two pairs of scans with different slice thickness have been examined. The first source scan has 96 slices, the target scan has 106 slices, and slice thickness is 1.25 mm. The second source scan has 53 slices, the target scan has 49 slices, and slice thickness is 2.5 mm. During experiments with the first pair of scans, for each source slice, the most similar slice was found out in the target scan. The correct slice in the target scan is known in advance. Therefore, the registration error may be set to be the absolute difference in millimetres (mm) between the positions on the human body longitudinal axis of two target slices, determined by Eq. (

Results of the pointwise comparison (PW) with different scale invariance options.

O1 | 6.575 | 8.405 | 36.25 |

O2 | 9.102 | 11.627 | 47.50 |

O3 | 9.648 | 11.976 | 47.50 |

Results of the total least-squares (TLS) with different scale invariance options.

O1 | 9.974 | 9.226 | 38.75 |

O2 | 10.052 | 14.607 | 60.00 |

O3 | 8.737 | 13.507 | 60.00 |

While examining our new registration method, for each source slice, the registration error has been evaluated applying four different strategies PW, TLS, WTLS and WOLS of translation invariance and three options O1, O2, and O3 of the unification of scales (see Sections

In addition, the experiments were carried out using Pyramidal Implementation of the Lucas Kanade Feature Tracker (Bouguet,

Results of the weighted total least-squares (WTLS) with different scale invariance options.

O1 | 0.977 | 0.604 | 2.50 |

O2 | 0.469 | 0.657 | 2.50 |

O3 | 0.703 | 0.740 | 3.75 |

Results of the weighted ordinary least-squares (WOLS) with different scale invariance options.

O1 | 0.508 | 0.614 | 1.25 |

O2 | 0.326 | 0.549 | 1.25 |

O3 | 0.339 | 0.555 | 1.25 |

Results of the weighted ordinary least-squares (WOLS) with invariance option O3 of a pair of CT scans with 2.5 mm slice thickness.

O3 | 0.102 | 0.245 | 2.5 |

Results of the

28.53 | 15.54 | 48.75 |

This research is devoted to the analysis of transversal plane images, obtained by computer tomography scans. A method of the feature-based registration has been developed, where the model of the rib-bounded contour serves as the basis of the similarity criterion of images (slices). The model is flexible and describes the rib-bounded contour independently of the patient age, sex, and disease. We consider the slices where ribs are visible because many important internal organs are located here. The registration method applies translation, rotation, and scaling invariances. Several strategies of translation invariance and options of the unification of image scales are proposed. The method is examined on real CT scans seeking for its best performance. It works well independently of the radiocontrast injection.

The experiments have proved the efficiency of the new registration method, where the configuration of bone tissue is taken into account in the form of a mathematical model.

The results in Tables