**Transformations**

#### Reorientation

One rarely succeeds to reconstruct sections same as the ones in the medical atlases. The reason of it is not the “clumsiness” of the assistance. In many cases it is the condition of the patient which does not allow the optimal configuration of the camera. Or another frequent case is when several sections are needed from different points of view for the accurate recognition and the explanation of abnormalities. For the preparation of these sections, one has to familiarize themself with the three-dimensional plotting and three-dimensional transformation of the images composed from the section images.

#### Three-dimensional data

The transverse sections can be received from the reconstruction of projected images. Placing the transverse sections “below each other” a three dimensional data is composed.

The three-dimensional data is usually given in a left-handed coordinate system (X, Y, Z) fixed to the tomography. The x axis points horizontally to the right side of the patient who is lying on his back, the y axis points vertically down to the patient’s leg and the z axis is parallel with the axis of rotation. The elemental unity of three-dimensional data is called voxel.

On the screen the x axis is always horizontal and the y goes from the top to the bottom.

Beside the transverse section it is usual to display the frontal and sagittal sections as well. The sections and the order numbering of them are shown in the figure above.

A transzverzális metszeteken kívül szokásos a frontális és a szagittális képek megjelenítése is. Az egyes metszetek síkját, és a metszeti képek sorszámozását a mellékelt táblázat mutatja.

Section’s position | Nomination | Order of images |

Parallel with (X, Y) plane | Transverse | head → leg |

Parallel with (X, Z) plane | Frontal | front → back |

Parallel with (Y, Z) plane | Sagittal | right → left |

#### Linear transformation

Among the most important transformations the enlargement and the rotation can be executed by matrix multiplication, but the shifting cannot. In order to solve this problem the used (inhomogeneous) coordinates should be replaced by homogeneous coordinates. The procedure of this replacement starts with the introduction of a forth w coordinate beside the ordinary x, y, z. The coordinates of the points are written into a column vector.

The new homogeneous form of the x, y, z) point given by inhomogeneous coordinates is x, y, z, 1.

For the following discussion there is no need to know more about homogeneous coordinates.

The homogeneous coordinated form of the ordinary linear transformations can be written as a matrix multiplication. The execution of the transformations by matrix multiplication is useful, because most of the time there are more than one transformations in a row after each other, and due to the associativity of the multiplicative operation, there is an opportunity to calculate the entire effect by the multiplication of the matrices. So the really time consuming transformation should be executed only once.

#### Brain record reorientation

The patient should face into the upper direction on a transverse section. Otherwise a section should be chosen where the “forward” direction can be defined properly, than a rotation should be applied which makes this direction vertical.

A line on the rotated transverse section is marked going the required frontal section. The frontal section shows if the patient’s head was leant during the examination. To correct this, a direction should be marked which is wanted to be horizontal.

A vertical line of the received section should be marked going through the needed sagittal section. On the modified sagittal section the plane of hypophysis should be found and rotated into a horizontal position.

#### Left ventricle (myocardium) reorientation

A direction on the transverse section defined by the center of the left ventricle and the apex of the ventricle is marked and rotated into vertical position.

The modified sagittal section going through the marked line is made and a direction on it which goes between the center of the left ventricle and the apex of the ventricle is marked and rotated into vertical position.

#### Bull’s eye

The examination of myocardium’s perfusion is hardened by the uneven activity distribution and the necessity of the analysis of a bunch of images. Spatially close voxels can be found on another image. For the solution of this problem a so-called bull’s eye representation was introduced.

The starting point is the reoriented image of myocardium.

A disk is made from the transverse section going through the area of the heart’s apex and rings are made from the other sections. The higher the position of the section, the larger the corresponding ring’s radius is. The disk and the rings are placed concentrically attaching one another. The pixel values in the disk and in the ring are determined by the search in the direction of the radius on the transverse section for the value required to plot at the bull’s eye. This value is generally the maximal (average) value inside a certain interval. It can be imagined as the myocardium would be deformed into a disk.

Of course, with this interpretation some information losses, but it has benefits: the effects of the voxels that are spatially close to each other but placed on different sections appear closely to one another on the bull’s eye.

The disadvantage of the method is its sensitivity for the choice of the lowest transverse section, so sometimes the middle area of the bull’s eye is not taken into consideration.

The bull’s eye can be created in another way as well. Imagine a model composed by a hemisphere and a cylinder above it inside the myocardium.

The radius of the bull’s eye is R, the distance of the (x, y) point and the center is ρ, the length of the generatrix from the model’s apex to the top of the model is A, the length of the generatrix which goes to that point on the model’s wall from which the value of the (x, y) point is originated is a. Then A : R = a : ρ. And from this a = A * ρ / R.

If a > r π/2 then the point of the model is on the cylinder, otherwise on the hemisphere.

Ha a > r π/2, akkor a modell pontja a hengeren, különben a félgömbön van.