The present invention relates to a method for rotating at least two-dimensional image records with non-isotropic topical resolution.
Known literature contains a number of methods for rotating and shifting image records. These methods can be roughly divided into two categories. In the first category, the rotations and translations of a record are computed exclusively in the image space. In the second, the rotations and translations are computed completely or partially in Fourier space with the aid of Fourier transformation.
The methods of the first category are disclosed in the following examples: Starting with an existing image record, a rotated image record is obtained by executing the following two steps: In the first step, the coordinate points of the rotated image record are computed by multiplying the coordinates of the existing image record by a rotation matrix. The value of each rotated coordinate pointxe2x80x94this can be, for example, a matter of the percent value on a gray scalexe2x80x94is calculated in an interpolation process. A large number, more or less, of values of neighboring points of the existing image record, or all values, should be integrated into the interpolation process. The interpolations can be based on a Sinc interpolation, for instance. The following example should demonstrate that said interpolation processes entail a significant computing outlay. Assume a two-dimensional image with 256 pixels in two dimensions. This means that 65536 pixels must be rotated. In turn, this means that in the worst case for each of the 65536 pixels a respective interpolation must be performed with the same number of pixels. Altogether, this means a quite significant computing outlay and thus time outlay. For shifts of image records in accordance with methods of the first category, the steps described above for a rotation are performed accordingly.
The methods of the second category are exemplified next. There, Fourier transformation is specifically used to reduce the computing outlay. The Fourier transform F(k) of a one-dimensional image space function f(x) is generally defined as       F    ⁡          (      k      )        =            ∫              -        ∞                    +        ∞              ⁢                            f          ⁡                      (            x            )                          ·                  ⅇ                      j            ·            2            ·            π            ·            k            ·            x                              ⁢                        ⅆ          x                .            
For the inverse-transformation,       f    ⁡          (      x      )        =            ∫              -        ∞                    +        ∞              ⁢                            F          ⁡                      (            k            )                          ·                  ⅇ                                    -              j                        ·            2            ·            π            ·            k            ·            x                              ⁢                        ⅆ          k                .            
As the central feature of the Fourier transformation, the displacement set is of primary importance in the methods of second category. The displacement set for a one-dimensional Fourier transformation is as follows:
f(x-x0) in the image space corresponds to ejxc2x72xc2x7xcfx80xc2x7x0xc2x7kxc2x7F(k) in Fourier space.
Applied to multidimensional applications, this means that a shift of an image record by a random vector is imaged in Fourier space in an additional phase of the Fourier transform. Unlike methods of the first category, interpolations are not required for the translation of an image record. The interpolation is completed on the basis of the attributes of Fourier transformation semi-automatically with the transformation into Fourier space, with the introduction of a desired additional phase and the corresponding back-transformation. Rotations of image records can likewise be performed with the aid of Fourier transformation and its displacement set without interpolation. To accomplish this, a rotation matrix describing the rotation is dismantled into a product of corresponding shear matrices. The effect of shear matrices is expressed in simple shifts of rows or columns of an image record, respectively. These shifts can be calculated easily in Fourier space with the aid of the Fourier transformation and its displacement set. A method of another kind is described in the essay by William F. Eddy, Mark Fitzgerald and Douglas C. Noll, xe2x80x9cImproved Image Registration by Using Fourier Interpolationxe2x80x9d, MRM 36, pp. 923-931, 1996.
The case of a non-isotropic topical resolution, which is the more frequent case in many applications, can be handled by the above method of the second category only with additional outlay. Non-isotropic records are not problematic for methods of the first category. For example, in rotations an elliptical path is merely executed instead of a circular path. The high computing outlay for the necessary interpolations remains unchanged. In methods of the second category, the non-isotropic record must be converted into an isotropic auxiliary record in a first step. Then, a method of the second category can be applied. Finally, the record must be converted back into a non-isotropic record. These conversion processes, which are known as resampling processes, represent additional computing outlay, regardless of the type and manner of execution of the resampling processes. In practice, the registering of non-isotropic records is often avoided for these reasons.
It is thus an object of the invention to create a method which reduces the computing outlay in rotations and translations of non-isotropic image records.
This object is inventively achieved in accordance with the present invention in a method for rotating at least two-dimensional image records with non-isotropic topical resolution, said method comprising the steps of: describing a rotation of an image record using a rotation matrix; representing said rotation matrix as a product of at least two shear matrices, each shear matrix having at least one element that is dependent on the angle of said rotation and remaining elements which are exclusively zeroes and ones; multiplying a matrix element, which is dependent on said angle of said rotation, of at least one of said shear matrices by a factor; and performing said rotation of said image record in Fourier space without interpolations and without forming an isotropic auxiliary record, upon exploitation of a displacement set of the Fourier transformation by implementing said shear matrices as displacements of line elements of said image record.
In an embodiment, the present method provides the following steps:
describing a rotation of an image record with a rotation matrix;
representing the rotation matrix as a product of at least two shear matrices, each of which comprises exactly one element that is dependent on the angle of rotation, and whose remaining elements are exclusively zeroes and ones;
multiplying the matrix element, which depends on the angle of rotation, of at least one shear matrix by a factor; and
performing the rotation of the image record in Fourier space, without interpolations and without forming an isotropic auxiliary record, by exploiting the displacement set of the Fourier transformation by implementing the shear matrices as displacements of line elements of the image record.
By simple one-time multiplications, by factors, of matrix elements of shear matrices that describe rotation intensive transformations of non-isotropic image records into isotropic auxiliary image records, corresponding back-transformations are made superfluous for the purpose of rotating non-isotropic image records in Fourier space. The application of the invention in a computer system reduces the necessary computing time.
In an advantageous embodiment, the factor depends exclusively on at least one ratio of a topical resolution in a first dimension to the topical resolution in a second dimension, which ratio describes a non-isotropic topical resolution. With the aid of a factor that is selected in this way, the error between the image record that is rotated as desired and the image record that is rotated by applying the method is minimized.
In a particularly advantageous embodiment, said error equals zero for two- or three-dimensional image records.
An advantageous embodiment relates to image records that are generated by a computer or MR tomography device. Here, the present invention can be advantageously employed particularly in the field of functional MR tomography, where the image records that are registered in a time series are rotated and displaced continuously for the purpose of motion correction.