METHOD AND APPARATUS FOR THE RECONSTRUCTION OF MEDICAL IMAGE DATA USING FILTERED BACKPROJECTION

Abstract

A system and method are provided for the reconstruction of medical image data using filtered backprojection with the use of a wavelet transformation. A filter function is applied to at least one part of an object using projection data captured with a detection device prior to backprojection.

Claims

1. A method for a reconstruction of medical image data using filtered backprojection, the method comprising: identifying projection data of an object; and applying, by a processor, with a wavelet transformation, a filter function to at least one part of the object, wherein the filter function is applied in a wavelet space.

2. The method of claim 1, wherein identifying the projection data of the object comprises acquiring, by a detection device, the projection data of the object prior to the filtered backprojection.

3. The method of claim 1, further comprising: applying, prior to the applying of the filter function, a discrete wavelet transformation to the projection data; applying, after the applying of the filter function, a corresponding inverse wavelet transformation to the projection data.

4. The method of claim 3, wherein the wavelet transformation comprises a wavelet function, a scaling function, and a redundant decomposition that links wavelet coefficients and scaling coefficients together.

5. The method of claim 3, wherein approximation values of the projection data resulting from the applying of the corresponding inverse wavelet transformation are used for the filtered backprojection.

6. The method of claim 3, further comprising: removing, prior to the applying of the inverse wavelet transformation, one or more values in an edge area of the projection data and above a limit value of truncated, filtered projection data, at each scaling level.

7. The method of claim 1, wherein the filter function is a transformed form of a ramp filter in a frequency space.

8. The method of claim 1, wherein the filter function is a matrix representation of an original filter function defined in the position space, wherein one or more matrix elements below a threshold value in a corresponding matrix are removed or replaced with zero entries.

9. The method of claim 8, wherein the one or more matrix elements are outside of a main diagonal of the matrix.

10. An apparatus for reconstruction of medical image data using filtered backprojection, the apparatus comprising: a detection device configured to acquire projection data of an object; a memory configured to store the projection data; and a processor configured to apply, with a wavelet transformation, a filter function to at least one part of the object, wherein the filter function is applied in a wavelet space.

11. The apparatus of claim 10, wherein the memory is an electronic non-volatile storage device, and the electronic non-volatile storage device is further configured to store a matrix representation of the filter function in the wavelet space cleared of any values below a defined threshold value.

12. The apparatus of claim 10, wherein the processor is further configured to apply, prior to application of the filter function, a discrete wavelet transformation to the projection data and apply, after application of the filter function, a corresponding inverse wavelet transformation to the projection data.

13. The apparatus of claim 12, wherein the wavelet transformation comprises a wavelet function, a scaling function, and a redundant decomposition that links wavelet coefficients and scaling coefficients together.

14. The apparatus of claim 12, wherein one or more approximation values of the projection data resulting from application of the corresponding inverse wavelet transformation are used for the filtered backprojection.

15. The apparatus of claim 12, wherein the processor is further configured to remove, prior to application of the inverse wavelet transformation, one or more values in an edge area of the projection data and above a limit value of truncated, filtered projection data, at each scaling level.

16. The apparatus of claim 10, wherein the filter function is a transformed form of a ramp filter in the frequency space.

17. The apparatus of claim 10, wherein the filter function is a matrix representation of an original filter function defined in the position space, and wherein one or more matrix elements below a threshold value in a corresponding matrix, outside of a main diagonal of a matrix, are removed or replaced with zero entries.

Description

BRIEF DESCRIPTION OF THE FIGURES

[0022] FIG. 1 depicts an example graphical representation of a core of a conventional ramp filter matrix where the x-axis indicates spatial coordinates and the y-axis indicates associated values.

[0023] FIG. 2 depicts an example graphical representation of the core of FIG. 1 following a wavelet transformation, reshaping, and thinning out of the associated matrix using thresholding.

[0024] FIG. 3 depicts an example graphical representation of a sample dataset in the wavelet space after application of a ramp filter with artifacts with high values at the respective edges at different scaling levels.

[0025] FIG. 4 depicts an example graphical representation of the sample dataset of FIG. 3 cleared of artifacts.

DETAILED DESCRIPTION

[0026] Embodiments include a standard filtered backprojection (FBP) method. For FBP, an object to be examined is first examined, for example, using an X-ray computed tomography system. The examination (e.g., irradiation) of the object at different angles and the respective detection of radiation passing through the object provides in each case two-dimensional sets of projection data captured using a flat (e.g., subdivided into individual pixels) detection device. Only spatially limited subareas of the object may be examined or irradiated in order to keep the radiation exposure or the dose as low as possible for the object. As a result, only cut off or truncated projection data is available.

[0027] Conventional methods may apply a simple ramp filter in the position space or Fourier space to the projection data before the projection data is backprojected in a backprojection step into the volume of the examined object in order to obtain a reconstructed three-dimensional image of the object. Applying the ramp filter leads to an improved image or reconstruction quality, as a blurring or smearing of the projection data available only two-dimensionally is at least partially compensated by the spatial third dimension of the projection volume. For only section-wise examination of the object and hence truncated projection data at the respective edges of the examination section or area, cupping or capping artifacts occur at the edges during image reconstruction. The artifacts reduce the reconstruction quality, as the artifacts do not represent any actual properties of the examined object and are therefore a falsification.

[0028] Embodiments provide a better image or reconstruction quality using a wavelet transformation. The filter function is applied in the wavelet space. The filtering (e.g., the application of the ramp filter on the projection data) takes place on a line basis (e.g., separately for each line of the two-dimensional projection data or of the two-dimensional set of projection data captured by the detection device with pixels arranged in a line). The one-dimensional (e.g., line-based) filtering of an individual projection data line of the dimension.sup.N where N=2.sup.n may be depicted as a matrix multiplication:

p.sub.F=Rp   (1)

[0029] where R∈.sup.N×N designates the ramp filter matrix, p∈.sup.N designates a line of the captured projection data, and p.sub.F∈.sup.N designates the corresponding filtered projection data. n specifies a maximum possible degree or level or stage of the wavelet transformation. A basis or an affine system of functions that form a Hilbert basis (e.g., a complete orthonormal system in the function space L.sup.2() of the square-integrable functions) is constructed from a wavelet function Ψ and a scaling function φ, defined as follows:

Ψ.sup.j.sub.k(t)=2.sup.jΨ(2.sup.jt−k)

φ.sup.k.sub.k(t)=2.sup.jφ(2.sup.jt−k),   (2)

[0030] where j=1, 2, . . . , n represents an extension or a scaling level, and k∈ represents a position or translation. The basis may be used to perform a wavelet transformation or wavelet decomposition of the projection data line p to obtain the transformed projection data line {tilde over (p)}:

[00001]$\begin{array}{cc}\tilde{p}=\underset{j=0}{\overset{n}{.Math.}}.Math.\underset{k=1}{\overset{2^{n-j}}{.Math.}}.Math.\left(d_k^j.Math.{\Psi }_k^j+c_k^j.Math.{\varphi }_k^j\right),& \left(3\right)\end{array}$

[0031] where d.sup.j.sub.k designates respective wavelet coefficients, and c.sup.j.sub.k designates respective scaling coefficients at the scaling level j. The basis is a non-standard form or a redundant decomposition or representation that links the wavelet coefficients d.sup.j.sub.k and the scaling coefficients c.sup.j.sub.k, providing a convolution used in the position space depicted as a matrix multiplication in the wavelet space. In an embodiment, the two-dimensional ramp filter matrix R may also be represented using a corresponding wavelet transformation or wavelet decomposition as a transformed ramp filter matrix {tilde over (R)} as:

[00002]$\begin{array}{cc}\tilde{R}=\underset{l,l^{\prime }}{.Math.}.Math.{\alpha }_{k,k^{\prime }}^j.Math.{\psi }_k.Math.{\psi }_k.Math.\underset{l,l^{\prime }}{.Math.}.Math.{\beta }_{k,k^{\prime }}^j.Math.{\Psi }_k.Math.{\varphi }_k.Math.\underset{l,l^{\prime }}{.Math.}.Math.{\gamma }_{k,k^{\prime }}^j.Math.{\varphi }_k.Math.{\Psi }_k.Math.\underset{l,l^{\prime }}{.Math.}.Math.{\omega }_{k,k^{\prime }}^j.Math.{\varphi }_k.Math.{\varphi }_k.& \left(4\right)\end{array}$

[0032] where α.sup.j={α.sup.j.sub.i,l}, β.sup.j={β.sup.j.sub.i,l}, γ.sup.j={γ.sup.j.sub.i,l}, ω.sup.j={ω.sup.j.sub.i,l} designate the two-dimensional wavelet and scaling coefficients in matrix form where i,l=1, 2, . . . , 2.sup.n-j. The totals in equation 4 are calculated across all dyadic quadratic subregions or submatrices I×I′ where I=I′=I.sub.j,k=[2.sup.j(k−1), 2.sup.jk], the dimensions or side lengths of which are defined or determined by dyadic intervals, within the ramp filter matrix {tilde over (R)} or the coefficient matrix that represents the ramp filter in the wavelet space in a non-standard form. In the non-standard form, the transformed ramp filter matrix {tilde over (R)} may have a dimension of (2N−2)×(2N−2), where in a standard form, a dimension of N×N may be present.

[0033] FIG. 1 depicts an example graphical representation 1 of a core of an uncompressed ramp filter matrix in a conventional method. Function values different to zero across at least the entire area represented are detectable. Also detectable is the non-localization of the ramp filter or the core or the support.

[0034] In an embodiment, the wavelet transformation (e.g., using Daubechies wavelets) results in a localization of the ramp filter matrix {tilde over (R)} where relatively large values are concentrated along the relevant main diagonals or along the areas or ranges of the submatrices running along the main diagonals. Values below a determined or to be determined threshold value (e.g., outside of the main diagonals or the band-type areas surrounding the main diagonals) may be ignored (e.g., set to zero) without any considerable or significant deterioration in quality occurring as a result during image reconstruction.

[0035] Embodiments provide the core depicted as a graphical representation 2 in FIG. 2 after the wavelet transformation and thinning out of the corresponding matrix using thresholding with a significantly shorter or localized ramp filter or ramp filter core or support. All function values outside of a narrowly defined area are identically equal to zero. The results are interpreted as the correct reconstruction of a pixel without requiring knowledge of all of the pixels or projection data values present in the same line. As complete data or information may not be available as a result of the truncating of projection data (e.g., the incomplete or only area-wise representation of the object), a significantly improved reconstruction may be provided as a result of the localization of the ramp filter or the support of the core of the ramp filter matrix R.

[0036] As the localization is not limited to a single pixel or data point, artifacts 9 (see FIG. 3) having an undesired effect on the reconstruction quality may occur at the edges of the data or reconstruction area. The artifacts become evident as narrow peaks or areas with particularly high function values or gray values relevant for a graphical representation, as depicted in FIG. 3 based on a sample data set 3 in the wavelet space. The increasing distances between the peaks or artifacts 9 along the x-coordinates reflect the different scaling levels 4, 5, 6, 7, 8. In a heuristic process or act, the corresponding data points of the artifacts 9 are removed, further improving the reconstruction quality.

[0037] Starting from the sample data set 3 shown in FIG. 3, FIG. 4 depicts a cleansed sample data set 10 as a result of removing the artifacts 9. FIG. 4 depicts not only a considerably lower maximum function value but also a significantly greater uniformity or evenness of the function values at all scaling levels 4, 5, 6, 7, 8.

[0038] There is an approximation or estimate pF of the filtered projection line data pF that may be described in a composition or inverse wavelet transformation as:

{tilde over (p)}.sub.p=Σ.sup.n.sub.j=0Σ.sub.k=1.sup.2.sup.n−1({circumflex over (d)}.sub.k.sup.iΨ.sub.k.sup.j+ĉ.sub.k.sup.jφ.sub.k.sup.j),   (5)

[0039] with the filtered wavelet coefficients {circumflex over (d)}.sub.k.sup.j and scaling coefficients ĉ.sub.k.sup.j. The coefficients are calculated at each scaling level j as

{circumflex over (d)}.sup.j=α.sup.j(d.sup.j)+β.sup.j(c.sup.j)   (6)

ĉ.sup.j=γ.sup.j(d.sup.j)

[0040] where d.sup.j={d.sup.j.sub.k}, c.sup.j={c.sup.j.sub.k}, k=1, 2, . . . , 2.sup.n-j where j=1, 2, . . . , n.

[0041] To obtain a reconstructed image representation of the examined object, the approximation values p.sub.F backtransformed from the wavelet space (see equation 5) of the filtered projection data pF are used as input data for the backprojection according to the known FBP method. The act of backprojection is otherwise not altered.

[0042] Both the wavelet transformation and the selection or coordination of a suitable threshold value and the subsequent thresholding of the filter matrix or coefficient matrix may be carried out in advance using untruncated projection data. A correspondingly prepared or precalculated transformed and thinned-out or localized matrix may be stored, for example, in an electronic storage device so that in the case of a specific application of the method, it is not necessary for the acts to be repeated at the time of application or runtime. Computing time and effort may be saved and an as effective as possible reconstruction result may be achieved as a result of coordination of the noise behavior.

[0043] The application of wavelet based ramp filtering in image reconstruction of tomography data leads to an improved representation with the application on truncated projection data independently of any possible different embodiments. In the case of an application on untruncated projection data, in contrast to conventional methods with which, for example, the ramp filtering is applied in the position or Fourier space based on the known FDK algorithm, at least equivalent or equal representation is achieved.

[0044] The wavelet-based method may provide benefits even in the case of untruncated projection data and may lead to an improved representation or reconstruction as, for example, local distortion effects such as defective pixels or non-translucent (e.g., as metallic) objects or areas in the examined object have only a limited localized distorting influence. Further, the wavelet-based ramp filtering as a result of localization properties may also, for example, be used to achieve an improved image reconstruction in methods that use a heuristic extrapolation schema to complete truncated projection data.

[0045] It is to be understood that the elements and features recited in the appended claims may be combined in different ways to produce new claims that likewise fall within the scope of the present invention. Thus, whereas the dependent claims appended below depend from only a single independent or dependent claim, it is to be understood that these dependent claims may, alternatively, be made to depend in the alternative from any preceding or following claim, whether independent or dependent, and that such new combinations are to be understood as forming a part of the present specification.

[0046] While the present invention has been described above by reference to various embodiments, it may be understood that many changes and modifications may be made to the described embodiments. It is therefore intended that the foregoing description be regarded as illustrative rather than limiting, and that it be understood that all equivalents and/or combinations of embodiments are intended to be included in this description.