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The X-ray transform projection of 3D mother wavelet function.

Yang X, Guo J, Lu L, Zeng L - Comput Math Methods Med (2013)

Bottom Line: X-ray transform can be applied to the three dimensional (3D) CT, depicting the relationship between material density and ray projection.So, the 3D wavelet transform of a 3D image can be implemented by the 2D wavelet transform of its X-ray transform projection and it will contribute to the reduction complexity and computation time during image processing.What is more, it can also avoid noise transfer and amplification during the processing of CT image reconstruction.

View Article: PubMed Central - PubMed

Affiliation: College of Mathematics and Econometrics, Hunan University, Hunan 410082, China.

ABSTRACT
As we all know, any practical computed tomography (CT) projection data more or less contains noises. Hence, it will be inconvenient for the postprocessing of a reconstructed 3D image even when the noise in the projection data is white. The reason is that the noise in the reconstructed image may be nonwhite. X-ray transform can be applied to the three dimensional (3D) CT, depicting the relationship between material density and ray projection. In this paper, nontensor product relationship between the two dimensional (2D) mother wavelet and 3D mother wavelet is obtained by taking X-ray transform projection of 3D mother wavelet. We proved that the projection of the 3D mother wavelet is a 2D mother wavelet if the 3D mother wavelet satisfies certain conditions. So, the 3D wavelet transform of a 3D image can be implemented by the 2D wavelet transform of its X-ray transform projection and it will contribute to the reduction complexity and computation time during image processing. What is more, it can also avoid noise transfer and amplification during the processing of CT image reconstruction.

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Related in: MedlinePlus

One of 128 Shepp Logan's CT slices and its edge picture.
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fig4: One of 128 Shepp Logan's CT slices and its edge picture.

Mentions: For practical validation, a series of experiments was performed on various images including the sequence of 75 wheel hub's CT slices [30] (one slice of them is shown in a following paper) and sequence of 128 Shepp Logan's CT slices. 3D wavelet transformation method extracted the edge of 3D CT volume data directly, while their X-ray transformation projections were used to extract the edge of 2D slice. In contrast, we used Mallat's wavelet method extract edges. We show the 3D mapping of wheel hub and Shepp Logan in Figure 2. Mallat's wavelet method, 2D X-ray transformation projection method, and 3D wavelet method are used to extract the edges, with results shown in Figures 3 and 4. From the edge pictures on visual inspection, the experimental results are close to 3D wavelet method and 2D wavelet method (Mallat's wavelet method, 2D wavelet projection method), but 3D wavelet method costs more computation time than 2D wavelet method.


The X-ray transform projection of 3D mother wavelet function.

Yang X, Guo J, Lu L, Zeng L - Comput Math Methods Med (2013)

One of 128 Shepp Logan's CT slices and its edge picture.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3860119&req=5

fig4: One of 128 Shepp Logan's CT slices and its edge picture.
Mentions: For practical validation, a series of experiments was performed on various images including the sequence of 75 wheel hub's CT slices [30] (one slice of them is shown in a following paper) and sequence of 128 Shepp Logan's CT slices. 3D wavelet transformation method extracted the edge of 3D CT volume data directly, while their X-ray transformation projections were used to extract the edge of 2D slice. In contrast, we used Mallat's wavelet method extract edges. We show the 3D mapping of wheel hub and Shepp Logan in Figure 2. Mallat's wavelet method, 2D X-ray transformation projection method, and 3D wavelet method are used to extract the edges, with results shown in Figures 3 and 4. From the edge pictures on visual inspection, the experimental results are close to 3D wavelet method and 2D wavelet method (Mallat's wavelet method, 2D wavelet projection method), but 3D wavelet method costs more computation time than 2D wavelet method.

Bottom Line: X-ray transform can be applied to the three dimensional (3D) CT, depicting the relationship between material density and ray projection.So, the 3D wavelet transform of a 3D image can be implemented by the 2D wavelet transform of its X-ray transform projection and it will contribute to the reduction complexity and computation time during image processing.What is more, it can also avoid noise transfer and amplification during the processing of CT image reconstruction.

View Article: PubMed Central - PubMed

Affiliation: College of Mathematics and Econometrics, Hunan University, Hunan 410082, China.

ABSTRACT
As we all know, any practical computed tomography (CT) projection data more or less contains noises. Hence, it will be inconvenient for the postprocessing of a reconstructed 3D image even when the noise in the projection data is white. The reason is that the noise in the reconstructed image may be nonwhite. X-ray transform can be applied to the three dimensional (3D) CT, depicting the relationship between material density and ray projection. In this paper, nontensor product relationship between the two dimensional (2D) mother wavelet and 3D mother wavelet is obtained by taking X-ray transform projection of 3D mother wavelet. We proved that the projection of the 3D mother wavelet is a 2D mother wavelet if the 3D mother wavelet satisfies certain conditions. So, the 3D wavelet transform of a 3D image can be implemented by the 2D wavelet transform of its X-ray transform projection and it will contribute to the reduction complexity and computation time during image processing. What is more, it can also avoid noise transfer and amplification during the processing of CT image reconstruction.

Show MeSH
Related in: MedlinePlus