Transconvolution

The term transconvolution denotes a numerical method from the field of medical imaging , in particular emission computed tomography , which enables subsequent manipulation of the point spread function (PSF) in images that have already been recorded.

Properties of an image such as the spatial resolution or the representability of small objects are determined by the PSF of the imaging system used for the image acquisition . Different imaging systems with different PSFs therefore also provide slightly different images of one and the same object.

Based on known PSFs of different tomographic systems, the transconvolution method can convert an image that was recorded on a specific tomograph as if it had been recorded by another tomograph. The process can thus ensure the comparability of images that were originally recorded on different systems.

definition

For two different tomographs with different point spread functions and the imaging can be described as convolution as: ${\displaystyle psf_{1}}$ ${\displaystyle psf_{2}}$

${\displaystyle obj*psf_{1}=img_{1}}$

${\displaystyle obj*psf_{2}=img_{2}}$

where " " represents the convolution operator and stands for the two slightly different images of the same object generated by the respective tomograph . ${\displaystyle *}$ ${\displaystyle img_{1}}$ ${\displaystyle img_{2}}$ ${\displaystyle obj}$

The relationship immediately follows from the two equations:

${\displaystyle img_{1}*psf_{1}^{-1}*psf_{2}=img_{2}}$

where stands for the inverse of the point spread function . ${\displaystyle psf_{1}^{-1}}$ ${\displaystyle psf_{1}}$

While the inverse point spread function diverges and cannot be determined numerically, the full term can be approximately calculated using numerical methods under certain boundary conditions. ${\displaystyle psf_{1}^{-1}}$ ${\displaystyle psf_{1}^{-1}*psf_{2}}$

The transconvolution function is now defined as ${\displaystyle tf}$

${\displaystyle tf=psf_{1}^{-1}*psf_{2}}$

with the resulting relationship

${\displaystyle img_{1}*tf=img_{2}}$

Knowing the PSFs of the respective tomographs, it is thus possible to convert an image generated by the first tomograph as if it had been recorded by the second tomograph. Of course, the method is subject to certain limits, in particular the calculated can not represent spatial frequencies that are no longer detected by the, i. e. the spatial resolution of an image cannot be increased at will. ${\displaystyle img_{1}}$ ${\displaystyle img_{2}}$ ${\displaystyle img_{2}}$ ${\displaystyle psf_{1}}$

Use in medical imaging

The second point spread function does not have to represent a real tomograph, but can be defined directly and then represents a virtual tomograph with corresponding properties. Based on the definition of a standardized virtual tomograph and the determination of the imaging properties of different real tomographs, the transconvolution method allows a uniform and quantitatively comparable representation of the image data recorded by the different tomographs or systems, as if all measurements had been made equally by the standardized virtual system. The method thus supports quantitative comparisons of images that were recorded by different imaging systems and in particular by different clinical tomographs. ${\displaystyle psf_{2}}$

Another application of the transconvolution method in positron emission tomography allows the handling of the different image blurring that is caused by the different positron ranges of the positron- emitting radionuclides ( β ⁺ decay ). In particular, this enables the use of different radionuclides in the calibration compared to the subsequent imaging.

literature

• T. Weitzel, F. Corminboeuf, B. Klaeser, T Krause: Cross-calibration of positron emission tomographs for multicenter studies: solid-state phantom and transconvolution. In: Bulletin of the Swiss Society for Radiation Biology and Medical Physics. Oct. 2010, p. 9ff. SGSMP Bulletin 72
• T. Weitzel, F. Corminboeuf, B. Klaeser, T. Krause, T Beyer: Transconvolution and virtual PET: A new concept for quantification of PET in multi-center trials. In: Journal of Nuclear Medicine. 51, 2010, p. 115.
• GA Prenosil, T. Weitzel, M. Hentschel, B. Klaeser, T Krause: Transconvolution and the virtual positron emission tomograph - A new method for cross calibration in quantitative PET / CT imaging. In: Medical Physics. 40, June 2013, p. 062503.
• GA Prenosil, M. Hentschel, M. Fürstner, T. Krause, T. Weitzel, B. Klaeser: Technical Note: Transconvolution based equalization of positron energy effects for the use of 68Ge / 68Ga phantoms in determining 18F PET recovery. In: Medical Physics. , 44, 2017 p. 3761

Web links

• GA Prenosil, T. Weitzel, M. Hentschel, B. Klaeser, T. Krause: Transconvolution and the virtual positron emission tomograph - a new method for cross calibration in quantitative PET ∕ CT imaging. In: Medical physics. Volume 40, number 6, June 2013, p. 062503, doi : 10.1118 / 1.4805112 , PMID 23718608 .
• Patent publication WO2011123964 (Google Patents)