Introduction.

   Book for Physicists  »  Introduction to the mathematics of medical imaging  »  Mathematical Methods of Linear Model Based Image Processing Procedures  »  Introduction.

The current chapter summarizes those mathematical tools and procedures playing important role in the analysis and description of the information transfer and signal/noise condition in a particular system (may be 1D, 2D, 3D imaging systems) both in the real parameter space and in the frequency space, where the quantity and quality of the parameter change are considered. It is possible to make a survey of the system response either in the real parameter space or in the frequency space by the influence of various inputs. Consequently the system quality can be improved like noise elimination level, resolution, distortion effects,… etc. –synthesis -. The main features of the system can be described by the evaluation of the various input functions (effects) and can be determined the system parameters and they differences from the references and standards –analysis-. The following figures and illustrations may take even more understandable how connect the above mentioned methods to the image processing subjects (see Figure.1., 2., 3.).

Figure.1. Tc99m radio-pharmaceutical accumulation is presented in a thyroid. „Original Image” shows the 2D raw data made by the imaging system. Near the raw data (right side) is displayed the processed (2D filtered) image. The processed result show clearly the effect of the noise, i.e. how does the noise disturb the real pharmaceutical distribution (in this particular case the noise level is comparable with the signal).'

Figure.2. An example for applying optimal filtering for tomography studies (SPECT) is presented. The figure shows on the left side images the effect of the improper pre-filtering (too many high frequency objects did extraordinary edge enhance), while on the right side is displayed the result with the selection of the proper pre-filtering.

Figure.3. Determination of radio-pharmaceutical concentration response function (RIR = Residual Impulse Response) in time by the following boundary condition: the bolus activity will enter into the examined organs (currently into the kidneys) in an instant (i.e. like an input impulse activity). The bio-chemical process follows the boundary conditions approximately well, because the bolus process controlled by heart is in the sec. range, while the kidney filtration and clearance pharmaceutical-kinetic process (i.e. the kidney function response) takes in minutes range.