Center Filters are lens- or lens-rage specifically calculated high-end filters. They compensate for lens-type inherent falloff / vignetting towards the edges.
When working with ultra-wide-angle lenses, the falloff of illumination towards the corners is sometimes disturbing. For this reason, concentrically graduated Center Filters have been designed to compensate for this vignetting. These filters have transparency that increases gradually from the center to the edges of the filter, where it reaches full transparency. The transmission is independent of the wavelength of the visible spectrum.
To avoid overexposure and consider the film emulsions' exposure latitude in the analog area, the optic manufacturers do not fully compensate for the lens's vignetting. This working hypothesis was adapted to the digital world. Specification 1.5, 2.5, etc. indicates the extension factor for the exposure.
As Center Filters must be on par with the lens quality, they are manufactured at the same level as optics. The price reflects this level of complexity.
The brightness drop / vignetting of digital backs can also be compensated by a digital correction function. This process is achieved by image analysis or, in the more complex and more accurate case, by a correction image with digital analysis. The latter variant (LCC Lens Cast Calibration, Scene Calibration) analyzes the edge falloff and color shifts utilizing a separate neutral image taken with an opaque white disc.
Since this process has to be offset against the image(s), the correction image is taken with the same aperture and shift adjustment as the original. This correction benefits from the existing dynamic range. For an optimal result, it may therefore be necessary to combine both techniques.