Image Formation. Cannell, M., B., McMorland, A., & Soeller, C. Image Formation.
The transformation of information from a real object to blurred image can be expressed mathematically using an operation called convolution. Deconvolution is the reverse operation, whose purpose in microscopy is to remove the contribution of out-of-focus objects from the image plane as well as (partially) reverse aberrations arising from imperfections in the optical train. It should be intuitive that, by using information about the imaging process, deconvolution techniques should be able to improve the quality of the image above that which could be achieved by any other method which does not provide extra information (beyond that contained in the image plane itself). The transformation carried out by the microscope on data from the object can be defined by a mathe-matical function called the point spread function (PSF). The PSF is the multi-dimensional image of a point in space and it can be practically measured by imaging very small objects (which must be less than the wavelength of light in size) or computed from the physical and optical properties of the imaging system (see Fig. 25.1). Given imperfections in the imaging system (or put another way, limited knowledge of the real optical system) the latter approach is inherently limited, but assumptions about the proper-ties of the PSF can help some blind deconvolution methods. Put simply: i(x, y, z, t) = o(x, y, z, t) ƒ psf(x, y, z, t) (1) where x, y, z, t are the dimensions of space and time and i is the recorded image, o the actual underlying object. The ƒ operator is convolution (see below and Fig. 25.2). If we take the Fourier trans-form (F) of this equation, the ƒ is replaced by multiplication: Fi(x, y, z, t) = Fo(x, y, z, t) ¥ Fpsf(x, y, z, t) (2)