APPENDIX B
DECONVOLUTION ALGORITHMS*
Mathematically, the general relationship between image i(x, y, z), object o(x, y, z) and point
spread function h(x, y, z) is expressed as a spatial convolution, i.e. a triple integration over space,
i(x, y, z) = Ih(x y r, z ;)o(, rq,)ddrd; + n(x, y, z), (B-l)
where n(x, y, z) is additive random noise. By denoting convolution operation as 0, the above
equation is abbreviated as i = h 0 o + n. This relationship indicates that the images taken from
the microscope deviate from their real counterparts, commonly referred to as optical blur.
Deconvolution is the mathematical operation to invert the convolution process, so that
images are reconstructed by reducing the noise and correcting for the optical blur. Once the
images are reconstructed, they are stacked to form a 3D data set, with each component
representing the fluorescence intensity at the corresponding point in object space. A variety of
deconvolution algorithms has been developed and they are summarized here.
B.1 Inverse Filter
An inverse filter algorithm such as a Wiener filter65 transforms the stack of raw images i(x,
y, z) in the spatial domain into I(u, v, w) in the Fourier domain. Since a convolution in the spatial
domain is equivalent to a point-by-point multiplication in the Fourier domain, i.e.,
FFT[h(x, y, z) o(x, y, z)] = FFT[h(x, y, z)]. FFT[o(x, y, z)], (B-2)
the image formation in Fourier domain is expressed as
I(u, v, w) = H(u, v, w) O(u, v, w) + N(u, v, w), (B-3)
where I(u, v, w), O(u, v, w), H(u, v, w) and N(u, v, w) are the respective counterparts of i, o, h and
n in the Fourier domain.
When noise is negligible compared to the coherent signal, the object is easily obtained by
dividing by H on both sides of Equation B-3,
O(u,v,w) I(,v,) (B-4)
H(u, v, w)
where "A" indicates an estimation. In cases when noise cannot be neglected, a Wiener inverse
filter is used to minimize degradation from noise. Assuming the ratio of the power spectrum of
the noise to the object is b, the estimate of the object becomes
0(u,v,w)= H(u,v,w) I(u,v,w), (B-5)
H(u,v,w)12 +b
where "*" indicates the complex conjugate. Once the reconstructed object is found in Fourier
domain, it may be applied the inverse Fourier transform, and the object is obtained in spatial
domain as,
6(x, y, z) = IFFT[O(u, v, w)]. (B-6)
B.2 Constrained Iterative Deconvolution
Many popular deconvolution algorithms, such as the Jansson Van-Cittert algorithm,66 the
Gold algorithm,67 and the Richardson Lucy (also known as Maximum Likelihood Estimation or
* Part of this chapter has been published in "Deconvolution Microscopy for Flow Visualization
in Microchannels", Analytical Chemestry, 2007, 79(6): p. 2576-2582.