How Can Convolution and Deconvolution be Defined for 3D Images?

I am trying to understand how convolution and deconvolution can be represented for 3D images/ stacks of data. I would prefer it, if you built the these concepts from 1D vectors to 3D matrices in terms of their equations. I am currently writing a report and am trying to represent 3D deconvolution as an equation.

Please do not hesitate to ask me any questions to clarify this question.

• Do you understand how 2-D convolution is represented mathematically? As for the deconvolution, it's usually represented as division in the frequency domain. – AnonSubmitter85 Feb 10 '16 at 15:51
• @AnonSubmitter85 Yeah, I was talking about it in the 3D time domain – SDG Feb 11 '16 at 7:10
• Yes, that was clear. But if you understand it in 2-D, where exactly are you having trouble extending it to 3-D? Perhaps you could edit your question to show this. – AnonSubmitter85 Feb 11 '16 at 18:11
• @AnonSubmitter85 Well, assuming math.vt.edu/people/dlr/m2k_opm_disfour2.pdf is the equatin for 2D convolution, how would I extend it to 3D for convolution and deconvolution? – SDG Feb 13 '16 at 13:12
• dsp.stackexchange.com/questions/2969/… This might be helpful – Andrey Rubshtein Feb 13 '16 at 21:17

For one variable, we have

$$y(i) = \sum_m x(i-m) \cdot h(m).$$

For two variables it's

$$y(i,j) = \sum_m \sum_n x(m,n) \cdot h(i-m,j-n).$$

For three:

$$y(i,j,k) = \sum_m \sum_n \sum_p x(m,n,p) \cdot h(i-m,j-n,k-p).$$

• Perfect Answer! – SDG Feb 14 '16 at 10:12