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Well, in fact I'm not sure exactly which narrow scientific domain these terms derive from, but I'll try to explain them to you concerning their application in DSP.
Temporal domain is used for representing and subsequent plotting of the signal in coordinate system "time-magnitude". It's often opposed to frequency domain based on classical Fourier transform. Besides, nowadays there have been already introduced numerous ideas of chhossing time-frequency domain for signals' analysis. It may be obtained by different distributions (Wigner-Ville,etc.), as well as time-frequency Fourier analysis (spectrogram), Wavelet-algorithms and Empirical Mode Decomposition.
Concerning the spatial domain it's not so familiar and obvious for me, nevertheless I've often met this term in image processing, especially image filtration (including denoising). Spatial filters are widely used for processing multidimensional images.
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