How to Handle Noise

The preceding ML estimators have assumed no noise to be present in the image data im. However, the incorporation of noise presents little problem. The estimators will simply be revised by having something added to them in all cases. They therefore will keep the forms previously derived. This is shown next. 

Let the image data suffer from noise sm. Then the imaging equations are [compare Eq. (18)] 

Now we have two sets of unknowns, object nm and noise em. The proper ML principle to use is therefore that the nm and em that occurred are both (jointly) most likely, 

Of these two contributors to the maximum, the first is already given by Eq. (17). It takes on the particular forms previously given under different states of object class p(ql9. qM). Hence, this part of the ML principle is already known. 

The second contributor to Eq. (53) has to do with the type of noise assumed to be present. To keep things simple, we shall assume it to be additive and independent Gaussian but with generally position-dependent variance e . This describes, for example, Johnson noise and other noise types that enter into the output of the spectrometer. Thus, we have as the probability PN of noise 

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