Why do we substract a background image and not divide it?

Question

Subtracting a background image for background correction is the standard way of doing background correction to obtain the foreground.

I am wondering if there are situations, where it would make sense to divide with the background image instead.

• For example, it seems that having a uniform image intensity after division could be beneficial in interpreting an image. E.g in microscopy images.
• It also seems physically more intuitive to have uniform baseline intensity after background correction/division than to have zero baseline intensity due to subtraction.

Answer 1

Division happens, indirectly, in cases where the image formation model is multiplicative. Concepts behind background varies on whether one only considers smooth fluctuations, or one includes noise. For the later, there is a lot of literature on multiplicative noise.

important example(s]: speckle noise [...] proper shadows due to undulations on the surface of the imaged objects, shadows cast by complex objects like foliage and Venetian blinds, dark spots caused by dust in the lens or image sensor, and variations in the gain of individual elements of the image sensor array.

For a combined case, the principle of homomorphic filtering considers a nonlinear transformation on the image $f$ to cope with two main components (one slow, illumination $i$ and one fast, reflection $r$). In the most basic case, $f=i\times r$ is passed through a logarithm before some linear filtering (and an inverse exponential on the results).This allows to reduce the noise without compressing the dynamic range too much.

It is suitable for instance to shading effects on surfaces, where one end is darker than the other, for instance, from Homomorphic filtering – part 2 (Mathworks):

In the mathematical morphology community, you can also find the LIP concept, for Logarithm Image Processing.

• Multiplicative Noise Removal Using Variable Splitting and Constrained Optimization
• Logarithmic Image Processing: Additive Contrast, Multiplicative Contrast, and Associated Metrics
• Image restoration with multiplicative noise
• Side work: [Multiplicative Calculus in Biomedical Image Analysis](https://doi.org/10.1007/s10851-011-0275-1

Answer 2

I’ve used division in the context of time series spectra where you divide the power of an individual spectrum by a long term background average and it works well but there are some caveats.

the denominator has to be nonzero and have much lower variance.

dividing a zero mean Gaussian by a zero mean Gaussian results in a Cauchy distribution which has no moments.

You need to have something which is a very stable background.

Answer 3

Division can be used to correct for gain variations over the image plane. For example an image from a camera may have dark corners, which is called vignetting. This can be corrected for by dividing the picture by a photo of a uniform white surface.