Sigmoid

Computes the sigmoid function element-wise. A linear transformation is applied first on the argument of the sigmoid function. The resulting total transform is given by

$f(x)=(Max-Min)\ast {\displaystyle \frac{1}{1+e^{-(x-\beta )/\alpha }}}+Min$

Every output element is equal to f(x), where x is the intensity of the homologous input element, and alpha and beta are user-provided constants.

Consider an exampel of an image. The $\beta$ value can be thought of the offset on the pixel value that you are trying to isolate. E.g., if the object you are trying to isolate is at a pixel intensity above 150, you would choose a $\beta$ value that is around that value. The $\alpha$ value can be thought of as the scaling or variance of the sigmoid. Lower $\alpha$ values will make the pixel range of your intensity sharper. Your $\alpha$ value would decide how much of the noise you would want to include in your transformation. Smaller $\alpha$ values (0.25, 0.5) would zero out most of the noise, but it might make your actual signal thicker and not specific enough. Conversely, larger alpha values (e.g. 3+) might have a smoother signal but might include more noise. Negative alpha values can be thought of as using a positive alpha value and then inverting the image.

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