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

\(f(x) = (Max-Min) * \dfrac{1}{1+e^{-(x - \beta) / \alpha }} + Min\)

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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