Image derivatives are susceptible to noise. To counteract this effect, you can regularize the image or data by a Gaussian kernel of standard deviation σ. The default value is σ=0.

In RidgeFilter[image,σ], σ is the scale of the ridges that is used to compute the derivatives in the Hessian. By default, σ=1 is used.

Nevertheless, combining the information from the two pages of the docs, I still can't tell what exactly the functions are doing. It sounds to me like they simply run a GaussianFilter over the image first. I can obtain results at the same scale by using 2σ as the radius of the Gaussian filter, but then I still get a lot of small scale artefacts that are missing when I use DerivativeFilter or RidgeFilter directly:

img = ColorConvert[ExampleData[{"TestImage", "Mandrill"}], "Grayscale"]
σ = 10;
ImageAssemble @ {
  ImageAdjust@DerivativeFilter[img, {1, 1}, σ], 
  ImageAdjust@DerivativeFilter[GaussianFilter[img, 2σ], {1, 1}]
}

GaussianFilter[image,r] uses r = σ/2.

All of this seems highly confusin, and there must be more to it than simply smoothing the input up front.

Image derivatives are susceptible to noise. To counteract this effect, you can regularize the image or data by a Gaussian kernel of standard deviation σ. The default value is σ = 0.

In RidgeFilter[image, σ], σ is the scale of the ridges that is used to compute the derivatives in the Hessian. By default, σ = 1 is used.

Nevertheless, combining the information from the two pages of the docs, I still can't tell what exactly the functions are doing. It sounds to me like they simply run a GaussianFilter over the image first. I can obtain results at the same scale by using 2 σ as the radius of the Gaussian filter, but then I still get a lot of small scale artefacts that are missing when I use DerivativeFilter or RidgeFilter directly:

img = ColorConvert[ExampleData[{"TestImage", "Mandrill"}], "Grayscale"]
σ = 10;
ImageAssemble @ {
  ImageAdjust @ DerivativeFilter[img, {1, 1}, σ], 
  ImageAdjust @ DerivativeFilter[GaussianFilter[img, 2 σ], {1, 1}]
}

GaussianFilter[image, r] uses r = σ/2.

All of this seems highly confusing, and there must be more to it than simply smoothing the input up front.

So there must be more to it then simply smoothing the input up front.

I'm actually completely baffled that I need to double σ (instead of halving it) since the docs for GaussianFilter call this parameter the radius r and they even say explicitly:

GaussianFilter[image,r] uses r = σ/2.

All of this seems highly confusin, and there must be more to it than simply smoothing the input up front.