Hi Geert
From a mathematical point of view, yes, it is exactly the same. The inverse ATrous Waveletets transform is the addition of every layer, plus the residual. This is the key point of our scale separation method.
In practice, you may find some differences in both procedures, due the "image interpretation" of the data, instead of more abstract wavelet layers. More explicitly, wavelet layers have a "zero mean value". They have positive and negative values, according to dark or bright features, compared to the surroundings (at a particular scale level). The residual layer, on the other side, is a strongly blurred image, with the same mean value as the original image.
So, if you apply the ATWT process to a image, deleting the residual layer, you'll end with an image with a lot of zeros. All the dark features are clipped, and only bright features remain.
If you clone the image, and supress all wavelet layers but the residual, the residual will be displayed as it is, no clipping. Now, if you subtract it to the original image (using PixelMath), using the Rescale option enabled, you'll get the first layers, rescaled to the normalized range. The "zero value" will be shifted somewhere in that range, usually near the center. The result will look gray, with darker or brighter features preserved. If you disable the rescale option, you'll get the same result as in the first procedure. Dark features will be clipped, and the "zero mean value" of those layers will remain at zero.
