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Below are a couple quick and easy tricks for reducing noise
and removing gradients in an image.
Noise Reduction
A fast trick for reducing noise in an image is to use the
Despeckle filter in Photoshop. The filter alone will result in lost detail
so there are a couple additional steps to keep the image looking sharp but with
less noise.

Above: Unprocessed single exposure of the
Horsehead Nebula. Note the graininess. Image by James McGaha.
Procedure
Open the desired image. Duplicate the image to another
layer (Ctrl-J). Make sure the new layer is the active layer.
Select Filter > Noise > Despeckle. You might run
the filter several times. Press Ctrl-F to rerun the filter.
For this example, the Despeckle filter was run three times.

Above: Image after three iterations of the
Despeckle filter. Note the grain is gone, but the sharpness of the image
is lost.
Change the layer blend mode to Lighten. This will let
the sharper stars from the lower image show through, but the softness in the
nebula and background is retained.

Above: The new top layer (on which the
Despeckle filter was run) set to Lighten mode

Above: Image after setting blend mode to
Lighten. Note the reduction in noise compared to the original image.
Gradient Removal
The basic method for this trick is to apply a radial gradient
to an image to reverse the effect of light falloff due to vignetting. The
technique assumes that the vignetting is circular and uniform, which is not
necessarily true. But for a quick and dirty gradient removal, this trick
works pretty well with most images.

Above: Unprocessed single exposure of the
Andromeda Galaxy. Note the darkening of the corners of the image due to
vignetting. Image by James McGaha.
Procedure
Begin by sampling the sky background from both the center and
the corner of the image. Select the Eyedropper tool (keyboard shortcut
I). Alt-click on the center of the image to set the background color.
Be sure to avoid sampling the object itself. In this example, Andromeda
takes up most of the center of the frame. The background was sampled from
just outside the left edge of the galaxy. This will be close enough for
this technique.
Select the foreground color by sampling a corner of the image
with the Eyedropper.
Create a new layer (Shift-Ctrl-N). Select the
Gradient tool (keyboard shortcut G). Be sure the radial gradient is
selected from the top menu. Create a radial gradient in the new layer by
clicking in the center of the image and dragging outward to the corner.

Above: The gradient will essentially be the
opposite of the sky background, dark in the center and light at the corners
Change the layer blend mode to Screen.

Above: Layer blend mode set to Screen
The resulting image will have a flatter background, but the
background will be bright.

Above: Image after setting layer blend mode
to Screen. Note the background is even in brightness but is too bright
overall.
Flatten the image (Ctrl-E). Open the Levels
window (Ctrl-L). Move the black point slider (left slider) up to
the toe of the histogram curve.

Above: Setting Levels to darken the
background

Above: Image after Levels adjustment
The image can now be processed normally. For tips, see
the Aesthetic Image Adjustments section. Below is
the final image after full Photoshop processing.

Above: Final processed image of Andromeda.
Single 10 minute exposure. Image by James McGaha.

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