OK, I know what's happening. The problem is that you have converted your image to the sRGB V4 beta profile, which you're using also as your default profile in PixInsight. The sRGB V4 profile is not identical to the sRGB IEC61966-2.1 profile, which is the web standard. The difference is small, but causes visible changes when the images are reproduced on web browsers, with or without color management support.
I have made a small tutorial to describe the correct color management workflow that will guarantee consistent results for web deplyment.
First I have selected the sRGB V4 profile as my default profile for both RGB and grayscale images. I've done this to reproduce your working environment, so your FITS processed image can be correctly reproduced on my screen. In other words, I want to see exactly what you see on your computer.
On the screenshot above, you can see both your FITS image and the JPEG version you made for the web. After setting the sRGB V4 profile as my default, there's no difference between both images, as expected.
The next screenshot shows how I have converted the FITS image to the standard sRGB IEC61966-2.1 profile. This is the profile --and not the sRGB V4 profile-- that all web browsers use as the default standard profile for reproduction of graphical contents on the WWW.
If you look carefully at both images, you'll see now a small difference. The FITS image, after conversion to sRGB IEC61966-2.1, is slightly more contrasted (the background is a bit darker). This can be seen more accurately on the next screenshot.
Note that the image after conversion to sRGB IEC61966-2.1 (the image at the right hand side) is slightly darker (actually it is a bit more contrasted) than the original image before conversion, which was referred to the sRGB V4 profile.
The explanation of why this happens is a bit technical; skip it if you are not interested. To perform an ICC profile transformation of a grayscale image using a RGB profile, the image must be converted to the RGB color space first. This is as easy as just copying the original gray channel to the red, green and blue channels. After the ICC color space transformation, the image must be converted back to the grayscale space. The RGB->grayscale conversion consists of computing the L* (lightness) component of the CIE L*a*b* color space. This is carried out in the RGB working space (RGBWS) of the image. Since the default RGBWS is the sRGB IEC61966-2.1 space in PixInsight, there is a small difference because the original image was referred to sRGB V4, not to sRGB IEC61966-2.1.
If instead of sRGB V4 you were using sRGB IEC61966-2.1 as your default profile, this change would not exist when you convert grayscale images: the image would look
exactly the same before and after ICCProfileTransformation. Actually, you wouldn't need any ICC profile transformation at all in this case, as your working color space would be the same as the target space.
The next step is writing the image to disk in JPEG format. As this JPEG image is intended for web deployment, I have not embedded an ICC profile. Now we have a problem: since the default ICC profile is sRGB V4 in PixInsight (see the first screenshot above), if we load the JPEG image we won't see it correctly represented on the screen. This happens because the JPEG image is in the sRGB IEC61966-2.1 space, but it does not embed a profile, so there's no way to interpret its pixel values correctly.
So before loading the JPEG image, we'll reset the default profiles to sRGB IEC61966-2.1, as shown on the following screenshot.
See that the JPEG image looks perfectly now. It has no profile embedded, but our default profile is correct to interpret its pixel values. At this point, PixInsight is acting just as a normal web browser, which expects sRGB IEC61966-2.1 by default.
Finally, let's load the JPEG image in Safari and Firefox (up-to-date versions). Both browsers are working with default settings:
Can you see any difference? Thanks to the correct use of the standard sRGB IEC61966-2.1 profile, we have now achieved consistent results.
So the bottom line is: don't use the sRGB V4 beta profile, or other variants of sRGB. Use the standard sRGB IEC61966-2.1 profile and you'll have no problems.
As a bonus, I have played a little with wavelet-based noise reduction, this time on my main Linux workstation. The noise reduction routines implemented in the ATrousWaveletTransform tool are very powerful and accurate, and have the important advantage that they can be used with both linear and nonlinear images.
This is your image before noise reduction:
And here you have it after noise reduction applied to the first three wavelet layers:
All significant structures have been preserved and the image looks much better without noise, in my opinion. Hope this will help you.