sRGB, Adobe RGB, and ProPhoto RGB - Exploring the Gamut Limitations of Printing

[Doug Gray]

Often there is discussion of gamut clipping when printing images in sRGB or Adobe RGB from ProPhoto RGB and Lab. There are occasionally very significant shifts in color and luminosity associated with these processes and, in general, differences are chalked up only to limitations of the working space gamut being unable to exploit the entire printer gamut. While this is true, there are other differences as well.

The other factor discussed is just how often sRGB or aRGB gamuts are exceeded by non-synthetic images. But the a more immediate issue is whether the image gamut in Lab or ProPhoto RGB, as mapped into the printer, exceeds sRGB or Adobe RGB. It can be interesting to print an image with the full gamut a printer can deliver then print the same image clipped to, say, sRGB and compare them side by side. The obvious and simple approach is to convert the ProPhoto image to sRGB then print and compare this to one printed directly from the ProPhoto space. The differences in the two prints would be presumed to reflect limitations in the sRGB gamut. It makes the differences look larger than they are. This simple approach is flawed because converting to sRGB from a larger RGB space prior to printing clips all colors, whether printable or not, that are outside the smaller space. Clipping can change the luminance and hue, not just the saturation. Colors clipped to sRGB but that are outside of the printer's gamut (a large number of sRGB colors are outside print gamuts) undergo two conversions prior to printing. First to the smaller RGB space then to the printer's gamut. Traditionally, RGB spaces are converted to other RGB spaces by matrix math with the result being clipped at 0 and 255 forcing the image to the target gamut. OTOH, printer profiles are LUT based and color engines convert RGB to Lab which then uses the LUTs to estimate the printer's RGB levels. At least for an RGB input based printer. There is quite a bit of variation in how profile software vendors set the LUT values when outside the printer's gamut and this creates many of the differences between prints using these two processes beyond those caused by the printer's limited gamut.

So here's a process that will allow testing of prints where the only gamut clipping is that of the ProPhoto RGB to a smaller space. It presumes one has an accurate printer profile, typically a custom profile.

Assuming the test image is in 16 bit ProPhoto.
1. Convert the image to the printer's profile using Perceptual or Relative Colorimetry with or without BP correction using the settings normally used to print.
2. Then convert that image to ProPhoto RGB using Relative without BP correction. (this is critical, AtoB BP correction will not replicate colors correctly)
3. Print the ProPhoto RGB image using Relative Colorimetry w/o BP correction.
4. Convert this image to sRGB (or Adobe RGB) and print again using Relative Colorimetry w/o BP correction.

Step 2 assures that all the colors now in ProPhoto RGB are within the printer's gamut. The only differences between the two prints are now due to conversion of these colors and not from vagaries of a profile s/w coder's notions.

One of the interesting changes is the much smaller shifts in Andrew's "Printer Gamut Test File's"  saturated spheres. However, since much of that image is out of gamut for both aRGB and sRGB it makes for an interesting comparison between these three prints from a single image that has been limited to a printer's gamut first. The shift from aRGB to sRGB is very noticeable, from ProPhoto RGB to aRGB, not nearly as much.

[digitaldog]

The other factor discussed is just how often sRGB or aRGB gamuts are exceeded by non-synthetic images.

I haven't had much difficulty finding such real world images I've captured. Certainly lots do not fall into this camp.

The obvious and simple approach is to convert the ProPhoto image to sRGB then print and compare this to one printed directly from the ProPhoto space. The differences in the two prints would be presumed to reflect limitations in the sRGB gamut. It makes the differences look larger than they are. This simple approach is flawed because converting to sRGB from a larger RGB space prior to printing clips all colors, whether printable or not, that are outside the smaller space.

If the workflow is to move from raw data to some RGB working space, how do we avoid this? We have to select some RGB working space for encoding. We may have no idea when, where or how the image might be printed or the printers gamut.

Clipping can change the luminance and hue, not just the saturation. Colors clipped to sRGB but that are outside of the printer's gamut (a large number of sRGB colors are outside print gamuts) undergo two conversions prior to printing. First to the smaller RGB space then to the printer's gamut. Traditionally, RGB spaces are converted to other RGB spaces by matrix math with the result being clipped at 0 and 255 forcing the image to the target gamut. OTOH, printer profiles are LUT based and color engines convert RGB to Lab which then uses the LUTs to estimate the printer's RGB levels. At least for an RGB input based printer. There is quite a bit of variation in how profile software vendors set the LUT values when outside the printer's gamut and this creates many of the differences between prints using these two processes beyond those caused by the printer's limited gamut.

Agreed, all limitations that we often have no control over.

Assuming the test image is in 16 bit ProPhoto.

Again, if that's the case, that's the case. The question is, what about those who state: always encode in sRGB? Or even Adobe RGB (1998)?

[Doug Gray]

I haven't had much difficulty finding such real world images I've captured. Certainly lots do not fall into this camp.

I suspect it's more common to see images from commercial packaging, flowers, and such well outside sRGB but some others can surprise and be outside sRGB as well. Somewhat fewer are also outside of Adobe RGB.  What some don't realize is that the modifications made to colors as part of rendering from scene reference to output reference almost always involve increasing saturation. It's simply a byproduct of increasing contrast and is what people expect to see in pleasing photos.

If the workflow is to move from raw data to some RGB working space, how do we avoid this? We have to select some RGB working space for encoding. We may have no idea when, where or how the image might be printed or the printers gamut.  Agreed, all limitations that we often have no control over.
Again, if that's the case, that's the case. The question is, what about those who state: always encode in sRGB? Or even Adobe RGB (1998)?

I don't know why people say that. Work in whatever space you understand and have the tools to utilize. At times, for customers that require sRGB workproduct then it may make sense to use that start to finish. Otherwise, it's a harmful simplification.

The last part of my post deals with the very large difference when using ProPhoto to print or converting the same image to sRGB, or aRGB then printing. Take your Gamut Test File, The differences are quite noticeable and most of them are due to the conversions between ProPhoto and either aRGB or sRGB and not from a fundamental printer gamut limit.

Using Glossy on a 9500 II, I printed the Gamut test directly from the ProPhoto space using RC w/o BP correction. I then converted to sRGB and printed the same. These have fairly large differences in appearance as one would expect.

But I also tried an alternative path. I converted the original to the printer profile space selecting BC and RC, Then converted back to ProPhoto using RC w/o BC. This represented the actual ProPhoto RGB colors that are in the printer's gamut and will create the same image when printed directly using RC w/o BP. It's essentially a symmetrical roundtrip. When this image was converted to sRGB, making the entire image fall within sRGB and the printer's gamut, and then printed using RC w/o BP the results were much closer to the original print that came directly from ProPhoto RGB space.

When the same process was repeated with aRGB instead of sRGB, the two prints appear the same side by side where as converting the original to aRGB then printing still makes a print that differs noticeably.

Here are some Lab measurements at the bottom of the vertical green stripe: in L*, a*, and b*.

Printed directly from ProPhoto (ref):  60.11, -66.95, 56.07
Printed after converting to sRGB: 72.48, -53.37, 66.8  (DeltaE from ref:21.3
Printed after converting to sRGB using roundtrip from printer space: 61.27, -59.02, 57.49 (DeltaE from ref: 8.1)
Printed after converting to aRGB using roundtrip from printer space: 59.88, -68.22, 55.49 (DeltaE from ref: 1.41)

So perhaps, if one has to absolutely deliver an image in either sRGB or aRGB, using this protocol would produce images in those colorspaces that more closely match the direct print. It's much less color distortive than transitioning by first converting to either sRGB or aRGB when significantly outside of these gamuts.

[digitaldog]

I suspect it's more common to see images from commercial packaging, flowers, and such well outside sRGB but some others can surprise and be outside sRGB as well.

Yes, agreed, I've pointed this out by mapping images from raw data in ColorThink:

Everything you thought you wanted to know about color gamut

A pretty exhaustive 37 minute video examining the color gamut of RGB working spaces, images and output color spaces. All plotted in 2D and 3D to illustrate color gamut.

High resolution: http://digitaldog.net/files/ColorGamut.mov
Low Res (YouTube): http://www.youtube.com/watch?v=n0bxSD-Xx-Q


So we have scene gamut and then capture that only falls within sRGB, so what? We have scene gamut and captured data that exceeds Adobe RGB (1998) gamut, and requires ProPhoto RGB. Are you proposing we examine every capture to see where it falls, despite the gamut of the raw processor (in my case, ACR) using a ProPhoto RGB gamut processing color space? For what gain?

Somewhat fewer are also outside of Adobe RGB.

But they are outside that gamut. What are you proposing other than pick the next larger encoding working space?

What some don't realize is that the modifications made to colors as part of rendering from scene reference to output reference almost always involve increasing saturation. It's simply a byproduct of increasing contrast and is what people expect to see in pleasing photos.

Again, what are you proposing other than using that larger gamut encoding working space?

Work in whatever space you understand and have the tools to utilize.

What's to understand other than there are three main RGB working spaces of differing color gamut to select? You can select the largest gamut option and produce an iteration with the smaller gamut one for posting on the web. Best of both worlds. Considering we don't know the final output, what  benefit would be picking the smallest or even the middle sized gamut when we both agree, some images exceed both?

At times, for customers that require sRGB workproduct then it may make sense to use that start to finish.

IOW, output only to the internet and mobile devices. That's the only output where sRGB makes sense. Oh, and those silly labs that demand sRGB for their own benefit, not yours.

The last part of my post deals with the very large difference when using ProPhoto to print or converting the same image to sRGB, or aRGB then printing. Take your Gamut Test File, The differences are quite noticeable and most of them are due to the conversions between ProPhoto and either aRGB or sRGB and not from a fundamental printer gamut limit.

Yes, they are noticeable and for a good reason. One produces a superior print compared to the others on this end. Why would I use anything but the working space that produces the best appearing print?

But I also tried an alternative path.

I don't understand what or why anyone would do this. Let's stick to workflows. We have a pile of raw images. Some will fit within sRGB, some must be encoded in ProPhoto RGB to contain color that fall outside sRGB and Adobe RGB gamut. The only way to know this would be to encode them and examine the gamut of the image, something I did in my video. It's super time consuming! So do we pick the smaller gamut working space and clip colors we capture and can output or pick the larger one and move on? I can produce a ProPhoto RGB, high resolution Master, spend X number of minutes or hours editing it and end up with a low resolution sRGB copy for the web. I can print the larger gamut image using all the color I can capture and utilize for that one printer. Again, I'm trying to understand what you're proposing I (we) do differently when some images will be outside Adobe RGB color gamut, we have a printer who's color gamut exceeds Adobe RGB (1998) and we want to make a print of that data?

[Iliah]

On a side note, the colours on Kodak / Fuji films and papers fall out of Adobe RGB gamut. Even Fogra Glossy allows more than Adobe RGB encompasses. I wonder how we and our customers live(d) with that; and how that "give us the film look" plea coexists with sRGB workflow demands.

[Doug Gray]

I don't understand what or why anyone would do this. Let's stick to workflows. We have a pile of raw images. Some will fit within sRGB, some must be encoded in ProPhoto RGB to contain color that fall outside sRGB and Adobe RGB gamut. The only way to know this would be to encode them and examine the gamut of the image, something I did in my video. It's super time consuming! So do we pick the smaller gamut working space and clip colors we capture and can output or pick the larger one and move on? I can produce a ProPhoto RGB, high resolution Master, spend X number of minutes or hours editing it and end up with a low resolution sRGB copy for the web. I can print the larger gamut image using all the color I can capture and utilize for that one printer. Again, I'm trying to understand what you're proposing I (we) do differently when some images will be outside Adobe RGB color gamut, we have a printer who's color gamut exceeds Adobe RGB (1998) and we want to make a print of that data?

There is no reason not to use ProPhoto with a good color managed process for printing. Your videos and instructional materials are excellent on this.

So why would I suggest an additional workflow, that departs from usual practice,  when other customers require sRGB that matches your prints as visually close as possible?

Well, there is one fairly good reason. The largest, out of gamut, color distortion going from ProPhoto to sRGB is in the standard matrix conversion used in "export for" or "convert to" sRGB. This is particularly obvious with your Gamut Test file. The alternate workflow, which is completely automatic, provides a very good representation of the actual printed image in the smaller (sRGB or aRGB) gamut.  For instance, if you are selling prints that use the full gamut a printer is capable of, and your workspace is ProPhoto RGB, then you want the sRGB images, which you may post on the web or email to potential customers, to be as close as possible to what the full gamut print looks like. Just converting from ProPhoto RGB to sRGB will wash out a lot of lot of out of gamut colors that could be better represented while still remaining within sRGB as I demonstrated above using the green vertical bars.

[digitaldog]

There is no reason not to use ProPhoto with a good color managed process for printing. Your videos and instructional materials are excellent on this.

Just the opposite Doug. The use of ProPhoto RGB shows improvements in the actual printed output (albeit to the printer and paper/profile combo used) compared to sRGB and Adobe RGB (1998).
The proof is in the print!

So why would I suggest an additional workflow, that departs from usual practice,  when other customers require sRGB that matches your prints as visually close as possible?

sRGB is optional for one output use and suboptimal for all others. If you need sRGB, convert to sRGB from a larger color space which can be used for output to any number of printed needs.
I've got plenty of prints from plenty of images as do others that illustrate that ProPhoto RGB is providing better appearing prints. Again, the proof is in the print.

[Doug Gray]

Just the opposite Doug. The use of ProPhoto RGB shows improvements in the actual printed output (albeit to the printer and paper/profile combo used) compared to sRGB and Adobe RGB (1998).

Read it again. The double negative threw you off.

The proof is in the print!  sRGB is optional for one output use and suboptimal for all others. If you need sRGB, convert to sRGB from a larger color space which can be used for output to any number of printed needs.
I've got plenty of prints from plenty of images as do others that illustrate that ProPhoto RGB is providing better appearing prints. Again, the proof is in the print.

Apparently my point isn't getting across. If you have a ProPhoto RGB image with out of sRGB gamut parts, converting from ProPhoto to sRGB produces a great deal of unnecessary distortion that can't easily be tweaked to produce an sRGB image that closely resembles the printed image. Converting to the printer's profile using the same target profile settings the print from ProPhoto uses then converting this using RC to sRGB produces an image that much more closely resembles the original, full gamut, pleasing print.

If you then print that sRGB image it will be much closer to the original, large gamut print. As you say, the proof is in the print.

[digitaldog]

Read it again. The double negative threw you off.

The we are in agreement about using ProPhoto RGB for output to print.

Apparently my point isn't getting across. If you have a ProPhoto RGB image with out of sRGB gamut parts, converting from ProPhoto to sRGB produces a great deal of unnecessary distortion that can't easily be tweaked to produce an sRGB image that closely resembles the printed image. Converting to the printer's profile using the same target profile settings the print from ProPhoto uses then converting this using RC to sRGB produces an image that much more closely resembles the original, full gamut, pleasing print.

If you then print that sRGB image it will be much closer to the original, large gamut print. As you say, the proof is in the print.

Closer to what original? The original in my workflow is raw to ProPhoto RGB. Are you saying by converting that to sRGB, it's closer to sRGB? No argument. Why would I do or want that?

[Mark D Segal]

............Converting to the printer's profile using the same target profile settings the print from ProPhoto uses then converting this using RC to sRGB produces an image that much more closely resembles the original, full gamut, pleasing print.

If you then print that sRGB image it will be much closer to the original, large gamut print. As you say, the proof is in the print.

I've been following this discussion with some interest, and must say I believe I needn't worry about either workflow option if I always start from from my raw file, use Lightroom as my primary photo editor and repurpose the raw files as needed, be it for devices, or for any particular printer/paper combination. The algorithms within LR for doing these on-the-fly conversions seem to be producing coherent results, taking into account that for printing purposes one should be making final luminance and colour adjustments under soft-proof bespoke to the printer/paper combination in play. So, that's the practical aspect of interest to me. Also of interest would be to see an explanation of the principles underlying the added workflow conversion step Doug is recommending - conversion to the printer profile - before conversion to a smaller colour space. Of course each printer/paper combination has its own gamut space and all of them are smaller than ProPhoto. So the process Doug recommends essentially compresses the colour space for the image in two steps rather than one: from Pro-Photo to the Printer Profile space and from that space to sRGB. Why in principle is this better than a direct conversion from ProPhoto to sRGB? Or is it a result of empirical observation and the reason is as yet unclear?

(Typo corrected)

[digitaldog]

I've been following this discussion with some interest, and must say I believe I needn't worry about either workflow option if I always start from from my raw file, use Lightroom as my primary photo editor and repurpose the raw files as needed, be it for devices, or for any particular printer/paper combination.

We are in violent agreement. And if you're not using a LR workflow 100%, you do want to edit in Photoshop, I suspect you'd encode 16-bit ProPhoto RGB.

[Mark D Segal]

We are in violent agreement. And if you're not using a LR workflow 100%, you do want to edit in Photoshop, I suspect you'd encode 16-bit ProPhoto RGB.

Yes, you suspect correctly; but I would still like to hear from Doug about the basic principles underlying his deviation into the printer profile colour gamut on the way. Not that I am denying the usefulness of visual evidence, but I'd like to know whether there is a scientific (i.e. objective, verifiable, repeatable) basis for it.

[Doug Gray]

Closer to what original?

"the original, large gamut print,"  That original. The print you made from ProPhoto RGB.

The original in my workflow is raw to ProPhoto RGB. Are you saying by converting that to sRGB, it's closer to sRGB? No argument. Why would I do or want that?

No, I'm saying that to obtain a closer rendering in sRGB of your actual print from ProPhoto RGB that neither converting directly from RAW to sRGB or RAW to ProPhoto RGB then to sRGB should be done. To get the best rendering in sRGB of a large gamut image print from ProPhoto do not convert to sRGB from ProPhoto. Convert to the printer space first using the desired printer settings then convert that to sRGB.

The differences on images with a lot of out of gamut colors are not subtle.

Andrew, try this test yourself and compare the three prints:

Print your Gamut Test image using whatever settings you like, perhaps Perceptual.
Then convert the test image to sRGB and print it using the same settings.
Then, take the ProPhoto test image, convert it to printer RGB space using the same rendering. Then convert it to sRGB using RC, now print it using RC.

This last print will be a much closer match that the first approach which is what people typically do.

[digitaldog]

Andrew, try this test yourself and compare the three prints:
Print your Gamut Test image using whatever settings you like, perhaps Perceptual.
Then convert the test image to sRGB and print it using the same settings.

I've done that already. 

Then, take the ProPhoto test image, convert it to printer RGB space using the same rendering. Then convert it to sRGB using RC, now print it using RC.
This last print will be a much closer match that the first approach which is what people typically do.

I can't print an sRGB image, it has to be in output color space so I'm again confused. Further, you state: This last print will be a much closer match that the first approach which is what people typically do.
I don't understand the bit about "what people typically do" either. Like Mark, I'm trying to understand the workflow and ramifications you're trying to apply but I'm still confused.

[Doug Gray]

I've done that already.    I can't print an sRGB image, it has to be in output color space so I'm again confused. Further, you state: This last print will be a much closer match that the first approach which is what people typically do.

Um, really?  I can't imagine what those print labs that insist on sRGB files do to actually get prints. Voodoo maybe?

sRGB, like ProPhoto RGB IS an output colorspace. Printing from sRGB is exactly the same process (matrix color adaption for differing white points - sRGB v Lab PCS and ProPhoto which already use D50 -  and conversion to coefs. of the RGB primaries as printing from ProPhoto RGB.

I don't understand the bit about "what people typically do" either. Like Mark, I'm trying to understand the workflow and ramifications you're trying to apply but I'm still confused.

What people that work in larger gamut spaces "typically" do is convert from the larger space to the target space inside either LR or PS. Then they put it on the web. This works perfectly great when the image gamut is reasonably close to sRGB. The further out the image gamut, the more poorly it works in the sense that it diverges from the image that gets printed. That may, or may not look better but it is different and more different than it needs to be.

My suggested workflow addressed the issues that come up with an image far enough out of sRGB gamut that the crude gamut reduction by matrix clipping produces bad results.

[digitaldog]

Um, really?  I can't imagine what those print labs that insist on sRGB files do to actually get prints. Voodoo maybe?

No Voodoo, sRGB gets converted to the output color space for the printer. Just as I can't send sRGB either, hence my question to you about printing sRGB. There's no such thing as an sRGB printer. However, a lab can demand an sRGB file to convert to their output color space. Which is  pretty awful when the data greatly exceeds sRGB as does the printer.

sRGB, like ProPhoto RGB IS an output colorspace.

No it's not. It's an RGB working space, a special group of synthetically defined color spaces based on three simple attributes: white point, chromaticity values and gamma (TRC). How can ProPhoto RGB be an output color space when it defines "colors" (device values) that are both invisible to us and are impossible to output to any device? All about RGB working space: http://www.adobe.com/digitalimag/pdfs/phscs2ip_colspace.pdf

Printing from sRGB is exactly the same process (matrix color adaption for differing white points - sRGB v Lab PCS and ProPhoto which already use D50 -  and conversion to coefs. of the RGB primaries as printing from ProPhoto RGB.

I don't again understand what you're saying. You can't print sRGB, it has to be converted to some output color space which is often (well always in every output profile I've examined) larger in some areas of color space than sRGB. Red is sRGB, colored is the gamut of the output device, those who's color gamuts one usually finds at lab's demanding all data be sent to them in sRGB:

What people that work in larger gamut spaces "typically" do is convert from the larger space to the target space inside either LR or PS

There's no such provision in LR it's processing pipeline is always using ProPhoto RGB primaries and thus it's gamut. One can export to a smaller gamut color space in that product and that's useful when you need sRGB for the web or mobile devices. You can't print sRGB though LR, it's not possible. As Mark pointed out, it's always sending ProPhoto RGB gamut through it's print module. If you happen to have clipped all that nice wider gamut data and have sRGB rendered data, the damage is done. None the less, IF edits are applied to the rendered image, ProPhoto RGB is used on that clipped data. And raw data isn't clipped to sRGB thankfully.

This works perfectly great when the image gamut is reasonably close to sRGB. The further out the image gamut, the more poorly it works in the sense that it diverges from the image that gets printed. That may, or may not look better but it is different and more different than it needs to be.

The web and thus sRGB are the lowest common denominator and it's kind of foolish to throw the baby out with the bath water when you intend to print that data too.

[Iliah]

> The web and thus sRGB are the lowest common denominator

That's scratching the surface

The history of discarding colour space in browsers is no small reason why sRGB proliferated.

[digitaldog]

The history of discarding colour space in browsers is no small reason why sRGB proliferated.

I have to wonder if consumers, who appear to prefer 'saturated' colors will move to wider gamut displays as they become less expensive. If so, it will be interesting to see where sRGB goes in the future.

[Doug Gray]

I've been following this discussion with some interest, and must say I believe I needn't worry about either workflow option if I always start from from my raw file, use Lightroom as my primary photo editor and repurpose the raw files as needed, be it for devices, or for any particular printer/paper combination. The algorithms within LR for doing these on-the-fly conversions seem to be producing coherent results, taking into account that for printing purposes one should be making final luminance and colour adjustments under soft-proof bespoke to the printer/paper combination in play. So, that's the practical aspect of interest to me. Also of interest would be to see an explanation of the principles underlying the added workflow conversion step Doug is recommending - conversion to the printer profile - before conversion to a smaller colour space. Of course each printer/paper combination has its own gamut space and all of them are smaller than ProPhoto. So the process Doug recommends essentially compresses the colour space for the image in two steps rather than one: from Pro-Photo to the Printer Profile space and from that space to sRGB. Why in principle is this better than a direct conversion from ProPhoto to sRGB? Or is it a result of empirical observation and the reason is as yet unclear?

(Typo corrected)

Here's the actual conversions done that demonstrates why the two step process is better when converting an image from ProPhoto to sRGB when trying to match a print made directly from Andrew's Printer Gamt Test File.

Focusing in on the Green, vertical gradient column locate a point equidistant from the bottom and left/right sides (a good location to take a verifying reading with a spectro) the ProPhoto RGB values are (7,255,7).  This is, of course, an unrealizable color that can be thought of as an intense, supersaturated green. This is not even close to being reproducible in sRGB, aRGB, a print gamut, or anything in our reality.

Let's follow the two paths of color conversion.

The first is straightforward. Convert the color to sRGB then print it using either Perceptual or Relative colorimetry. Let's use Relative for simplicity noting that we could instead use Perceptual. The results in sRGB are (0,255,0) over the entire bottom third of the vertical green column. So the conversion results in clipping. Let's look at the details. PP(7,255,7) is, in XYZ (PCS working space for matrix profiles) is 13.6, 71.2, .128.  Running the matrix conversion to sRGB and scaling to sRGB's almost 2.2 gamma yields  sRGB(-221, 279, -108). Since us mortals, and Adobe, can't deal with RGB values beyond [0:255] they are simply clipped to sRGB(0,255,0). That's a big haircut.  But does that get us close to the printer's gamut? No, it does not. Virtually all colors near sRGB's three primaries cannot be printed. Printer gamuts exceed sRGB and aRGB only along the gamut triangle's sides.

So what happens when we print sRGB(0,255,0) and compare that to printing the ppRGB(7,255,7). First the sRGB(0,255,0) is converted to PCS (Lab) then the Lab values, adapted for BPC, use the 3D LUTs to interpolate a set of RGB values to send to the printer. This process is a function of the printer profile so YMWV. Often, printer profiles simply map out of gamut colors to the closest color on the gamut boundary.

Using my printer and profiles (a custom profile with Epson PPG printed on a Canon 9500 II) RC w BPC resulted in a Lab value of 72,-53, 67 as read by a spectro.

Now let's examine what happens when the image is printed from ProPhoto directly to the printer using RC w BPC. First the ppRGB(7,255,7) is converted to PCS (Lab clipped [-128:127]) then the BPC adapted 3D lookup is output to the printer and the image is printed. On my printer this produces a Lab value of (60, -67, 56) which is very far off (Deltas E:21) from the yellowish green printed by converting first to sRGB.

By going through the convert to printer then convert back to sRGB allows one to more closely match the printer's actual image as printed from RAW or ProPhoto RGB to an image intended for the web.

The problem's encountered in this is loss of accuracy due to inaccuracies in the printer's BtoA->AtoB. Each transition can be expected to introduce error ranging from tenths of a DeltaE to several DeltaEs at the gamut boundary. So, unless an image is significantly out of aRGB or sRGB, I don't recommend this. However, it's great for images with a lot of synthetic colors such as rainbow or the grainger color gradients. It's not accurate per se since the colors are out of gamut, but it does make for a much closer match between a print image and an image in sRGB intended for the web.

[digitaldog]

By going through the convert to printer then convert back to sRGB allows one to more closely match the printer's actual image as printed from RAW or ProPhoto RGB to an image intended for the web.

So the goal is to make the print as desaturated as an sRGB browser viewing that image on the web? Again, I don't see why one would do that.
Your technique isn't new and is useful for print in one case I can see; proofing (make my wider gamut Epson better match a contract proof).