Compute White Underprint Values

Question

I want to print this motif on a grey hoodie using DTG, how would I compute the color values for the white underprint to make the colors on grey fabric approximate the colors in the image? My intuition is, after converting to CMYK (assuming no pixel has positive values in all four channels), I subtract the K value of the background grey from the K value of each pixels, the negative K values give me exactly the intensity of the white underprint needed to represent the color. Is that somewhat correct? [EDIT] My back of an envelope calculation went like following. Assume for simplicity that every layer tranlucent print of color S ∈ {C, M, Y, K} lets light of color A ∈ {R, G, B} pass at rate S(A). Assume further that the fabric reflects light of color A at rate F(A), and a layer of white print on the fabric with opacity w reflects light of color A at rate w + (1 - w) F(A). So a standard CMYK conversion of the RGB image with corresponding values r, g, b would tell us what values c, m, y, k approximate a ~ (c C(A))² (m M(A))² (y Y(A))² (k K(A))² for (a, A) ∈ {(r, R), (g, G), (b, B)}. We want to find values c', m', y', k', w to approximate a ~ (c' C(A))² (m' M(A))² (y' Y(A))² (k' K(A))² (w + (1 - w) F(A)) for (a, A) ∈ {(r, R), (g, G), (b, B)}. To have the grey being transparent, we must further assume that approximating a := F(A) gives all zeros.

Answer 1

I can show you one way (out of several) to remove the gray background color from the image, but I'm not sure if the resulting image will work for a shirt print.

In RGB, just add a layer on top of the image filled with the same gray color as the image and set the new layer's Blend Mode to Divide.

Optionally use a Levels adjustment layer or similar to add some contrast to the image.

Now the problem is that if the printer adds solid white under each pixel which has some other color than white, the result won't look like the original image.

Here I try to show how it would look by first grouping the layers from before. Then pasting in a cotton fabric background and finally double-clicking the group and use Blend If to make every completely white pixel transparent. The logo contains some darker gray areas which I try to remove by altering the Levels layer, but this exposes the bad quality of the image. Perhaps your original is better than the compressed version I downloaded.

If the result was better you could perhaps add back some of the removed gray color, but I think the difference between gray fabric and gray print would always be visible. Even if you somehow managed to get the exact tone, there would still be a difference in texture.

Perhaps you should forget getting the exact colors of the original and just print the result from my first gif (with the gray removed) directly on a (not too dark) gray shirt. It could look a little like this:

It won't look like the original image, but it would be a "pure" solution without any trickery needed.

It will probably be a good idea to ask the manufacturer about which method would give the best result.

Edit due to edit in the original question

You have added some calculations to your question for calculating the values for the white channel. I don't fully understand. Maybe the math makes sense, but I think you misunderstand how the print is made physically.

I'm not an expert in DTG print (I work with offset and digital print), but I believe that like all other kinds of print you can really only print in 1-bit. Ink or no ink. So all tints are made with some kind of screen pattern. The CMYK colors are transparent so it works well to overlap the inks, but the white is opaque and is normally just used at 100%. (Others please correct me if I'm wrong.)

Let me show you an example of what I mean. Let's assume you wanted to print a magenta gradient on a gray shirt.

If you just printed the magenta directly on the fabric you wouldn't get the intended result.

So you want to add some white opaque ink underneath.

You hope to get the following result.

In reality we can't print tints of colors. We can only print in 1-bit. Either there is ink or there is no ink. So we have to use some kind of screen pattern to simulate tints. I don't know which pattern DTG uses, so here I've just used traditional halftone screening. The issues should be similar.

We would have to print the opaque white first.

And then print the transparent magenta on top.

As you see this method have some limitations. In some places you see the magenta on top of fabric other places it's on top of white. In some places the white doesn't have magenta on top and shines through. Even a tiny white dot lights up on the dark fabric and the gradient doesn't fade out seamlessly. It doesn't quite give the intended result.

Answer 2

If that is all that you have for the logo, good luck. Any printer that you send that to is going to print you a big rectangle and flood white behind it. There's nothing else that they can do. Even if they do try to print white just under the rainbow parts, that file would still print CMYK to make the grey rectangle and show up on the shirt. That's how the print works.

You can't do what you intend with that image. When you convert that to CMYK the "grey" isn't just in the K channel, it will be in all four with no easy way to separate it out.

If you want to print Just the rainbow haloing, or even to print the isolated OpenGL logo, you are going to have to recreate this in some way that doesn't have the grey in it, only the parts you want to print. Once you have that on a white background (Photoshop) or as vector art with gradients (Illustrator), the printer will have a setting in the software for "Pixel Match" or something similar that tells the printer to print white behind anywhere it's printing CMYK.