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# GPU friendly compression of 2D signal

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Hey

I want to try shade particles by compute a "small" number of samples, e.g. 10, in VS. I only need to compute the intensity of the light, so essentially it's a single piece of data in 2 dimensions.

Now I want to compress this data, pass it on to PS and decompress it there (the particle is a single quad and the data is passed through interpolators). I will accept a certain amount of error as long as there are no hard edges, i.e. blurred.

The compressed data has to be small and compression/decompression fast. Does anyone know of a good way to do this?

Maybe I could do something fourier based but I'm not sure of what basis functions to use.

Thanks

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Combine the results of the samples using Spherical Harmonics and output that in the VS.

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If you want it only for 2D, SH is more than necessary, but i'm unsure what you try to do.

However, i use code below to calculate the curvature directions of a mesh. The problem here is that curvature direction is the same on opposite sides, e.g. vec2(0.7, 0.7) equals vec2(-0.7, -0.7) so i can not simply add vectors to get average curvature direction.

Instead i express directions with a sine wave that has two lobes pointing forwards and backwards, phase is direction and amplitude is intensity. Now adding two sine waves always results in another single sine wave and this way i get an accurate result from summing any number of samples. (Same principle is used in SH and Fourier Transform).

So, if this sounds interesting to you, you could do the same for lighting, but you would want the lobe pointing only in one direction and not the opposite as well, which means replacing factors of 2 with 1 and adjusting some other things as well.

But: For lighting i would just sum up vector wise and accept the error coming from that. Also note that my approach does not have a constant band like SH, so the same amount of light coming from right and left would result to zero - might be worth to add this for lighting.

	struct Sinusoid
{
float phase;
float amplitude;

Sinusoid ()
{
phase = 0;
amplitude = 0;
}

Sinusoid (const float phase, const float amplitude)
{
this->phase = phase;
this->amplitude = amplitude;
}

Sinusoid (const float *dir2D, const float amplitude)
{
this->amplitude = amplitude;
phase = PI + atan2 (dir2D[1], dir2D[0]) * 2.0f;
}

float Value (const float angle) const
{
return cos(angle * 2.0f + phase) * amplitude;
}

void Add (const Sinusoid &op)
{
float a = amplitude;
float b = op.amplitude;
float p = phase;
float q = op.phase;

phase = atan2(a*sin(p) + b*sin(q), a*cos(p) + b*cos(q));
float t = a*a + b*b + 2*a*b * cos(p-q);
amplitude = sqrt(max(0,t));
}

float PeakAngle () const
{
return phase * -0.5f;
}

float PeakValue () const
{
return Value(PeakAngle ());
}

void Direction (float *dir2D, const float angle) const
{
float scale = (amplitude + Value (angle)) * 0.5f;
dir2D[0] = sin(angle) * scale;
dir2D[1] = cos(angle) * scale;
}

};

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Thanks for the replies!

SH could be an option and it did cross my mind. It's 2D indeed. The domain when using SH is usually a sphere. In my case I want to use it on a quad. Would that have an impact? I understand that spherical coords can be regarded as a square. I'm just wondering if the domain has any impact & if there would be some other basis functions more suited for an actual quad.

@JoeJ your suggestion looks similar to a fourier series which also crossed my mind. The fact that sin/cos operations are expensive on GPUs made me a little less keen. The general idea of treating the problem as some sort of curve is good though. I could use something like a power function, that could be encoded in 4 params - uv intensity multiplier & uv exponent, given that I pass on the actual colour of 1 of the corners (on the other hand this approach would only be able to depict gradients).

I'm not 100% of how much detail I need to encode in the quad but preferably as much as possible for as little cost

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2 hours ago, 51mon said:

Thanks for the replies!

SH could be an option and it did cross my mind. It's 2D indeed. The domain when using SH is usually a sphere. In my case I want to use it on a quad. Would that have an impact? I understand that spherical coords can be regarded as a square. I'm just wondering if the domain has any impact & if there would be some other basis functions more suited for an actual quad.

You can warp fourier transform around a circle, and traet the circle as a square. You can then decice how much bands you need: 1st. Band is a constant term, 2nd can encode a lode towards a single directions, adding more bands means you can approximate multiple lights more accurate.

SH is similar: The smallest 2 band version has one number for the constant term (1st band), and a 3D vector (2nd band) for a directional bump. (This band tells you the dominant light direction if you gather many samples, similar to my curvature example.)

So for 2D you should need 3 numbers: Constant term, and a 2D direction (or angle and amplitude like i did, but direction avoids the trig functions when decoding).

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Shortcut to sterp implementation.
Shortcut to code used to generate animations in this post.
An Alternative to Slerp
Slerp, spherical linear interpolation, is an operation that interpolates from one orientation to another, using a rotational axis paired with the smallest angle possible.
Quick note: Jonathan Blow explains here how you should avoid using slerp, if normalized quaternion linear interpolation (nlerp) suffices. Long store short, nlerp is faster but does not maintain constant angular velocity, while slerp is slower but maintains constant angular velocity; use nlerp if you’re interpolating across small angles or you don’t care about constant angular velocity; use slerp if you’re interpolating across large angles and you care about constant angular velocity. But for the sake of using a more commonly known and used building block, the remaining post will only mention slerp. Replacing all following occurrences of slerp with nlerp would not change the validity of this post.
In general, slerp is considered superior over interpolating individual components of Euler angles, as the latter method usually yields orientational sways.
But, sometimes slerp might not be ideal. Look at the image below showing two different orientations of a rod. On the left is one orientation, and on the right is the resulting orientation of rotating around the axis shown as a cyan arrow, where the pivot is at one end of the rod.

If we slerp between the two orientations, this is what we get:

Mathematically, slerp takes the “shortest rotational path”. The quaternion representing the rod’s orientation travels along the shortest arc on a 4D hyper sphere. But, given the rod’s elongated appearance, the rod’s moving end seems to be deviating from the shortest arc on a 3D sphere.
My intended effect here is for the rod’s moving end to travel along the shortest arc in 3D, like this:

The difference is more obvious if we compare them side-by-side:

This is where swing-twist decomposition comes in.

Swing-Twist Decomposition
Swing-Twist decomposition is an operation that splits a rotation into two concatenated rotations, swing and twist. Given a twist axis, we would like to separate out the portion of a rotation that contributes to the twist around this axis, and what’s left behind is the remaining swing portion.
There are multiple ways to derive the formulas, but this particular one by Michaele Norel seems to be the most elegant and efficient, and it’s the only one I’ve come across that does not involve any use of trigonometry functions. I will first show the formulas now and then paraphrase his proof later:
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Swing-Twist Interpolation
Replacing slerp with the swing and twist components is actually pretty straightforward. Let the Q_0 and Q_1 denote the quaternions representing the rod's two orientations we are interpolating between. Given the interpolation parameter t, we use it to find "fractions" of swing and twist components and combine them together. Such fractiona can be obtained by performing slerp from the identity quaternion, Q_I, to the individual components. So we replace: Slerp(Q_0, Q_1, t) with: Slerp(Q_I, S, t) Slerp(Q_I, T, t) From the rod example, we choose the twist axis to align with the rod's longest side. Let's look at the effect of the individual components Slerp(Q_I, S, t) and Slerp(Q_I, T, t) as t varies over time below, swing on left and twist on right:
And as we concatenate these two components together, we get a swing-twist interpolation that rotates the rod such that its moving end travels in the shortest arc in 3D. Again, here is a side-by-side comparison of slerp (left) and swing-twist interpolation (right):

I decided to name my swing-twist interpolation function sterp. I think it’s cool because it sounds like it belongs to the function family of lerp and slerp. Here’s to hoping that this name catches on.
And here’s my code implementation:
public static Quaternion Sterp ( Quaternion a, Quaternion b, Vector3 twistAxis, float t ) { Quaternion deltaRotation = b * Quaternion.Inverse(a); Quaternion swingFull; Quaternion twistFull; QuaternionUtil.DecomposeSwingTwist ( deltaRotation, twistAxis, out swingFull, out twistFull ); Quaternion swing = Quaternion.Slerp(Quaternion.identity, swingFull, t); Quaternion twist = Quaternion.Slerp(Quaternion.identity, twistFull, t); return twist * swing; } Proof
Lastly, let’s look at the proof for the swing-twist decomposition formulas. All that needs to be proven is that the swing component S does not contribute to any rotation around the twist axis, i.e. the rotational axis of S is orthogonal to the twist axis. Let vec{V_{R_para}} denote the parallel component of vec{V_R} to vec{V_T}, which can be obtained by projecting vec{V_R} onto vec{V_T}: vec{V_{R_para}} = proj_{vec{V_T}}(vec{V_R}) Let vec{V_{R_perp}} denote the orthogonal component of vec{V_R} to vec{V_T}: vec{V_{R_perp}} = vec{V_R} - vec{V_{R_para}} So the scalar-vector form of T becomes: T = [W_R, proj_{vec{V_T}}(vec{V_R})] = [W_R, vec{V_{R_para}}] Using the quaternion multiplication formula, here is the scalar-vector form of the swing quaternion: S = R T^{-1} = [W_R, vec{V_R}] [W_R, -vec{V_{R_para}}] = [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_R} + W_R (-vec{V_{R_para}})] = [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R (vec{V_R} -vec{V_{R_para}})] = [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_{R_perp}}] Take notice of the vector part of the result: vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_{R_perp}} This is a vector parallel to the rotational axis of S. Both vec{V_R} X(-vec{V_{R_para}}) and vec{V_{R_perp}} are orthogonal to the twist axis vec{V_T}, so we have shown that the rotational axis of S is orthogonal to the twist axis. Hence, we have proven that the formulas for S and T are valid for swing-twist decomposition. Conclusion
That’s all.
Given a twist axis, I have shown how to decompose a rotation into a swing component and a twist component.
Such decomposition can be used for swing-twist interpolation, an alternative to slerp that interpolates between two orientations, which can be useful if you’d like some point on a rotating object to travel along the shortest arc.
I like to call such interpolation sterp.
Sterp is merely an alternative to slerp, not a replacement. Also, slerp is definitely more efficient than sterp. Most of the time slerp should work just fine, but if you find unwanted orientational sway on an object’s moving end, you might want to give sterp a try.

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