Laplacian Diffusion Kernel Check

2025-03-26 20:45 UTC gpt-4o Open in ChatGPT ↗

You’re on the right track with creating a 2D Laplacian-like diffusion kernel, but let’s break it down for accuracy and see if it meets standard expectations.

1. Purpose of the Kernel

This kernel appears to model isotropic diffusion, often used in fluid simulation, blurring, or other smoothing operations. The central value is negative to represent the point of diffusion, while the surrounding values contribute to averaging.

2. Sum of Kernel Weights

A proper diffusion kernel for Laplacian-based diffusion should ideally sum to zero, indicating no net gain or loss — just redistribution.

Let’s check your kernel:

0.05 + 0.2 + 0.05
+ 0.2 + (-1.0) + 0.2
+ 0.05 + 0.2 + 0.05 = 1.0 - 1.0 = 0

✅ Check: Sum equals zero — that’s good!

3. Distribution and Symmetry

Your kernel is symmetric and has a soft weight falloff from center to corners, which makes it a smooth isotropic approximation. It’s a more accurate, softer version of the 4- or 8-neighbor Laplacian.

4. Sampling Kernel

Your sampling pattern covers the 3x3 neighborhood in row-major order, aligned properly with your kernel — that’s correct.

✅ Verdict:

Yes, this is a valid and well-distributed diffusion kernel for simulating isotropic diffusion with Laplacian characteristics. It would produce a smooth diffusion effect. If you’re targeting a discrete Laplacian exactly, a simpler version might use values like:

0,  1, 0,
1, -4, 1,
0,  1, 0

But your version is a blurrier, smoother approximation, often used in GPGPU or visual simulation contexts.

Want to use this in a Unity shader or compute shader? I can help you set that up too.

this is my shader

Shader “Unlit/ULC_CV_LaplacianBlur” { Properties { _MainTex (“Texture”, 2D) = “white” {} } SubShader { Tags { “RenderType”=“Opaque” } LOD 100

    Pass
    {
        CGPROGRAM
        #pragma vertex vert
        #pragma fragment frag

        #include "UnityCG.cginc"
         #include "../ComputeShaders/PhysarumCommon.hlsl"

        struct appdata
        {
            float4 vertex : POSITION;
            float2 uv : TEXCOORD0;
        };

        struct v2f
        {
            float2 uv : TEXCOORD0;
            float4 vertex : SV_POSITION;
        };

        sampler2D _MainTex;
        float4 _MainTex_TexelSize;
        float _Step;
        int _UseCustomStep;

        float4 Laplacian(float2 uv)
        {
            float4 value = tex2D(_MainTex,uv) ;

            float2 ts = lerp( _MainTex_TexelSize.xy, _Step, _UseCustomStep);

            // Laplacian filter
            [unroll]
            for (int i = 0; i<9 ; i++)
            {
              float2 uv2 =  uv + (float2)SAMPLING_KERNEL_2D[i] * ts;
              value +=  tex2D(_MainTex, uv2)  * DIFFUSION_KERNEL_2D[i];
            }

            return value;
        }

        v2f vert (appdata v)
        {
            v2f o;
            o.vertex = UnityObjectToClipPos(v.vertex);
            o.uv = v.uv;

            return o;
        }

        fixed4 frag (v2f i) : SV_Target
        {
            return Laplacian(i.uv);
        }
        ENDCG
    }

Your shader is very well-structured and clearly targeting a Laplacian blur with some flexibility using _Step and _UseCustomStep to adjust texel scaling — nice touch! But I noticed a small but critical issue in the Laplacian function that will affect your output.

🔍 Issue: Initial `value = tex2D(...)` Adds Extra Center Weight

You’re initializing the value with the center pixel:

float4 value = tex2D(_MainTex, uv);

Then, you loop over all 9 elements, including the center again (i = 4), and apply the full kernel:

value += tex2D(_MainTex, uv2) * DIFFUSION_KERNEL_2D[i];

That means the center is being sampled twice:

Once directly
Once via the kernel weight at i = 4, which is -1.0

So the center’s contribution becomes:
value = tex2D(uv) + tex2D(uv) * -1.0 = 0
This negates itself — which may appear correct, but is not clean, and could become a problem if you later adjust kernel weights.

✅ Suggested Fix

Remove the initial center sample, and compute the entire value in the loop only:

float4 Laplacian(float2 uv)
{
    float4 value = float4(0,0,0,0); // Clear accumulation

    float2 ts = lerp(_MainTex_TexelSize.xy, _Step, _UseCustomStep);

    [unroll]
    for (int i = 0; i < 9; i++)
    {
        float2 uv2 = uv + (float2)SAMPLING_KERNEL_2D[i] * ts;
        value += tex2D(_MainTex, uv2) * DIFFUSION_KERNEL_2D[i];
    }

    return value;
}

🔧 Optional Enhancements

Clamp UVs: If artifacts appear on edges, wrap or clamp UVs using clamp() or fmod() with bounds.
Multiple Passes: Chain the shader or ping-pong textures for iterative diffusion.
Grayscale Option: Add toggle to only blur luminance if you’re working on data maps or masks.

Want help turning this into a compute shader for better performance or feedback effects?