Unity FFT海水渲染效果展示

最近研究了一下FFT的海水渲染，发现这东西真的涉及到好多好多的数学知识，我基本就是囫囵吞枣，现在基本也忘得差不多了。我在这里挂几个大佬的文章，想研究的可以去看一下，我这个数学渣渣就不强行解释了。
fft海面模拟（一、二、三）
fft ocean注解

Ocean simulation part one: using the discrete Fourier transform
快速傅立叶变换(FFT)的海面模拟
Shader相册第6期 — 实时水面模拟与渲染
【学习笔记】Unity 基于GPU FFT海洋的实现-理论篇
【学习笔记】Unity 基于GPU FFT海洋的实现-实践篇
我自己写了两个版本，一个是CPU得，另一个是GPU版本。
先看一下CPU版本：

这个波浪真得相当好看了，但是这个CPU版本的计算是和顶点绑定在一起的，模型的顶点数没办法太多，我这是64x64，再上去就要卡得不行了。
接下来的是GPU版本，这里就是照搬了后面两篇文章的做法，可以看看效果：

CSDN不能传超过4.5M的图片，下面是视频：

Unity ComputerShader FFT海水渲染

效果真的很不错。
我放一下CPU版本的代码，因为总感觉没代码怪怪的。
代码用的是第三第四篇文章的，涉及到的数学知识太多了，想实现的话还是需要花点时间和耐心得。

using System;
using System.Collections.Generic;
using System.Numerics;
using UnityEngine;
using Random = System.Random;
using Vector2 = UnityEngine.Vector2;
using Vector3 = UnityEngine.Vector3;public class Ocean
{public struct VertexOcean{public Vector3[] vertices;// vertexpublic Vector3[] normals;// normalpublic Vector2[] uvs;//UVpublic Vector2[] htildes;// htilde0public Vector2[] conjugates;// htilde0mk conjugatepublic Vector3[] originals;// original positionpublic Vector2[] uvs2;}public struct ComplexVectorNormal{// structure used with discrete fourier transformpublic Complex h;// wave heightpublic Vector2 D;// displacementpublic Vector3 n;// normal}float g = 9.81f;                                // gravity constantint N, Nplus1;                          // dimension -- N should be a power of 2float A;                                // phillips spectrum parameter -- affects heights of wavesVector2 w;                              // wind parameterfloat length;                           // length parameterRandom rand = new Random();Complex[] tilde, tildeSlopeX, tildeSlopeZ, tildeDx, tildeDz;FFT fft;                              // fast fourier transformpublic VertexOcean oceanData;                    // vertices for vertex buffer objectpublic List<int> indices;                  // indicies for vertex buffer objectint debug = 0;//M=N,L=Lx=Lzpublic Ocean(int N, float A, Vector2 w, float length){tilde = new Complex[N * N];tildeSlopeX = new Complex[N * N];tildeSlopeZ = new Complex[N * N];tildeDx = new Complex[N * N];tildeDz = new Complex[N * N];fft = new FFT(N);indices = new List<int>();this.N = N;Nplus1 = N + 1;this.A = A;this.w = w;this.length = length;oceanData = new VertexOcean();oceanData.vertices = new Vector3[Nplus1 * Nplus1];oceanData.normals = new Vector3[Nplus1 * Nplus1];oceanData.uvs = new Vector2[Nplus1 * Nplus1];oceanData.htildes = new Vector2[Nplus1 * Nplus1];oceanData.conjugates = new Vector2[Nplus1 * Nplus1];oceanData.originals = new Vector3[Nplus1 * Nplus1];oceanData.uvs2 = new Vector2[Nplus1 * Nplus1];int index;Complex htilde0, htilde0mk_conj;for (int m = 0; m < Nplus1; m++){for (int n = 0; n < Nplus1; n++){index = m * Nplus1 + n;htilde0 = hTilde_0(n, m);htilde0mk_conj = Complex.Conjugate(hTilde_0(-n, -m));oceanData.htildes[index].x = (float)htilde0.Real;oceanData.htildes[index].y = (float)htilde0.Imaginary;oceanData.conjugates[index].x = (float)htilde0mk_conj.Real;oceanData.conjugates[index].y = (float)htilde0mk_conj.Imaginary;oceanData.originals[index].x = oceanData.vertices[index].x = (n - N / 2.0f) * length / N;oceanData.originals[index].y = oceanData.vertices[index].y = 0;oceanData.originals[index].z = oceanData.vertices[index].z = (m - N / 2.0f) * length / N;oceanData.normals[index].x = 0.0f;oceanData.normals[index].y = 1.0f;oceanData.normals[index].z = 0.0f;oceanData.uvs[index].x = (float)n / N;oceanData.uvs[index].y = (float)m / N;}}for (int m = 0; m < N; m++){for (int n = 0; n < N; n++){index = m * Nplus1 + n;indices.Add(index);indices.Add(index + Nplus1);indices.Add(index + Nplus1 + 1);indices.Add(index);indices.Add(index + Nplus1 + 1);indices.Add(index + 1);}}}//高斯随机数public Complex GaussRandomVariable(){float s = 0, u = 0, v = 0;while (s > 1 || s == 0){u = (float)rand.NextDouble() * 2 - 1;v = (float)rand.NextDouble() * 2 - 1;s = u * u + v * v;}s = Mathf.Sqrt(-2 * Mathf.Log(s) / s);return new Complex(u * s, v * s);}//菲利普频谱public float Phillips(int n, int m){Vector2 k = new Vector2(Mathf.PI * (2 * n - N) / length, Mathf.PI * (2 * m - N) / length);float k_Length = k.magnitude;if (k_Length < 0.000001) return 0.0f;float k_Length2 = k_Length * k_Length;float k_Length4 = k_Length2 * k_Length2;float KdotW = Vector2.Dot(k.normalized, w.normalized);float KdotW2 = KdotW * KdotW;float w_Length = w.magnitude;float L = w_Length * w_Length / g;float L2 = L * L;float damping = 0.001f;float l2 = L2 * damping * damping;//这里为什么要再乘个 Mathf.Exp(-k_Length * l2)?float phillips = A * Mathf.Exp(-1.0f / (k_Length2 * L2)) / k_Length4 * KdotW2* Mathf.Exp(-k_Length2 * l2);return phillips;}public Complex hTilde_0(int n, int m){Complex r = GaussRandomVariable();return r * Mathf.Sqrt(Phillips(n, m) / 2.0f);}Complex hTilde(float t, int n, int m){int index = m * Nplus1 + n;Complex htilde0 = new Complex(oceanData.htildes[index].x, oceanData.htildes[index].y);Complex htilde0mk_conj= new Complex(oceanData.conjugates[index].x, oceanData.conjugates[index].y);float omegat = Dispersion(n, m) * t;float cos = Mathf.Cos(omegat);float sin = Mathf.Sin(omegat);Complex c0 = new Complex(cos, sin);Complex c1 = new Complex(cos, -sin);Complex res = htilde0 * c0 + htilde0mk_conj * c1;return res;}public float Dispersion(int n, int m){float w0 = 2.0f * Mathf.PI / 200.0f;float kx = Mathf.PI * (2 * n - N) / length;float kz = Mathf.PI * (2 * m - N) / length;//wo是用来做缩放的？return Mathf.Floor(Mathf.Sqrt(g * Mathf.Sqrt(kx * kx + kz * kz) / w0)) * w0;}public ComplexVectorNormal ComputerHDN(Vector2 x,float t){Complex h = new Complex();Vector2 D = Vector2.zero;Vector3 normal = Vector3.zero;Complex c, htilde;Vector2 k;float kx, kz, k_Length, KdotX;for (int m = 0; m < N; m++) {kz = 2.0f * Mathf.PI * (m - N / 2.0f) / length;for (int n = 0; n < N; n++){kx = 2.0f * Mathf.PI * (n - N / 2.0f) / length;k = new Vector2(kx, kz);k_Length = k.magnitude;KdotX = Vector2.Dot(k, x);c = new Complex(Mathf.Cos(KdotX), Mathf.Sin(KdotX));htilde = hTilde(t, n, m) * c;h += htilde;normal += new Vector3(-kx * (float)htilde.Imaginary, 0.0f, -kz * (float)htilde.Imaginary);if (k_Length < 0.000001) continue;D += new Vector2(kx / k_Length * (float)htilde.Imaginary, kz / k_Length * (float)htilde.Imaginary);}}normal = new Vector3(0f, 1f, 0f) - normal;ComplexVectorNormal complex=new ComplexVectorNormal();complex.h = h;complex.D = D;complex.n = normal.normalized;return complex;}public void EvaluateWaves(float t){float kx, kz, len, lambda = -1.0f;int index, index1;for (int m = 0; m < N; m++){kz = Mathf.PI * (2.0f * m - N) / length;for (int n = 0; n < N; n++){kx = Mathf.PI * (2.0f * n - N) / length;len = Mathf.Sqrt(kx * kx + kz * kz);index= m * N + n;tilde[index] = hTilde(t, n, m);tildeSlopeX[index] = tilde[index] * new Complex(0, kx);tildeSlopeZ[index] = tilde[index] * new Complex(0, kz);if(len<0.000001f){tildeDx[index] = new Complex(0f, 0f);tildeDz[index] = new Complex(0f, 0f);}else{tildeDx[index] = tilde[index] * new Complex(0f, -kx / len);tildeDz[index] = tilde[index] * new Complex(0f, -kz / len);}}}for (int m = 0; m < N; m++){fft.ComputeFFT(tilde, ref tilde, 1, m * N);fft.ComputeFFT(tildeSlopeX, ref tildeSlopeX, 1, m * N);fft.ComputeFFT(tildeSlopeZ, ref tildeSlopeZ, 1, m * N);fft.ComputeFFT(tildeDx, ref tildeDx, 1, m * N);fft.ComputeFFT(tildeDz, ref tildeDz, 1, m * N);}for (int n = 0; n < N; n++){fft.ComputeFFT(tilde, ref tilde, N, n);fft.ComputeFFT(tildeSlopeX, ref tildeSlopeX, N, n);fft.ComputeFFT(tildeSlopeZ, ref tildeSlopeZ, N, n);fft.ComputeFFT(tildeDx, ref tildeDx, N, n);fft.ComputeFFT(tildeDz, ref tildeDz, N, n);}int sign;int[] signs = { 1, -1 };Vector3 normal;Vector3[] offsets = ComputeOffset();debug = 0;for (int m = 0; m < N; m++){for (int n = 0; n < N; n++){index = m * N + n;index1 = m * Nplus1 + n;//如果n+m为奇数，则sign=signs[1]=-1；如果n+m为偶数，则sign=signs[0]=1sign = signs[(n + m) & 1];tilde[index] = tilde[index] * sign;tildeDx[index] = tildeDx[index] * sign;tildeDz[index] = tildeDz[index] * sign;Vector3 offset = new Vector3(lambda * (float)tildeDx[index].Real, (float)tilde[index].Real,lambda * (float)tildeDz[index].Real);oceanData.vertices[index1].y = offset.y;oceanData.vertices[index1].x = oceanData.originals[index1].x + offset.x;oceanData.vertices[index1].z = oceanData.originals[index1].z + offset.z;// normaltildeSlopeX[index] = tildeSlopeX[index] * sign;tildeSlopeZ[index] = tildeSlopeZ[index] * sign;normal = new Vector3(-(float)tildeSlopeX[index].Real, 1f, -(float)tildeSlopeZ[index].Real);oceanData.normals[index1] = normal;Vector2 jaco=new Vector2(ComputeJacobian(n, m, offsets),0);//DebugLog(jaco.x.ToString());oceanData.uvs2[index1] = jaco;int number;if (n == 0 && m == 0){number = index1 + N + Nplus1 * N;oceanData.vertices[number].y = offset.y;oceanData.vertices[number].x = oceanData.originals[number].x + offset.x;oceanData.vertices[number].z = oceanData.originals[number].z + offset.z;oceanData.normals[number] = normal;oceanData.uvs2[number] = jaco;}if (n == 0){number = index1 + N;oceanData.vertices[number].y = offset.y;oceanData.vertices[number].x = oceanData.originals[number].x + offset.x;oceanData.vertices[number].z = oceanData.originals[number].z + offset.z;oceanData.normals[number] = normal;oceanData.uvs2[number] = jaco;}if (m == 0){number = index1 + Nplus1 * N;oceanData.vertices[number].y = offset.y;oceanData.vertices[number].x = oceanData.originals[number].x + offset.x;oceanData.vertices[number].z = oceanData.originals[number].z + offset.z;oceanData.normals[number] = normal;oceanData.uvs2[number] = jaco;}}}}void DebugLog(string a){if (debug > 500) return;debug++;Debug.Log(a);}Vector3[] ComputeOffset(){Vector3[] offsets = new Vector3[N*N];float lambda = -1.0f;int index;int[] signs = { 1, -1 };for (int m = 0; m < N; m++){for (int n = 0; n < N; n++){index = m * N + n;int sign = signs[(n + m) & 1];offsets[index] = new Vector3(lambda * (float)(tildeDx[index] * sign).Real,(float)(tilde[index] * sign).Real,lambda * (float)(tildeDz[index] * sign).Real);}}return offsets;}float ComputeJacobian(int n, int m, Vector3[] offsets){int up = m - 1 < 0 ? (N - 1) * N + n : (m - 1) * N + n;int down = m + 1 > N - 1 ? n : (m + 1) * N + n;int left = n - 1 < 0 ? m * N + (N - 1) : m * N + n - 1;int right = n + 1 > N - 1 ? m * N : m * N + n + 1;Vector3 ddx = offsets[down] - offsets[up];Vector3 ddz = offsets[right] - offsets[left];float jacobian = (1.0f + ddx.x) * (1.0f + ddz.z) - ddx.z * ddz.x;return jacobian;}
}

using System.Numerics;
using UnityEngine;public class FFT
{public struct ComplexList{public Complex[] complices;}int N, which;int log2N;float pi2=2*Mathf.PI;uint[] reversed;ComplexList[] T;ComplexList[] c = new ComplexList[2];public FFT(int N){this.N = N;c[0].complices = new Complex[N];c[1].complices = new Complex[N];log2N = (int)(Mathf.Log(N) / Mathf.Log(2));//Debug.Log(log2N);reversed = new uint[N];for (int i = 0; i < N; i++){reversed[i] = Reverse((uint)i);//Debug.Log(reversed[i]);}int pow2 = 1;T = new ComplexList[log2N];for (int i = 0; i < log2N; i++){T[i].complices = new Complex[pow2];for (int j = 0; j < pow2; j++){T[i].complices[j] = ComputeT(j, pow2 * 2);}pow2 *= 2;}which = 0;}//bitreverse算法public uint Reverse(uint i){uint res = 0;for(int j=0;j<log2N;j++){res = (res << 1) + (i & 1);i >>= 1;}return res;}public Complex ComputeT(int x, int n){//Debug.Log(Mathf.Cos((float)pi2 * x / n));return new Complex(Mathf.Cos(pi2 * x / n), Mathf.Sin(pi2 * x / n));}public void ComputeFFT(Complex[] input, ref Complex[] output,int stride,int offset){for (int i = 0; i < N; i++) {c[which].complices[i] = input[reversed[i] * stride + offset];}int loops = N >> 1;int size = 1 << 1;int sizeOver2 = 1;int w = 0;for (int i = 1; i <= log2N; i++){//0,1which ^= 1;for (int j = 0; j < loops; j++){for (int k = 0; k < sizeOver2; k++){c[which].complices[size * j + k] = c[which ^ 1].complices[size * j + k] +c[which ^ 1].complices[size * j + sizeOver2 + k] * T[w].complices[k];}for (int k = sizeOver2; k < size; k++){c[which].complices[size * j + k] = c[which ^ 1].complices[size * j - sizeOver2 + k] -c[which ^ 1].complices[size * j + k] * T[w].complices[k - sizeOver2];}}loops >>= 1;size <<= 1;sizeOver2 <<= 1;w++;}for (int i = 0; i < N; i++){output[i * stride + offset] = c[which].complices[i];}}
}

using UnityEngine;public class OceanManager : MonoBehaviour
{public int N = 32;public float WaveHeight=1f;                                public Vector2 Wind=new Vector2(1f,1f);public float Length=1f;Ocean _ocean;Mesh _mesh;void Start(){_ocean = new Ocean(N, WaveHeight, Wind, Length);_mesh = GetComponent<MeshFilter>().mesh;}private void FixedUpdate(){//Debug.Log(Time.time);_ocean.EvaluateWaves(Time.time);_mesh.vertices = _ocean.oceanData.vertices;_mesh.triangles = _ocean.indices.ToArray();_mesh.uv = _ocean.oceanData.uvs;_mesh.normals = _ocean.oceanData.normals;_mesh.uv2 = _ocean.oceanData.uvs2;//_mesh.RecalculateNormals();}
}

Shader用的是最后两篇的代码，加了一个深度渐变

Shader "Learn/GPUWater"
{Properties{_OceanColor ("Ocean Color",COLOR) = (1,1,1,1)_SpecualrColor ("Specular Color",COLOR) = (1,1,1,1)[HDR]_BubbleColor ("Bubble Color",COLOR) = (1,1,1,1)_MainTex ("Texture", 2D) = "white" {}_Gloss("Gloss", Range(8,200)) = 1_DepthVisibility ("_Depth Visibility", Float) = 1}SubShader{Tags { "RenderType"="Transparent" "Queue"="Transparent" "IgnoreProjector"="True"}LOD 100Pass{Blend SrcAlpha OneMinusSrcAlpha//Cull OffCGPROGRAM#pragma vertex vert#pragma fragment frag#include"UnityCG.cginc"#include"Lighting.cginc"struct appdata{float4 vertex : POSITION;float2 uv : TEXCOORD0;};struct v2f{float2 uv : TEXCOORD0;float4 vertex : SV_POSITION;float3 worldPos:TEXCOORD1;float4 screenPos:TEXCOORD2;};sampler2D _MainTex;sampler2D _Displace;sampler2D _Normal;float4 _MainTex_ST;fixed4 _BubbleColor;fixed4 _OceanColor;fixed4 _SpecualrColor;half _Gloss;half _DepthVisibility;sampler2D _CameraDepthTexture;v2f vert(appdata v){v2f o;o.uv = TRANSFORM_TEX(v.uv, _MainTex);float4 dispalce=tex2Dlod(_Displace,float4(o.uv,0,0));v.vertex+=float4(dispalce.xyz,0);o.worldPos=mul(unity_ObjectToWorld,v.vertex).xyz;o.vertex = UnityWorldToClipPos(o.worldPos);o.screenPos=ComputeScreenPos(o.vertex);COMPUTE_EYEDEPTH(o.screenPos.z);return o;}fixed4 frag(v2f i) : SV_Target{float3 lightDir=normalize(UnityWorldSpaceLightDir(i.worldPos));float3 viewDir=normalize(UnityWorldSpaceViewDir(i.worldPos));float3 halfDir=normalize(lightDir+viewDir);float4 normalAndBubbles=tex2D(_Normal,i.uv);float3 normal=normalize(UnityObjectToWorldNormal(normalAndBubbles.xyz).rgb);float depth=SAMPLE_DEPTH_TEXTURE_PROJ(_CameraDepthTexture, UNITY_PROJ_COORD(i.screenPos));float sceneZ=max(0,LinearEyeDepth(depth)- _ProjectionParams.g);float partZ=max(0,i.screenPos.z- _ProjectionParams.g);float depthGap = sceneZ - partZ;float x=clamp(depthGap/_DepthVisibility,0,1);half NdotL=saturate(dot(lightDir,normal));fixed3 diffuse = tex2D(_MainTex, i.uv).rgb*NdotL*_OceanColor.rgb;fixed3 bubbles=_BubbleColor.rgb*saturate(dot(lightDir,normal));diffuse=lerp(diffuse,bubbles,normalAndBubbles.a);fixed3 specular=_LightColor0.rgb*_SpecualrColor.rgb*pow(max(0,dot(normal,halfDir)),_Gloss);fixed3 col=diffuse+specular;return fixed4(col,_OceanColor.a*x);}ENDCG}}
}

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