# Rain Again

The task is simple – add some kind of detail (PBR) in the rain. So this we have from the start. Add a special option for materials to get wet (wood, asphalt and other rough surfaces become darker when it’s raining) Change the ground shader, underfoot the same bare ground. Slate in the rain becomes wet and darker but obviously  not shine as a metal.

float4 position_with_offset = float4( view_space_position + normal * 1.3f, 1.0f );
float4 projected_position = mul( view_to_shadow, position_with_offset );
float roughness = gb.roughness;
float fresnel = gb.fresnel;
float metalness = saturate( ( dot( fresnel, 0.33 ) * 1000 - 500 ) );
float porosity = saturate( ( roughness - 0.5) / 0.1 );
float factor = lerp( 0.1, 1, metalness * porosity );
float3 to_eye = -view_space_position;
diffuse_result *= lerp ( 0.85, ( factor * 5 ), rain_mask ); Add some details and dof. Perhaps that’s it.   Looks better and this is well.

In-game result with tone mapping.

# Energy Conservation

Energy conservation is a restriction on the reflection model that requires that the total amount of reflected light cannot be more than the incoming light (0).

$$\int_{\Omega} f(\vec{x}, \phi, \theta) L_i \cos \theta \,\delta\omega\leq L_i$$

• f – BRDF(any?)

We have to calculate energy conservation for Diffuse and Specular and than combined model need  to fit this:

$$C_d C_s \leq 1$$

This means that if you want to make material with more specular, you may have to reduce the diffuse. Left: A sphere with a high diffuse intensity, and low specular intensity. Right: High specular intensity, low diffuse. Middle: Maximum diffuse AND specular intensity. Note how it looks blown out and too bright for the scene (1) The upper part of the table shows the nor­mal­ized reflec­tion den­sity func­tion (RDF). This is the prob­a­bil­ity den­sity that a pho­ton from the incom­ing direc­tion is reflected to the out­go­ing direc­tion, and is the BRDF times . Here, is the angle between and , which is, for the assumed view posi­tion, also the angle between and , resp. and .

The lower part of the table shows the nor­mal­ized nor­mal dis­tri­b­u­tion func­tion (NDF) for a micro-​facet model. This is the prob­a­bil­ity den­sity that the nor­mal of a micro-​facet is ori­ented towards . It is the same expres­sion in spher­i­cal coor­di­nates than that for of the Phong RDF, just over a dif­fer­ent vari­able, , the angle between and . The height­field dis­tri­b­u­tion does it slightly dif­fer­ent, it nor­mal­izes the pro­jected area of the micro-​facets to the area of the ground plane (adding yet another cosine term). (3)  With and without energy conservation /π (4)

Beware to overdo the result with minimal maximum.  # IBL

Image Based Lights (IBLs) allow to represent the incident lughtning surrounding a point. Often use Diffuse and Specular cubemaps. To use commom PBR idea BRDF shading we need to solve the the radiance integral. We will use importance sampling (0). The following equation describes this numerical integration:

$$\int_H L_i(l) f(l,v)cos\theta_l\mathrm{d}l \approx \frac{1}{N}\sum\limits_{k=1}^N\frac{L_i(l_k)f(l_k,v)cos\theta_{l_k}}{p(l_k,v)}$$ How Radiance traces rays to determine the incident illumination on a surface from an IBL environment (1)

The main problem is that even with importance sampling we still need to take a lot of samples! We can use MIP maps to reduce sample count, but counts still need to be greateer that 16 for approriate quality. As it based on blending between many environment maps per pixel for local reflections, we can only practically afford a single sample for each.

Unreal 4 Split Sum Approximation

$$\frac{1}{N}\sum\limits_{k=1}^N\frac{L_i(l_k)f(l_k,v)cos\theta_{l_k}}{p(l_k,v)} \approx {\left( \frac{1}{N}\sum\limits_{k=1}^N)L_i(l_k \right) }{\left( \frac{1}{N}\sum\limits_{k=1}^N\frac{f(l_k,v)cos\theta_{l_k}}{p(l_k,v)} \right) }$$

First sum pre-calculate for different roughness values and store the results in the mip-map levels of cubemap (typical approach). In Unreal4 minor difference is they convolve the environment map with the GGX distribution of their shading model using importance sampling.

Second sum includes Environment BRDF – we may keep the same as integrating specular BRDF with a $$L_i(l_k) = 1$$

In remember me (2) they also approximate by considering two parts

• The lighting is pre-integrated offline with the normal distribution function (NDF) of the BRDF and the cosine from Lambert’s law for different Specular Power values.
• The result is stored in a prefiltered mipmap radiance environment map (PMREM) which is a cubemap texture where each mipmap store the result of the pre-integration for a lower roughness (example selecting mip as a linear function of roughness-  texCUBElod(uv, float4(R, nMips – gloss * nMips))(3) )

Local cubemaps blending (3)

Let’s use Hammersley (4) to generate psuedo random for 2-D coordinates (this will give use the well spaced coord), the GGX importance sample function (5)  will gathers into a region thats going to contribute the most for the given roughness. HLSL Epic Karis :

float radicalInverse_VdC(uint bits)
{
bits = (bits << 16u) | (bits >> 16u);
bits = ((bits & 0x55555555u) << 1u) | ((bits & 0xAAAAAAAAu) >> 1u);
bits = ((bits & 0x33333333u) << 2u) | ((bits & 0xCCCCCCCCu) >> 2u);
bits = ((bits & 0x0F0F0F0Fu) << 4u) | ((bits & 0xF0F0F0F0u) >> 4u);
bits = ((bits & 0x00FF00FFu) << 8u) | ((bits & 0xFF00FF00u) >> 8u);

return float(bits) * 2.3283064365386963e-10; // / 0x100000000
}
float2 Hammersley(uint i, uint n)
{
}

ImportanceSampleGGX (6)

float3 ImportanceSampleGGX( float2 Xi, float Roughness, float3 N )
{

float a = pow(Roughness + 1, 2);
float Phi = 2 * PI * Xi.x;
float CosTheta = sqrt( (1 - Xi.y) / ( 1 + (a*a - 1) * Xi.y ) );
float SinTheta = sqrt( 1 - CosTheta * CosTheta );
float3 H = 0;
H.x = SinTheta * cos( Phi );
H.y = SinTheta * sin( Phi );
H.z = CosTheta;

float3 UpVector = abs(N.z) < 0.999 ? float3(0,0,1) : float3(1,0,0);
float3 TangentX = normalize( cross( UpVector, N ) );
float3 TangentY = cross( N, TangentX );
// Tangent to world space
return TangentX * H.x + TangentY * H.y + N * H.z;
}

We can use and need to try couple aproxx.

// Smith aproxx
float GGX(float nDotV, float a)
{
float aa = a*a;
float oneMinusAa = 1 - aa;
float nDotV2 = 2 * nDotV;
float root = aa + oneMinusAa * nDotV * nDotV;
return nDotV2 / (nDotV + sqrt(root));
}

// shlick aproxx
float GGX(float NdotV, float a)
{
float k = (a * a) / 2;
return NdotV / (NdotV * (1.0f - k) + k);
}

float G_Smith(float a, float nDotV, float nDotL)
{

return GGX(nDotL,a) * GGX(nDotV,a);
}

The finished function

float2 IntegrateBRDF( float Roughness, float NoV )
{
float3 V;
V.x = sqrt( 1.0f - NoV * NoV ); // sin
V.y = 0;
V.z = NoV;
// cos
float A = 0;
float B = 0;
const uint NumSamples = 128;
[loop]
for( uint i = 0; i < NumSamples; i++ )  {  	float2 Xi = Hammersley( i, NumSamples ); 	float3 H = ImportanceSampleGGX( Xi, Roughness, float3(0,0,1) ); 	float3 L = 2 * dot( V, H ) * H - V; 	float NoL = saturate( L.z ); 	float NoH = saturate( H.z ); 	float VoH = saturate( dot( V, H ) ); [branch] 	if( NoL > 0 )
{
float G = G_Smith( Roughness, NoV, NoL );
float G_Vis = G * VoH / (NoH * NoV);
float Fc = pow( 1 - VoH, 5 );
A += (1 - Fc) * G_Vis;
B += Fc * G_Vis;
}
}
return float2( A, B ) / NumSamples;
}

Example of function call

float env_brdf = IntegrateBRDF(roughness, NoV);

Pre-calculate the first sum for different roughness values and store the results in the mip-map
levels of a cubemap.

float3 PrefilterEnvMap( float Roughness, float3 R )
{
float3 N = R;
float3 V = R;
float3 PrefilteredColor = 0;
const uint NumSamples = 128;

for( uint i = 0; i < NumSamples; i++ )  { 	float2 Xi = Hammersley( i, NumSamples ); 	float3 H = ImportanceSampleGGX( Xi, Roughness, N ); 	float3 L = 2 * dot( V, H ) * H - V; 	float NoL = saturate( dot( N, L ) );   if( NoL > 0 )
{
PrefilteredColor += probe_decode_hdr( t_probe_cubemap.SampleLevel( s_base, vec, mip_index ) ).rgb * NoL;
TotalWeight += NoL;
}
}
return PrefilteredColor / TotalWeight;
} Final function:

float3 ApproximateSpecularIBL( float3 SpecularColor , float Roughness, float3 N, float3 V )
{
float NoV = saturate( dot( N, V ) );
float3 R = 2 * dot( V, N ) * N - V;
float3 PrefilteredColor = PrefilterEnvMap( Roughness, R );
float2 envBRDF = IntegrateBRDF( Roughness, NoV );
return PrefilteredColor * ( SpecularColor * envBRDF.x + envBRDF.y );
}

# BRDF

Common Cook-Torrance microfacet specular BRDF (0)

$$f(l, v) = \frac{D(h) F(v, h) G(l, v, h)}{4(n\cdot l)(n\cdot v)}$$

• l – light direction
• v – view direction
• f (l,v)– brdf
• D(h) – normal distribution function
• F(v,h) – fresnel function
• G(l,v,h) – geometric shadowing term

For D I like Trowbridge-Reitz GGX

$$D_{GGX}(v) = \frac{\alpha^2}{\pi((n \cdot m)^2 (\alpha^2 – 1) + 1)^2}$$

$$\alpha = roughness^2$$ Unreal alpha, matches Burley’s remapping(1) $$\alpha = (1 – smoothness * 0.7)^6$$ Ryse alpha, matches (2) $$(1-smoothness)^4$$ For G best Smith-Schlick

$$G(l, v, h) = G_{1}(l) G_{1}(v)$$

$$G_{Schlick}(v) = \frac{n\cdot v}{(n\cdot v)(1 – k) + k }$$

$$G_{Schlick}(l) = \frac{n\cdot l}{(n\cdot l)(1 – k) + k }$$

$$k = \frac{(0.8 + 0.5*\alpha)^2}{2}$$

Crytek Ryse – avoid highlights getting too hot on smooth surfaces and reduce gain on rough materials at grazing angles

$$k = 0.5*\alpha \text{ – fit for me}$$

Fresnel function Schlick (3)

$$F_{Schlick}(\mathbf{v}, \mathbf{h}) = F_0 + (1 – F_0) ( 1 – (v \cdot h) )^5$$ # Tone mapping – Perceived brightness

Simplest function that modify luminance of the color: (0)

$$T(color) = \frac{color}{ 1 + \frac{luma}{range} }$$

• T – tone mapping function
• color – be tone mapped
• luma – luminance of color
• range -range to tone map into.
• inverse  would be 1 –

Luma – perceived brightness

Luminance standard – 0.2126*R + 0.7152*G + 0.0722*B (1)
Luminance po var 1 – 0.299*R + 0.587*G + 0.114*B (2)
Luminance po var 2 – sqrt( 0.299*R^2 + 0.587*G^2 + 0.1*B^2 ) : SLOW (3)

Approximation:

$$Y = \frac{(R + R + B + G + G + G)}{6}$$

$$Y = (R + R + R + B + G + G + G +G) >> 3$$

Reduce fireflies:

$$weight(s) = \frac{1}{ 1 + luma }$$

• s – samples

Average:

$$F(average)=\frac{sample * x}{weight(s)x}$$

• x – number of samples

Expensive function, better results –  more than color range linear best for us.

$$T(color) = \left\{ \begin{array}{l l} color & \quad \text{if luma \leq a}\\ \frac{color}{luma} \left( \frac{ a^2 – b*luma }{ 2a – b – luma } \right) & \quad \text{if luma \gt a} \end{array} \right.$$

$$T_{inverse}(color) = \left\{ \begin{array}{l l} color & \quad \text{if luma \leq a}\\ \frac{color}{luma} \left( \frac{ a^2 – ( 2a – b )luma }{ b – luma } \right) & \quad \text{if luma \gt a} \end{array} \right.$$

• 0 to a – linear
• a to b –  tone mapped
• if a = 0, b=range – will be the same as first two functions

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Game Designer, Scripter, Product Manager, Animator

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