Modified Phong BRDF #

Introduction #

Phong described in 1975 a lighting model that consist of a diffuse and a specular term [1]. Since the model does not have a physical basis Lafortune et al. proposed a modified version that is called the Modified Phong BRDF [2].

A reference image: Modified Phong BSDF in Mitsuba 0.5.0 #

A nice property that is shared by all physical based renderers is that they want to render the same physical correct image. To get a reference image for our expected result, we can pick a physical based renderer that supports the Modified Phong BRDF and create a reference image.

Mitsuba 0.5.0 supports the Modified Phong BRDF. Please note that newer versions of Mitsuba (e.g. 3.x) do not support the Modified Phong BRDF anymore, since this model is a bit outdated and replaced by more advanced models. Anyways, the Modified Phong BRDF is a good starting point to understand the basics of how to sample BRDFs.

In the Mitsuba 0.5.0 scene file a Modified Phong BRDF can be defined via:

<bsdf type="phong" id="modified_phong_brdf">
    <float name="exponent" value="30"/>
    <rgb name="diffuseReflectance" value="0.5"/>
    <rgb name="specularReflectance" value="0.2"/>
</bsdf>

When this material is applied to a sphere within a Cornell Box the output can look like this:

The whole scene description can be found here.

Bidirectional reflection distribution function (BRDF) #

The bidirectional reflection distribution function (BRDF) describes how a material reflects light. It is formally defined by [2]:

• \(x\) defines the position on the surface
• \(\omega_i\) (sometimes written as \(\Theta_i\) ) is the direction from which the light comes in defined in polar coordinates \((\varphi_i,\theta_i)\) , where \(\varphi_i\) and \(\theta_i\) describe the azimuth and evaluation angles. We do not need to define a radius value for this polar coordinate, since we are only interested in the direction.
• \(\omega_o\) (sometimes written as \(\Theta_o\) ) is the outgoing light direction defined in polar coordinates \((\varphi_o,\theta_o)\) , where \(\varphi_o\) and \(\theta_o\) describe the azimuth and evaluation angles. We do not need to define a radius value for this polar coordinate, since we are only interested in the direction.
• \(\cos{\theta_i}\) takes account of Lambert`s cosine law. The cosine between \(\omega_o\) and the surface normal at point \(x\)
• \(d\omega_i\) is a differential solid angle “around” \(\omega_i\)
• \(L_i(x,\omega_i)\) is the radiance incoming to point \(x\) along the direction \(\omega_i\) through \(d\omega_i\)
• \(dL_o(x,\omega_o)\) is the differential amount of reflected light towards \(\omega_o \)

There are materials that reflect light dependent on its wavelength \(\lambda\) . This wavelength dependency was ignored in the above definition. The above formula works only for a one specific wavelength and not a light spectrum. Let’s assume we consider only radiation with a single constant frequency from the visible spectrum, i. e. monochromatic light.

Helmholtz reciprocity #

According to the Helmholtz reciprocity the incoming and outgoing direction of light can be interchanged:

https://github.com/JiayinCao/SORT/blob/6769f98f7125cdc34b2a389b99f7790dfebe6b23/src/unittests/bxdf.cpp#L41C1-L56C2

// A physically based BRDF should obey the rule of reciprocity
void checkReciprocity(const Bxdf* bxdf) {
    spinlock_mutex mutex;
    ParrallRun<8,128>( [&]() {
        const Vector wi = UniformSampleSphere(sort_rand_float(), sort_rand_float());
        const Vector wo = UniformSampleSphere(sort_rand_float(), sort_rand_float());

        const auto f0 = bxdf->F(wo, wi) * absCosTheta(wo);
        const auto f1 = bxdf->F(wi, wo) * absCosTheta(wi);

        std::lock_guard<spinlock_mutex> lock(mutex);
        ASSERT_NEAR(f0.r, f1.r, 0.001f);
        ASSERT_NEAR(f0.g, f1.g, 0.001f);
        ASSERT_NEAR(f0.b, f1.b, 0.001f);
    });
}

Definition #

The Modified Phong BRDF consist of an diffuse ( \(f_{r,d}\) ) and a specular ( \(f_{r,s}\) ) part:

where

• \(\alpha\) defines the angle between the perfect specular direction and the outgoing direction \(\omega_o\)
• \(k_d\) defines the reflection coefficient for the diffuse part of the BRDF
• \(k_s\) defines the reflection coefficient for the specular part of the BRDF
• \(n\) is a power exponent for the specular part. Higher values make the make specular reflection part sharper

To preserve energy conservation \(k_d + k_s \le 1\) must hold.

Implementation of a Modified Phong BRDF #

For a Modified Phong BRDF we need to implement the BSDF interface.

class ModifiedPhongBRDF : public BSDF {
    Color3f sample(BSDFSample& sample, const Point2f& sample_point);
    Scalar pdf(const BSDFSample& sample);
    Color3f evaluate(const BSDFSample& sample);
};

Evaluate #

Lets first focus on the implementation on the evaluate method. How would a simple test look like?

Given

Now lets compute \(f_r\) :

A unit test for this can look like this:

TEST(Phong, evaluate) {
    PropertySet ps;
    ps.add_property("diffuseReflectance", Color3f{.5f});
    ps.add_property("specularReflectance", Color3f{.5f});
    ps.add_property("exponent", 10.f);

    Phong phong{ps};

    BSDFSample3f sample;
    sample.wo = Vector3f{0.f, 0.f, 1.f};
    sample.wi = Vector3f{0.f, 0.f, 1.f};

    Color3f result = phong.evaluate(sample);

    float abs_error = 0.001f;
    EXPECT_THAT(result.red(), testing::FloatNear( 7.f / (2.f * pi_v<float>), abs_error));
    EXPECT_THAT(result.green(), testing::FloatNear( 7.f / (2.f * pi_v<float>), abs_error));
    EXPECT_THAT(result.blue(), testing::FloatNear( 7.f / (2.f * pi_v<float>), abs_error));
}

No lets think about other test cases:

PDF #

Calculus #

Speed is the derivation of time after location:

In the following diagram the “blue” line shows a 1D motion. Depending on the time \(t\) the position \(x\) changes:

TODO: FIX THIS GRAPH - note this is WIP and not correct

The red line shows the corresponding velocity for each point in time.

The average speed \(\bar{v}\) can be computed by

Partial differentiation #

References #

• https://computergraphics.stackexchange.com/questions/5502/phong-brdf-in-mitsuba-tungsten-and-agi
• https://www.youtube.com/@VrKomarov/videos
• https://www.youtube.com/watch?v=aolI_Rz0ZqY&t=1762s
• https://www.youtube.com/watch?v=XZ3w_jec1v8
• https://github.com/cs440-epfl
• https://github.com/voithos/muon/blob/main/muon/materials.cc
• https://github.com/ange-yaghi/manta-ray/blob/master/src/dielectric_media_interface.cpp
• https://github.com/JiayinCao/SORT/blob/master/src/unittests/bxdf.cpp
• https://www.open-std.org/JTC1/SC22/WG21/docs/papers/2023/p2996r1.html
• https://ashvardanian.com/posts/google-benchmark/
• https://www.youtube.com/watch?v=zduSFxRajkE Let’s build the GPT Tokenizer

1. B. Phong, Illumination for computer generated pictures, Seminal graphics: pioneering efforts that shaped the field, 1975. [Online]. Available: https://api.semanticscholar.org/CorpusID:1439868
2. E. Lafortune and Y. Willems, Using the modified Phong reflectance model for physically based rendering. Celestijnenlaan 200A, 3001 Heverlee, Belgium: Departement Computerwetenschappen, KU Leuven, 1994. [Online]. Available: https://www.cs.princeton.edu/courses/archive/fall03/cs526/papers/lafortune94.pdf