# Assignment 3 Solution

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## Description

Fill in the function estimate_direct_lighting in pathtracer.cpp.

This function is called by trace_ray in order to get an estimate of the direct lighting at a point after a ray hit. At a high level, it should sum over every light source in the scene, taking samples from the surface of each light, computing the incoming radiance from those sampled directions, then converting those to outgoing radiance using the BSDF at the surface.

The starter code at the top of the function uses the input intersection normal to calculate a local coordinate space for the

object hit point. In this frame of reference, the normal to the surface is (0, 0, 1) (so z is “up” and xy are two other arbitrary perpendicular directions). We compute this transform because the BSDF functions you will define later expect the normal to

be (0, 0, 1).

Additionally, in this coordinate system the cosine of the angle between a direction vector w_in and the normal vector Vector3D(0,0,1) is simply w_in.z . (This is clear if you remember that the cosine of the angle between two unit vectors

is equal to their dot product.)

We store the transformation from local object space to world space in the matrix o2w. This matrix is orthonormal, thus o2w.T() is both its transpose and its inverse — we store this world to object transform as w2o.

Matrix3x3 o2w;

make_coord_space(o2w, isect.n);

Matrix3x3 w2o = o2w.T();

For your later convenience, we store a copy of the hit point in its own variable:

const Vector3D& hit_p = r.o + r.d * isect.t;

Finally, we calculate the outgoing direction w_out in the local object frame. Remember that this should be opposite to the direction that the ray was traveling.

const Vector3D& w_out = w2o * (-r.d);

You need to implement the following steps:

1. Loop over every SceneLightin the PathTracer‘s vector of light pointers scene->lights .

1. For each light, check if it is a delta light. If so, you only need to ask the light for 1 sample (since all samples would be the same). Else you should ask for ns_area_lightsamples in a loop.
2. To get each sample, call the SceneLight::sample_L() This function requests the ray hit point

hit_p and returns both the incoming radiance as a Spectrum as well as 3 values by pointer. The values returned by pointer are

a probabilistically sampled

wi unit vector giving the direction from

hit_p

to the light source,

a distToLight

float giving the distance from

hit_p

to the light source, and

a pdf float giving the probability density function evaluated at the returned wi direction.

The wi direction returned by compute it in object space bysample_Lis in world space. In order to pass it to the BSDF, you need to calculating Vector3D w_in = w2o * wi .

1. Check if the zcoordinate of w_in is negative — if so, you can continue the loop since you know the sampled light point lies behind the surface.

 6. Cast a shadow ray from the intersection point towards the light, testing against the  bvh to see if it intersects the scene anywhere. Two subtleties: You should set the ray’s max_tto be the distance to the light, since we don’t care about intersections behind the light source. You should offset the origin of the ray from the hit point by the tiny global constant  EPS_D(i.e., add EPS_D * wi to hit_p  to get the shadow ray’s origin). If you don’t do this, the ray will frequently intersect  the same triangle at the same spot again because of floating point imprecision. 7. If the ray does not intersect the scene, you can calculate the BSDF value at  w_out and w_in. Accumulate the result (with the correct multiplicative factors) inside  L_out.

Factors to remember when summing up the radiance values in  L_out:

sample_L returns an incoming radiance. To properly multiply this by the BSDF, you need to convert it to irradiance using a cosine term.

You are summing up some number of samples per light. Make sure you average them correctly.

The sample_Lroutine samples from a light. It does this using some probability distribution which can bias the sample towards certain parts of the light — you need to account for this fact by using the returned pdf value when summing radiance values.

Now you should be able to render nice direct lighting effects such as area light shadows and ambient occlusion, albeit without full global illumination. However, you will only be able to render files with Lambertian diffuse BSDFs, as we have not yet implemented any of the other BSDFs.

At this point, here are the results of the commands

./pathtracer -t 8 -s 64 -l 32 -m 6 ../dae/sky/dragon.dae

And

./pathtracer -t 8 -s 64 -l 32 -m 6 ../dae/sky/CBbunny.dae

In this task, you need to fill in the function estimate_indirect_lighting in pathtracer.cpp.

This function is called by trace_ray in order to get an estimate of the indirect lighting at a point after a ray hit. At a high level, it should take one sample from the BSDF at the hit point and recursively trace a ray in that sample direction.

We provide with the same setup code as in the direct lighting function:

Matrix3x3 o2w;

make_coord_space(o2w, isect.n);

Matrix3x3 w2o = o2w.T();

Vector3D hit_p = r.o + r.d * isect.t;

Vector3D w_out = w2o * (-r.d);

As in part 3, w2o transforms world space vectors into a local coordinate frame where the normal is (0, 0, 1), the unit vector in the z direction. Thus w_out is the outgoing radiance direction in this local frame.

You need to implement the following steps:

1. Take a sample from the surface BSDF at the intersection point using bsdf->sample_f(). This function requests the outgoing radiance direction w_out and returns the BSDF value as a Spectrum as well as 2 values by pointer. The values returned by pointer are the probabilistically

sampled w_in unit vector giving the incoming radiance direction (note that unlike the direction returned by sample_L, this w_in vector is in the object coordinate frame!) and a pdf float giving the probability

density function evaluated at the return w_in direction.

1. Use Russian roulette to decide whether to randomly terminate the ray. We recommend basing this on the reflectance of the BSDF Spectrum, calculated with the spectrum’s illum()method — the lower the reflectance, the more likely you should be to terminate the ray and return an empty Spectrum.

You can use the coin_flip method from random_util.h to generate randomness. Remember that

you need to divide by one minus the Russian roulette probability (i.e., the probability of not terminating) when you return a radiance value. Note: you may want to multiply illum() by some

large constant (>10) and then add it by some constant (e.g. 0.05) to make sure that rays are not terminated too early, resulting in high noise.

1. If not terminating, create a ray that starts from EPS_Daway from the hit point and travels in the direction of o2w * w_in (the incoming radiance direction converted to world coordinates). This ray should have depth depth – 1 . (Note that for the ray depth cutoff to work, you should initialize your camera rays to have depth max_ray_depth.)

1. Use trace_rayto recursively trace this ray, getting an approximation for its incoming radiance.

The includeLe parameter should be set to isect.bsdf->is_delta() since emission was

not included in the  direct lighting calculation for delta BSDFs.

1. Convert this incoming radiance into an outgoing radiance estimator by scaling by the BSDF and a cosine factor and dividing by the BSDF pdfand one minus the Russian roulette termination probability.

You still can only render diffuse BSDFs until completing Part 5, however, you should now see some nice color bleeding in Lambertian scenes. You should be able to correctly render images such as

./pathtracer -t 8 -s 64 -l 16 -m 5 ../dae/sky/CBspheres_lambertian.dae

Reflection

Implement the BSDF::reflect() function.

We recommend taking advantage of the object coordinate space that BSDF calculations occur in. This means that the origin is the intersection point and the z axis lies along the normal vector. In this situation, reflection should be a trivial one-line transformation of the x, y, and z coordinates.

Refraction

To see pretty glass objects in your scenes, you’ll first need to implement the helper function BSDF::refract() that takes a wo direction and returns the wi direction that results from refraction.

As in the reflection section, we can take advantage of the fact that our BSDF calculations always take place in a canonical “object coordinate frame” where the z axis points along the surface normal. This allows us to take advantage of the spherical coordinate Snell’s equations. In other words, our wo vector

starts out with coordinates:

ωo . x = sin θ cos ϕ, ωo . y = sin θ sin ϕ, ωo . z = ± cos θ.

Note: we put a ± sign on the z coordinate because when wostarts out inside the object with index of refraction greater than 0, its z coordinate will be negative. The surface normal always points out into air.

When z is positive, we say that we are entering the non-air material, else we are exiting. This is depicted in the figure.

For the transmitted ray, ϕ = ϕ + π, so cos ϕ = − cos ϕ and sin ϕ = − sin ϕ. As seen in the

figure, define η to be the ratio of old index of refraction to new index of refraction. Then Snell’s law states thatθ). sin θ′ = η sin θ. This implies that cos θ = √1 − sin2 θ′ = √1 − η2 sin2 θ = √1 − η2(1 − cos2

In the case where 1 − η2(1 − cos2 θ) < 0, we have total internal reflection–in this case you return false and the wi field is unused.

Be careful when implementing this in code, since you are only provided with a single  ior value in the refract

 function: when entering, this is the new index of refraction of the material ω i is pointing to, else it is the old index of refraction of the material that o itself is inside. In other words, η=1/ ior when entering and η = ior when exiting. ω In spherical coordinates, our equations for ϕ′ and θ′   tell us that ωi . x = −ηωo . x,   ωi . y = −ηωo . y, , ωi . z =  √1 − η2(1 − ωo . z2 ), where we are indicating that ωi . z has the opposite sign of ωo . z.

Some results:

CBgems.dae

CBdragon.dae