I fixed a problem with the EXPECT_THROW macro (I had uploaded the wrong file). Also, I added the ability to stub out results for function arguments passed by pointer. I would recommend downloading the zip file to take a look since there were a few files that had to change this time. I added one new example to show the new functionality.

Recently at work I was chatting with a colleague, and the topic of energy conservation for specular reflections came up. This reminded me that I’ve been sitting on a blog post for a while about just this subject, so I thought it was time to finish it.

First of all, I’d like to start by looking at the standard diffuse reflection model. In games, the typical formula for calculating diffuse reflection from a particular light is:

Where Cd is the diffuse material color, Li is the light color, N is the normal, and L is the normalized direction to the light. What’s the problem with this? Well, it’s not energy conserving. In itself, this isn’t really a problem since we don’t calculate multiple bounces of light in games, so we’re not adding energy to the scene as light bounces around like would happen in a ray tracer. It’s a good starting point for discussion though.
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In my previous post, I wrote very briefly about an  important improvement to the irradiance caching algorithm - irradiance gradients - and I’m going to expand on rotational gradients this time.

Gradients

The gradient of a function represents both the direction and rate of change of that function as the inputs vary. For a one dimensional function this is simply the derivative of the function. As you move into higher dimensions, you need to consider which coordinate system the inputs for the function are specified in, as this will change how you need to calculate the gradient.

For now, I’m just going to focus on calculating the gradient of a function defined using normalized spherical coordinates. Unfortunately, there’s no real standard way to define spherical coordinates, and despite similar looking symbols, the values are often interchanged. I’m going to define the spherical coordinates on the unit sphere as azimuthal value φ [0, π), and polar value θ [0, 2π).

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Solving the rendering equation with even just one bounce of indirect lighting can take a long time. The majority of time spent rendering a frame is in estimating the lighting integral. For example, rendering a single bounce of indirect lighting at 720p resolution with 256 sample rays for a Monte Carlo estimator requires about 237 million rays to be cast. This doesn’t even include the rays needed for sampling the lights for direct lighting, so in practice, the total will be even higher.

One interesting observation made by Greg Ward in his Siggraph ‘88 paper is that contrary to direct lighting, where shadows and lights can cause harsh changes, the indirect lighting on a surface tends to vary relatively slowly. One way to picture why this is, is to imagine the computing average color from the what you can see from each of your eyes. Even though each eye has a slightly different view on the world, the images they see are nearly similar, and so the average color is also nearly the same.

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I’ve made a small update to MockItNow to allow you to throw exceptions when replaying function calls. You basically record the function call as normal, and provide the exception object that you want to throw during the replay using the EXPECT_THROW macro. You can also make a function default to throwing an exception at registration time using REGISTER_THROW.

If you want to see a couple of examples of this feature, take a look at the bottom of the sample file here.

A couple of days ago, I stacked the random assortment of books I had lying on my desk at work, and for some reason I decided to take a photo. It’s a bit of a mixed bag of good and not-so-good books.

Every now and then I know I’m in trouble because I have to whip out the yellow maths book to remind myself of something that I have forgotten. I enjoy the challenge of reading the quaternions book, but it’s pretty hard to motivate myself to really get into that one.

My favourite at the moment is probably Real-Time Rendering (the latest edition which I have at home to be more specific), and my least favourite is probably Clean Code.

Clean Code seems to take the good ideas from other books (like C++ Coding Standards), and then adds some more which don’t make as much sense (I can’t think of a specific example off-hand). The customer reviews on Amazon.com are pretty good though. I definitely had an ‘uh-oh’ moment when I read the buzzword-laden full title - Clean Code: A Handbook Of Agile Software Craftmanship.

A couple of days ago, I compared the images my ambient occlusion integrator produced with those of Modo using similar settings. I noticed immediately how much ‘cleaner’ the render from Modo was. Clearly there was an issue with the way I was picking my samples, so I set about improving things.

My approach for generating the ambient occlusion rays was to generate uniform random samples over the hemisphere about the normal. Based on two random numbers in the range [0,1), I calculate the normalized sample direction using the following function:

Vector3 Sample::UniformSampleHemisphere(float u1, float u2)
{
	const float r = Sqrt(1.0f - u1 * u1);
	const float phi = 2 * kPi * u2;

	return Vector3(Cos(phi) * r, Sin(phi) * r, u1);
}

This generates points on a hemisphere from uniform variables u1 and u2, where each point has equal probability of being selected. The following image was generated with 256 random uniform samples:

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For the first time in about three years, I’ve had two weeks off work. I’ve spent a lot of time just relaxing and taking a break from things, but I’ve also been able to get back to doing some graphics work. Ever since Vivendi bought Activision, the project that I was leading has been “put on hold”, so I’ve been back on the game team. It’s not as fun for me, that’s for sure, but luckily, I have my code at home to play with, so all is not lost! With the holidays, I’ve found some motivation to get back to it.

What have I been doing? Well, as I was approaching the break, I read through the course notes from the Practical Global Illumination with Irradiance Caching course at Siggraph last year. I thought the course itself was really good, and very clearly presented. After blitzing through the notes again, I thought I’d have a go at writing a ray tracer. It seemed simple enough at the time, but like most things, the devil is in the details.

The first thing I did was to set up a really simple single-threaded ray tracer that just displayed the color of the surface it hit. This was fairly quick to get up and running once I had written a few supporting classes for the cameras and shapes. It’s not very glamorous, but it’s a start:

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This is just a quick note to say that I’ve updated MockItNow on Google Code to allow you to define storage types on a per-class basis using the DECLARE_STORAGE_TYPE macro. I did this so that the Mocker can deal with abstract class parameters. Please note that the macro must be declared at global scope because it uses partial template specialization.

I updated the download, and the source. You can see the new test at the end of the file here, and the only other affected file is Storage.h.

Thanks to Lance for pointing this problem out.