Cardinal Peak

The Basics of 3D Image Acquisition

One of our clients is heavily involved in 3D video and has been for several years. However, several are just now starting to think about it because of the uptick of interest in the consumer electronics world. Enough questions have been posed to us recently that it seemed worthwhile to me to pull together a few basic facts regarding 3D stereopair imaging and stereo disparity.

First, we need a simple model of a lens. Consider the diagram below:

In this picture, the long horizontal line that passes through the center of the lens is called the lens axis. The lens has the property that rays that pass through the center of the lens are undeviated. Therefore, the ray from the top of the tree, at a distance l to the left of the lens, passes straight through the center of the lens. (The tree has a height of h.) The lens also has the property that rays that arrive perpendicular to the lens are refracted to pass through the focal point of the lens. The focal point lies on the lens axis and is a distance f from the center of the lens. The intersection of these two rays shows where the image of the tree will be formed. You can see that the image of the tree is upside down, and has a new height h’. The image is formed a distance d to the right of the focal point.

By using similar triangles we see first that

Using a different pair of similar triangles we also see that

Solving the first equation above for h’, substituting the result into the second equation and simplifying, we derive the following relationship:

This is the fundamental equation of a simple lens. It shows that as the object gets further and further from the lens, i.e. as l increases, the distance of the image of the object from the focal plane decreases, i.e. d gets smaller. We can assume that the camera’s image sensor is located at a distance f from the lens, is perpendicular to the lens axis, and that all objects more than a certain distance away from the lens will be in focus. In other words, the image of all sufficiently distant objects will appear on the focal plane where the image sensor is located.

In the case of 3D video, two cameras are used to acquire a sequence of stereopair images, one from the left camera and one from the right. Different stereo geometries are possible, but the most common one is to place the two cameras horizontally apart from each other by a distance i, and to keep their focal planes coplanar. The diagram below illustrates this configuration:

The horizontal line at the bottom is the focal plane; it is clear from the diagram that the focal planes are coplanar. The lenses are a distance f from the focal plane and are separated by a distance of i from each other. We assume that a small object (or a point on a larger object) is located a distance l from the lens plane and a distance m to the right of the axis of the right lens. We want to know where the image of that object appears in the left and the right camera. In particular, we want to know if we overlaid the left image on top of the right image, how far apart would the images appear? Mathematically, we want to know the disparity, which we define to be

where s1 and s2 are the distances from the image point to the intersection of the lens axis with the focal plane for the left and the right cameras respectively. Note that we are assuming that the object being imaged is far enough away that its image forms on the focal plane.

Using our favorite trick of similar triangles we have the following two equations:

and

Solving the first equation for s1, the second equation for s2, taking the difference and simplifying yields

Although this expression was derived for an object to the right of the axis of the right camera, it is easy to show in a similar manner that it is also true for an object between the axes of the two cameras as well as for an object to the left of the axis of the left camera.

So what does this equation tell us? First, it says that for this particular camera geometry, the disparity is only a function of the separation between the two cameras, i, and the distance of the object from the lens plane, l. Second the equation tells us that the disparity increases as we increase the separation between the cameras. Finally, it tells us that the disparity decreases as the object gets further away from the cameras, approaching zero for objects an infinite distance away. (You can see this when you watch 3D content without wearing the special 3D glasses: The “distant” objects can be seen by the naked eye, whereas the near objects appear blurry to the naked eye, because the value of ρ is greater.)

It should be clear from this equation that if a stereopair is available, and corresponding points can be found in the left and right pictures, that the disparity between those points can be measured, and the distance to the point can be computed.

Mike Perkins, Ph.D., is a managing partner of Cardinal Peak and an expert in algorithm development for video and signal processing applications.

Thoughts on 3D after NAB

I just returned from this year’s NAB show, where I was bombarded with 3D demos in virtually every booth. Most of the factors driving this 3D superabundance originate outside of the broadcast industry itself. First, TV manufacturers are hot on 3D as a way to get everyone who just bought an HDTV to upgrade to a new 3D enabled display. Cinema owners like 3D because they can charge more for the tickets. The Blu-ray consortium recently standardized a method for storing 3D video on BD discs, and hopes to enable and piggyback on the efforts of the TV manufacturers as a way to drive sales of 3D Blu-ray players and finally displace DVDs. Hollywood has begun producing more 3D movies too, including the phenomenally successful Avatar (which will be released on 3D Blu-ray very soon). So naturally the broadcast industry needs to be prepared to author and carry 3D signals.

Most of the demos I saw were nothing more than “here, put on these glasses”…in other words, “me too” type demos. (Although after seeing all this 3D interest I did wonder if Philips regretted terminating their auto stereographic display effort last year!).

Nevertheless, I did see two problems addressed that I found technically interesting. First, real-time 2D to 3D conversion (see here and here), and second, automatic 3D quality monitoring.

I worked a lot on the problem of compressing stereopairs as part of my Ph.D. research, and I also spent time thinking about 3D video quality assessment. However, I had never considered the problem of real-time 2D to 3D conversion, so the show got me thinking. It’s a pretty tricky problem!

Converting a 2D video stream to 3D can be partitioned into two fundamental steps. First, creating a depth map for each video image, and second, using the depth map to construct a second viewpoint. Although both steps are challenging, the first step feels substantially harder to me.

With regards to the first step, a sequence of 2D video images must be analyzed to extract a depth map. Several special cases are worth discussing, but I’ll only mention two. First, consider the case where the camera is stationary and a 3D object moves through the field of view. The closer points on that object will have frame-to-frame pixel displacements that are larger than those for object points that are further away. Therefore, one useful approach for deriving information for a depth map would be the following: a) segment the image into two regions: moving and stationary; b) segment the moving areas into distinct objects using various clues such as color and proximity; c) find distinct matching points on the moving objects in two different frames; d) determine depths for those matching points based on the measured point displacements; e) interpolate the depth map for non-matched moving object points.

As a second special case, consider the situation where nothing is moving in the video sequence for many frames in a row. In this case, occlusion becomes a major depth cue. If one object is in front of another, then it will occlude the background object, and it must be closer. If an image can be segmented into objects, and an occlusion map can be deduced, then different depths can be assigned to different objects based on where they lie in the occlusion map. Other clues that may be algorithmically exploitable could stem from perspective considerations applied to the edges of identified objects.

Many powerful depth clues will be hard to take advantage of algorithmically—although humans can exploit them easily—because they involve recognizing objects. For example, we can easily recognize two humans in a picture, and determine whether or not they are adults or children. We know that if two adult males appear in the picture, and one appears substantially taller than the other (and isn’t holding a basketball), then the shorter one is further away. I suspect that taking advantage of this sort of knowledge is beyond the capability of today’s real-time (and non real-time!) processing. Nevertheless, I was amazed at how well the systems I saw at the show appeared to work.

With regards to the second step, given an image and a depth map, a second view can be created from the first by displacing each pixel of the original image with a disparity value corresponding to its depth. In practice it won’t be that easy. Why? Because after displacing pixels with their appropriate disparities, gaps will appear in the new image. These gaps result from image detail that is visible in one image but not in the other, so the gaps will need to be interpolated or otherwise synthesized in some reasonable way.

With regards to 3D video quality assessment, I just want to interject a note of caution. I was encouraged to see that several vendors have made progress in developing systems that automatically approximate the “mean opinion scores” that subjective human evaluation tests would assign to various image sequences. However, when dealing with 3D video, the sum is greater than the parts. If the algorithmic approach implemented is to naively apply 2D image quality assessment to the left and right pictures independently, and then average the scores together, the result is likely to not correspond at all to a human’s subjective viewing experience. For those of you who wear glasses, like me, you can experience this directly if one of your eyes is better than the other. Take off our glasses, look at the world around you, and you will see it with the resolution of your better eye; but you will still have stereoscopic vision. This effect will ultimately need to be taken into account in automatic systems that purport to algorithmically assess the quality of a 3D image sequence.