3d Reconstruction From 2d Images

[Glynis Waughtal]

IfI take a picture with a camera, so I know the distance from the camera to the object, such as a scale model of a house, I would like to turn this into a 3D model that I can maneuver around so I can comment on different parts of the house.

If I sit down and think about taking more than one picture, labeling direction, and distance, I should be able to figure out how to do this, but, I thought I would ask if someone has some paper that may help explain more.

Right now I am considering showing the house, then the user can put in some assistance for height, such as distance from the camera to the top of that part of the model, and given enough of this it would be possible to start calculating heights for the rest, especially if there is a top-down image, then pictures from angles on the four sides, to calculate relative heights.

As mentioned, the problem is very hard and is often also referred to as multi-view object reconstruction. It is usually approached by solving the stereo-view reconstruction problem for each pair of consecutive images.

Performing stereo reconstruction requires that pairs of images are taken that have a good amount of visible overlap of physical points. You need to find corresponding points such that you can then use triangulation to find the 3D co-ordinates of the points.

Stereo reconstruction is usually done by first calibrating your camera setup so you can rectify your images using the theory of epipolar geometry. This simplifies finding corresponding points as well as the final triangulation calculations.

you can calculate the fundamental and essential matrices using only matrix theory and use these to rectify your images. This requires some theory about co-ordinate projections with homogeneous co-ordinates and also knowledge of the pinhole camera model and camera matrix.

Finding corresponding points is the tricky part that requires you to look for points of the same brightness or colour, or to use texture patterns or some other features to identify the same points in pairs of images. Techniques for this either work locally by looking for a best match in a small region around each point, or globally by considering the image as a whole.

If you already have the fundamental matrix, it will allow you to rectify the images such that corresponding points in two images will be constrained to a line (in theory). This helps you to use faster local techniques.

For specific implementations you can use Google Scholar to search through the current literature. Here is one highly cited paper comparing various techniques: A Taxonomy and Evaluation of Dense Two-Frame Stereo Correspondence Algorithms.

This whole stereo reconstruction would then be repeated for each pair of consecutive images (implying that you need an order to the images or at least knowledge of which images have many overlapping points). For each pair you would calculate a different fundamental matrix.

Of course, due to noise or inaccuracies at each of these steps you might want to consider how to solve the problem in a more global manner. For instance, if you have a series of images that are taken around an object and form a loop, this provides extra constraints that can be used to improve the accuracy of earlier steps using something like bundle adjustment.

As you can see, both stereo and multi-view reconstruction are far from solved problems and are still actively researched. The less you want to do in an automated manner the more well-defined the problem becomes, but even in these cases quite a bit of theory is required to get started.

If it's within the constraints of what you want to do, I would recommend considering dedicated hardware sensors (such as the XBox's Kinect) instead of only using normal cameras. These sensors use structured light, time-of-flight or some other range imaging technique to generate a depth image which they can also combine with colour data from their own cameras. They practically solve the single-view reconstruction problem for you and often include libraries and tools for stitching/combining multiple views.

Research has made significant progress and these days it is possible to obtain pretty good-looking 3D shapes from 2D images. For instance, in our recent research work titled "Synthesizing 3D Shapes via Modeling Multi-View Depth Maps and Silhouettes With Deep Generative Networks" took a big step in solving the problem of obtaining 3D shapes from 2D images. In our work, we show that you can not only go from 2D to 3D directly and get a good, approximate 3D reconstruction but you can also learn a distribution of 3D shapes in an efficient manner and generate/synthesize 3D shapes. Below is an image of our work showing that we are able to do 3D reconstruction even from a single silhouette or depth map (on the left). The ground-truth 3D shapes are shown on the right.

The approach we took has some contributions related to cognitive science or the way the brain works: the model we built shares parameters for all shape categories instead of being specific to only one category. Also, it obtains consistent representations and takes the uncertainty of the input view into account when producing a 3D shape as output. Therefore, it is able to naturally give meaningful results even for very ambiguous inputs. If you look at the citation to our paper you can see even more progress just in terms of going from 2D images to 3D shapes.

I watched a lecture from one of the main researchers of the project (Roger Hubbold), and the image results are quite amazing! Although is a complex and long problem. It has a lot of tricky details to take into account to get an approximation of the 3d data, take for example the 3d information from wall surfaces, for which the heuristic to work is as follows: Take a photo with normal illumination of the scene, and then retake the picture in same position with full flash active, then subtract both images and divide the result by a pre-taken flash calibration image, apply a box filter to this new result and then post-process to estimate depth values, the whole process is explained in detail in this paper (which is also posted/referenced in the project website)

The essence of an image is a projection from a 3D scene onto a 2D plane, during which process the depth is lost. The 3D point corresponding to a specific image point is constrained to be on the line of sight. From a single image, it is impossible to determine which point on this line corresponds to the image point. If two images are available, then the position of a 3D point can be found as the intersection of the two projection rays. This process is referred to as triangulation. The key for this process is the relations between multiple views which convey the information that corresponding sets of points must contain some structure and that this structure is related to the poses and the calibration of the camera.

In recent decades, there is an important demand for 3D content for computer graphics, virtual reality and communication, triggering a change in emphasis for the requirements. Many existing systems for constructing 3D models are built around specialized hardware (e.g. stereo rigs) resulting in a high cost, which cannot satisfy the requirement of its new applications. This gap stimulates the use of digital imaging facilities (like a camera). An early method was proposed by Tomasi and Kanade.[2] They used an affine factorization approach to extract 3D from images sequences. However, the assumption of orthographic projection is a significant limitation of this system.

Camera calibration consists of intrinsic and extrinsic parameters, without which at some level no arrangement of algorithms can work. The dotted line between Calibration and Depth determination represents that the camera calibration is usually required for determining depth.

By the stage of Material Application you have a complete 3D mesh, which may be the final goal, but usually you will want to apply the color from the original photographs to the mesh. This can range from projecting the images onto the mesh randomly, through approaches of combining the textures for super resolution and finally to segmenting the mesh by material, such as specular and diffuse properties.

In auto-calibration or self-calibration, camera motion and parameters are recovered first, using rigidity. Then structure can be readily calculated. Two methods implementing this idea are presented as follows:

With a minimum of three displacements, we can obtain the internal parameters of the camera using a system of polynomial equations due to Kruppa,[6] which are derived from a geometric interpretation of the rigidity constraint.[7][8]

Recently, new methods based on the concept of stratification have been proposed.[10] Starting from a projective structure, which can be calculated from correspondences only, upgrade this projective reconstruction to a Euclidean reconstruction, by making use of all the available constraints. With this idea the problem can be stratified into different sections: according to the amount of constraints available, it can be analyzed at a different level, projective, affine or Euclidean.

Usually, the world is perceived as a 3D Euclidean space. In some cases, it is not possible to use the full Euclidean structure of 3D space. The simplest being projective, then the affine geometry which forms the intermediate layers and finally Euclidean geometry. The concept of stratification is closely related to the series of transformations on geometric entities: in the projective stratum is a series of projective transformations (a homography), in the affine stratum is a series of affine transformations, and in Euclidean stratum is a series of Euclidean transformations.

Simple counting indicates we have 2 n m \displaystyle 2nm independent measurements and only 11 m + 3 n \displaystyle 11m+3n unknowns, so the problem is supposed to be soluble with enough points and images. The equations in homogeneous coordinates can be represented:

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