Rigid motion segmentation

In general, motion can be considered to be a transformation of an object in space and time. If this transformation preserves size and shape of the object it is known as a Rigid Transformation. Rigid transform can be rotational, translational or reflective. We define rigid transformation mathematically as:

${\displaystyle \digamma :A\to A}$

Figure 1: Rigid Motion in 3-D

where F is a rigid transform if and only if it preserves isometry and space orientation.

In the sense of motion, rigid transform is the movement of a rigid object in space. As shown in Figure 1: this 3-D motion is the transformation from original co-ordinates (X,Y,Z) to transformed co-ordinates (X',Y',Z') which is a result of rotation and translation captured by rotational matrix R and translational vector T respectively. Hence the transform will be:

${\displaystyle {\begin{bmatrix}X'\\Y'\\Z'\\\end{bmatrix}}=R\cdot {\begin{bmatrix}X\\Y\\Z\\\end{bmatrix}}+T}$

where,

${\displaystyle R={\begin{bmatrix}r11&r12&r13\\r21&r22&r23\\r31&r32&r33\\\end{bmatrix}}and\ T={\begin{bmatrix}T_{x}\\T_{y}\\T_{z}\\\end{bmatrix}}}$

R has 9 unknowns which correspond to the rotational angle with each axis and T has 3 unknowns (Tx,Ty,Tz) which account for translation in X,Y and Z directions respectively. This motion(3-D) in time when captured by a camera(2-D) corresponds to change of pixels in the subsequent frames of the video sequence. This transformation is also known as 2-D rigid body motion or the 2-D Euclidean transformation. It can be written as:

${\displaystyle X'=R\cdot X+t}$

where,

${\displaystyle X={\begin{bmatrix}X\\Y\\\end{bmatrix}},\ X'={\begin{bmatrix}X'\\Y'\\\end{bmatrix}},\ R={\begin{bmatrix}\cos(\theta )&&-\sin(\theta )\\\sin(\theta )&&\cos(\theta )\\\end{bmatrix}}and\ t={\begin{bmatrix}t_{x}\\t_{y}\\\end{bmatrix}}}$

X→ original pixel co-ordinate.

X'→ transformed pixel co-ordinate.

R→ orthonormal rotation matrix with R ⋅ R^T = I and |R| = 1.

t→ translational vector but in the 2D image space.

To visualize this consider an example of a video sequence of a traffic surveillance camera. It will have moving cars and this movement does not change their shape and size. Moreover, the movement is a combination of rotation and transformation of the car in 3D which is reflected in its subsequent video frames. Thus the car is said to have a rigid motion.

Figure 2: Motion Segmentation Algorithm

Image segmentation techniques are interested in segmenting out different parts of the image as per the region of interest. As videos are sequences of images, motion segmentation aims at decomposing a video in moving objects and background by segmenting the objects that undergo different motion patterns. The analysis of these spatial and temporal changes occurring in the image sequence by separating visual features from the scenes into different groups lets us extract visual information. Each group corresponds to the motion of an object in the dynamic sequence. In the simplest case motion segmentation can mean extracting moving objects from a stationary camera but the camera can also move which introduces the relative motion of the static background. Depending upon the type of visual features that are extracted, motion segmentation algorithms can be broadly divided into two categories. The first is known as direct motion segmentation that uses pixel intensities from the image. Such algorithms assume constant illumination. The second category of algorithms computes a set of features corresponding to actual physical points on the objects. These sparse features are then used to characterize either the 2-D motion of the scene or the 3-D motion of the objects in the scene. There are a number of requirements to design a good motion segmentation algorithm. The algorithm must extract distinct features (corners or salient points) that represent the object by a limited number of points and it must have the ability to deal with occlusions. The images will also be affected by noise and will have missing data, thus they must be robust. Some algorithms detect only one object but the video sequence may have different motions. Thus the algorithm must be multiple object detectors. Moreover, the type of camera model, if used, also characterizes the algorithm. Depending upon the object characterization of an algorithm it can detect rigid, non-rigid motion or both. Moreover, algorithms used to estimate single rigid-body motions can provide accurate results with robustness to noise and outliers but when extended to multiple rigid-body motions they fail. In case of view-based segmentation techniques described below, this happens because the single fundamental matrix assumption is violated as each motion will now be represented by means of a new fundamental matrix corresponding to that motion.

As mentioned earlier that there is no particular way to distinguish Motion Segmentation techniques but depending on the basis of the segmentation criterion used in the algorithm it can be broadly classified as follows:[2]

Further motion detection algorithms can also be classified depending upon the number of views: two and multi view-based approaches namely. The two-view based approaches are usually based on epipolar geometry. Consider two perspective camera views of a rigid body and find its feature correspondences. These correspondences are seen to satisfy either an epipolar constraint for a general rigid-body or a homography constraint for a planar object. Planar motion in a sequence is the motion of the background, facade or the ground.[15] Thus it is a degenerate case of rigid body motion together with general rigid body objects e.g. cars. Hence in a sequence we expect to see more than one type of motion, described by multiple epipolar constraints and homographies. The view based algorithms are sensitive to outliers but recent approaches deal with outliers by using random sample consensus (RANSAC)[16] and enhanced Dirichlet process mixture models.[3][17] Other approaches use global dimension minimization to reveal the clusters corresponding to the underlying subspace. These approaches use only two frames for motion segmentation even if multiple frames are available as they cannot use multi frame information. Multiview-based approaches utilize the trajectory of feature points unlike two-view based approaches.[18] A number of approaches have been provided which include Principle Angles Configuration (PAC)[19] and Sparse Subspace Clustering (SSC)[20] methods. These work well in two or three motion cases. These algorithms are also robust to noise with a tradeoff with speed, i.e. they are less sensitive to noise but slow in computation. Other algorithms with a multi-view approach are spectral curvature clustering (SCC), latent low-rank representation-based method (LatLRR)[21] and ICLM-based approaches.[22] These algorithms are faster and more accurate than the two-view based but require greater number of frames to maintain the accuracy.

Motion segmentation is a field under research as there are many issues which provide scope of improvement. One of the major problems is of feature detection and finding correspondences. There are strong feature detection algorithms but they still give false positives which can lead to unexpected correspondences. Finding these pixel or feature correspondences is a difficult task. These mismatched feature points from the objects and the background often introduce outliers. The presence of image noise and outliers further affect the accuracy of structure from motion (SFM) estimation. Another issue is that of motion models or motion representations. It requires the motion to be modeled or estimated in the given model used in the algorithm. Most algorithms perform 2-D motion segmentation by assuming the motions in the scene can be modeled by 2-D affine motion models. Theoretically, this is valid because 2-D translational motion model can be represented by general affine motion model. However, such approximations in modeling can have negative consequences. The translational model has two parameters and the affine model has 6 parameters so we estimate four extra parameters. Moreover, there may not be enough data to estimate the affine motion model so the parameter estimation might be erroneous. Some of the other problems faced are:

• Prior knowledge about the objects or about the number of objects in the scene is essential and it is not always available.
• Blurring is a common issue when motion is involved.
• Moving objects can create occlusions, and it is possible that the whole object can disappear and reappear in the scene.
• Measurement of 3-D feature correspondences in the images can be noisy in terms of pixel coordinates.

Robust algorithms have been proposed to take care of the outliers and implement with greater accuracy. The Tomasi and Kanade factorization method is one of the methods as mentioned above under factorization.

Motion segmentation has many important applications.[1] It is used for video compression. With segmentation, it is possible to eliminate the redundancy related to the repetition of the same visual patterns in successive images. It can also be used for video description tasks, such as logging, annotation and indexing. By using Automatic object extraction techniques video content with object-specific information can be segregated. Thus concept can be used by search engines and video libraries. Some specific applications include:

• Video surveillance in security applications
• Sports scene analysis
• Road safety applications in intelligent vehicles
• Video indexing
• Traffic monitoring
• Object recognition

• Vision Lab covers GPCA, RANSAC (RANdom SAmple Consensus) and Local Subspace Affinity (LSA), JCAS (Joint Categorization and Segmentation), Low-Rank Subspace Clustering (LRSC) and Sparse Representation Theory. A link to a few implementations using Matlab by the Vision Lab at The Johns Hopkins University