Lucas–Kanade method

The Lucas- Kanade method to calculate the optical flow goes back to the two researchers Bruce D. Lucas and Takeo Kanade. They proposed this method for the first time before 1981. The method is a popular method that finds wide application today. The additional condition that is needed to calculate the optical flow, the assumption of equality of the river in the local environment of the central pixel for which the flow is determined.

Mathematical Foundations

The Lucas- Kanade method is based on the basic equation of the optical flow. The river for two 3D image volumes ( 2D or nD cases are similar) is given by. In a small environment, which has its center in the voxel, the river is considered to be constant. This assumption is generally true when the time increments between frames are selected to be small enough., Denote the partial derivatives of the image in the -, -, direction and time. Numbered one with the voxels, then a system of equations can be established:

We receive more than three equations for the three desired flow variables. There is an over-determined system. The following applies:

The on certain system can now be solved using the least squares method:

Or

Or

The sum here runs from i = 1 to n

The flow can thus be determined on the images by calculating the derivatives ( gradient = ). To give greater prominence to the central voxel, one often uses a weighting formula W (i, j, k), with. For this purpose, a Gaussian function can be used. Other extensions of the Lucas- Kanade method to use statistical methods to better deal with noise.

This method is also applied in a hierarchical process in which the flow is first computed on a coarser scale, and is then refined successively on a finer and finer scale.

Properties

One of the properties of the Lucas- Kanade method is that it provides (such as other local methods for the calculation of optical flow ) is not dense flow (ie, not sparse dense ). The flow information fades quickly with the distance from the edges ( edges or corners). The advantage of the method is relatively robust to noise and minor defects in the image.