Velocity Moments

In the field of computer vision, velocity moments are weighted averages of the intensities of pixels in a sequence of images, similar to image moments but in addition to describing an object's shape also describe its motion through the sequence of images. Velocity moments can be used to aid automated identification of a shape in an image when information about the motion is significant in its description. There are currently two established versions of velocity moments: Cartesian and Zernike.
Cartesian velocity moments.
Cartesian moments for single images.
A Cartesian moment of a single image is calculated by
where formula_2 and formula_3 are the dimensions of the image, formula_4 is the intensity of the pixel at the point formula_5 in the image, and formula_6 is the basis function.
Cartesian velocity moments for sequences of images.
Cartesian velocity moments are based on these Cartesian moments. A Cartesian velocity moment formula_7 is defined by
where formula_2 and formula_3 are again the dimensions of the image, formula_11 is the number of images in the sequence, and formula_12 is the intensity of the pixel at the point formula_5 in image formula_14.
formula_15 is taken from Central moments, added so the equation is translation invariant, defined as
where formula_17 is the formula_18 coordinate of the centre of mass for image formula_14, and similarly for formula_20.
formula_21 introduces velocity into the equation as
where formula_23 is the formula_18 coordinate of the centre of mass for the previous image, formula_25, and again similarly for formula_20.
After the Cartesian velocity moment is calculated, it can be normalised by
where formula_28 is the average area of the object, in pixels, and formula_29 is the number of images. Now the value is not affected by the number of images in the sequence or the size of the object.
As Cartesian moments are non-orthogonal, so are Cartesian velocity moments, so different moments can be closely correlated. These velocity moments do however provide translation and scale invariance (unless the scale changes within the sequence of images).
Zernike velocity moments.
Zernike moments for single images.
A Zernike moment of a single image is calculated by
where formula_31 denotes the complex conjugate, formula_32 is an integer between formula_33 and formula_34, and formula_35 is an integer such that formula_36 is even and formula_37. For calculating Zernike moments, the image, or section of the image which is of interest is mapped to the unit disc, then formula_4 is the intensity of the pixel at the point formula_5 on the disc and formula_40 is a restriction on values of formula_18 and formula_20. The coordinates are then mapped to polar coordinates, and formula_43 and formula_44 are the polar coordinates of the point formula_5 on the unit disc map.
formula_46 is derived from Zernike polynomials and is defined by
Zernike velocity moments for sequences of images.
Zernike velocity moments are based on these Zernike moments. A Zernike velocity moment formula_50 is defined by
where formula_11 is again the number of images in the sequence, and formula_12 is the intensity of the pixel at the point formula_5 on the unit disc mapped from image formula_14.
formula_21 introduces velocity into the equation in the same way as in the Cartesian velocity moments and formula_57 is from the Zernike moments equation above.
Like the Cartesian velocity moments, Zernike velocity moments can be normalised by
where formula_28 is the average area of the object, in pixels, and formula_29 is the number of images.
As Zernike velocity moments are based on the orthogonal Zernike moments, they produce less correlated and more compact descriptions than Cartesian velocity moments. Zernike velocity moments also provide translation and scale invariance (even when the scale changes within the sequence).