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
m p q = ∑ x = 1 M ∑ y = 1 N x p y q P x y {\displaystyle m_{pq}=\sum _{x=1}^{M}\sum _{y=1}^{N}x^{p}y^{q}P_{xy}}
where M {\displaystyle M} and N {\displaystyle N} are the dimensions of the image, P x y {\displaystyle P_{xy}} is the intensity of the pixel at the point ( x , y ) {\displaystyle (x,y)} in the image, and x p y q {\displaystyle x^{p}y^{q}} is the basis function.
Cartesian velocity moments for sequences of images Cartesian velocity moments are based on these Cartesian moments. A Cartesian velocity moment v m p q μ γ {\displaystyle vm_{pq\mu \gamma }} is defined by
v m p q μ γ = ∑ i = 2 i m a g e s ∑ x = 1 M ∑ y = 1 N U ( i , μ , γ ) C ( i , p , g ) P i x y {\displaystyle vm_{pq\mu \gamma }=\sum _{i=2}^{images}\sum _{x=1}^{M}\sum _{y=1}^{N}U(i,\mu ,\gamma )C(i,p,g)P_{i_{xy}}}
where M {\displaystyle M} and N {\displaystyle N} are again the dimensions of the image, i m a g e s {\displaystyle images} is the number of images in the sequence, and P i x y {\displaystyle P_{i_{xy}}} is the intensity of the pixel at the point ( x , y ) {\displaystyle (x,y)} in image i {\displaystyle i} .
C ( i , p , q ) {\displaystyle C(i,p,q)} is taken from Central moments, added so the equation is translation invariant, defined as
C ( i , p , q ) = ( x − x i ¯ ) p ( y − y i ¯ ) q {\displaystyle C(i,p,q)=(x-{\overline {x_{i}}})^{p}(y-{\overline {y_{i}}})^{q}}
where x i ¯ {\displaystyle {\overline {x_{i}}}} is the x {\displaystyle x} coordinate of the centre of mass for image i {\displaystyle i} , and similarly for y {\displaystyle y} .
U ( i , μ , γ ) {\displaystyle U(i,\mu ,\gamma )} introduces velocity into the equation as
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