Algorithm 1 Video segmentation algorithm based on multi-frame homography constraints

1: Input: video sequence 2: Initialize: frames number T; temporal window interval t; trajectory classification thresholdand ${\tau }_L$ and ${\tau }_H$; initial superpixel classification parameters T1 and T2

3: Trajectory classification based on multi-frame homography model

a: Calculate the long term trajectory $\mathbf{\Lambda }=\left\{{\mathbf{\Lambda }}_i,i=1,\dots ,n\right\}$ of input video

b: Estimate the homography matrix set ${\mathit{H}}^{multi}=\left\{{\mathit{H}}^k,k=1,\dots ,T-t\right\}$

c: for $i=1,\dots ,n$ do

Use Equation (3) to select the corresponding homography matrix set ${\mathit{H}}_i^{multi}$ of trajectory ${\mathbf{\Lambda }}_i$

Use Equation (4) to estimate the average projection error ${\epsilon }_i$ of the trajectory ${\mathbf{\Lambda }}_i$

end for

d: Use Equation (5) to classify the motion trajectory.  e: Use motion boundary to refine the spatial accuracy of trajectory classification.

4: Pixel labeling based on Markov Random Fields model  f: Oversegment the input video to get the superpixel set ${\displaystyle \sum r_t^i}$  g: for t = 1:T do   Use Equation (8)–(11) to calculate the unary potential $A_t^c\left(r_t^i\right)$ of superpixel $r_t^i$   Use Equation (12) and (13) to calculate the pairwise potential $S_t^{i,j}\left(r_t^i,r_t^j\right)$ and $T_t^{i,j}\left(r_t^i,r_{t+1}^j\right)$ of superpixel $r_t^i$   end for

h: Use graph cut algorithm to solve the energy function minimization problem

5: Output: Pixel level object segmentation result for each frame of input video