Implementing Lucas-Kanade Optical Flow algorithm in Python

In this article an implementation of the Lucas-Kanade optical flow algorithm is going to be described. This problem appeared as an assignment in this computer vision course from UCSD. The inputs will be sequences of images (subsequent frames from a video) and the algorithm will output an optical flow field (u, v) and trace the motion of the moving objects. The problem description is taken from the assignment itself.


Problem Statement

Single-Scale Optical Flow

  • Let’s implement the single-scale Lucas-Kanade optical flow algorithm. This involves finding the motion (u, v) that minimizes the sum-squared error of the brightness constancy equations for each pixel in a window.  The algorithm will be implemented as a function with the following inputs:

     def optical_flow(I1, I2, window_size, tau) # returns (u, v)

  • Here, u and v are the x and y components of the optical flow, I1 and I2 are two images taken at times t = 1 and t = 2 respectively, and window_size is a 1 × 2 vector storing the width and height of the window used during flow computation.
  • In addition to these inputs, a theshold τ should be added, such that if τ is larger than the smallest eigenvalue of A’A, then the the optical flow at that position should not be computed. Recall that the optical flow is only valid in regions where

has rank 2, which is what the threshold is checking. A typical value for τ is 0.01.

  • We should try experimenting with different window sizes and find out the tradeoffs associated with using a small vs. a large window size.
  • The following figure describes the algorithm, which considers a nxn (n>=3) window around each pixel and solves a least-square problem to find the best flow vectors for the pixel.


  • The following code-snippet shows how the algorithm is implemented in python for a gray-level image.
import numpy as np
from scipy import signal
def optical_flow(I1g, I2g, window_size, tau=1e-2):

    kernel_x = np.array([[-1., 1.], [-1., 1.]])
    kernel_y = np.array([[-1., -1.], [1., 1.]])
    kernel_t = np.array([[1., 1.], [1., 1.]])#*.25
    w = window_size/2 # window_size is odd, all the pixels with offset in between [-w, w] are inside the window
    I1g = I1g / 255. # normalize pixels
    I2g = I2g / 255. # normalize pixels
    # Implement Lucas Kanade
    # for each point, calculate I_x, I_y, I_t
    mode = 'same'
    fx = signal.convolve2d(I1g, kernel_x, boundary='symm', mode=mode)
    fy = signal.convolve2d(I1g, kernel_y, boundary='symm', mode=mode)
    ft = signal.convolve2d(I2g, kernel_t, boundary='symm', mode=mode) +
         signal.convolve2d(I1g, -kernel_t, boundary='symm', mode=mode)
    u = np.zeros(I1g.shape)
    v = np.zeros(I1g.shape)
    # within window window_size * window_size
    for i in range(w, I1g.shape[0]-w):
        for j in range(w, I1g.shape[1]-w):
            Ix = fx[i-w:i+w+1, j-w:j+w+1].flatten()
            Iy = fy[i-w:i+w+1, j-w:j+w+1].flatten()
            It = ft[i-w:i+w+1, j-w:j+w+1].flatten()
            #b = ... # get b here
            #A = ... # get A here
            # if threshold τ is larger than the smallest eigenvalue of A'A:
            nu = ... # get velocity here

    return (u,v)


Some Results

  • The following figures and animations show the results of the algorithm on a few image sequences. Some of these input image sequences / videos are from the course and some are collected from the internet.
  • As can be seen, the algorithm performs best if the motion of the moving object(s) in between consecutive frames is slow. To the contrary, if the motion is large, the algorithm fails and we should implement / use multiple-scale version Lucas-Kanade with image pyramids.
  • Finally,  with small window size,  the algorithm captures subtle motions but not large motions. With large size it happens the other way.

Input Sequences



Output Optical Flow with different window sizes

window size = 15


window size = 21



Input Sequences

Output Optical Flow


Input Sequences (hamburg taxi)


Output Optical Flowtaxi_opt


Input Sequences


Output Optical Flow

Input Sequences


Output Optical Flowseq_opt

Input Sequences    fount3.gif


Output Optical Flowfount_opt

Input Sequences

Output Optical Flow

Input Sequencessynth

Output Optical Flowsynth_opt

Input Sequencescars1
Output Optical Flowcars1_optcars1_cmap

Input Sequencescars2

Output Optical Flowcars2_opt

Output Optical Flowcars2_opt2cars2_cmap

Input Sequences



Output Optical Flow with window size 45

Output Optical Flow with window size 10

Output Optical Flow with window size 25
Output Optical Flow with window size 45cars3_opt2_45



3 thoughts on “Implementing Lucas-Kanade Optical Flow algorithm in Python

  1. Hi there,
    Your tutorial appears to be the closest online to what I’m trying to do, but I don’t understand the steps you commented out in the pseudocode for finding A, b, and nu. Could you elucidate just so I understand how to code up the LK algorithm in cases where opencv isn’t available?



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