We show that optical flow, i.e., the apparent motion of the time-varying brightness over the image plane of an imaging device, can be estimated by means of simple differential techniques. Linear algebraic equations for the two components of optical flow at each image location are derived. The coefficients of these equations are combinations of spatial and temporal derivatives of the image brightness. The equations are suggested by an analogy with the theory of deformable bodies and are exactly true for particular classes of motion or elementary deformations. Locally, a generic optical flow can be approximated by using a constant term and a suitable combination of four elementary deformations of the time-varying image brightness, namely, a uniform expansion, a pure rotation, and two orthogonal components of shear. When two of the four equations that correspond to these deformations are satisfied, optical flow can more conveniently be computed by assuming that the spatial gradient of the image brightness is stationary. In this case, it is also possible to evaluate the difference between optical flow and motion field—that is, the two-dimensional vector field that is associated with the true displacement of points on the image plane. Experiments on sequences of real images are reported in which the obtained optical flows are used successfully for the estimate of three-dimensional motion parameters, the detection of flow discontinuities, and the segmentation of the image in different moving objects.

Wataru Suzuki, Atsushi Hiyama, Noritaka Ichinohe, Wakayo Yamashita, Takeharu Seno, and Hiroshige Takeichi J. Opt. Soc. Am. A 37(12) 1958-1964 (2020)

References

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The first column reports the true displacement over the image plane (in pixels/frame), and the second column reports the standard deviation σ_{n} of the added Gaussian noise (noise-free sequences correspond to the case of a standard deviation equal to zero). The third, fourth, fifth, sixth, and seventh columns show the estimated component of image motion that was obtained by averaging the optical flow as computed by Eqs. (2.3) and (3.9), Eqs. (2.3) and (3.10), Eqs. (2.3) and (3.11), Eqs. (2.3) and (3.12), and Eqs. (4.1), respectively, through the procedures described in this paper. The average is computed over a square of 30 × 30 pixels centered in the moving square to minimize boundary effects. Similar results have been obtained for the vertical component. The indeterminacy in the third, fourth, fifth, sixth, and seventh columns is the standard error multiplied by a factor S^{2} (with S = 2σ, where σ = 3 is the standard deviation of the spatial Gaussian filter used), which takes into account the principle that only the estimates that are separated by S = 6 along both the vertical and the horizontal axes can be thought of as roughly independent.

Table 2

Estimates of the Time to Collision for the Translation of Fig. 3^{a}

Results are computed through Eqs. (4.1) and Eqs. (2.3) and (3.9) with respect to the values directly measured in the scene and for different values of the standard deviation of the final smoothing Gaussian filter (rightmost column). The raw optical flows do not depend on the final smoothing step. The time to collision is expressed in t_{d}, where t_{d} is the temporal displacement between consecutive frames and is taken as unitary. As the spatial displacement of the viewed object between consecutive frames is 1 cm, the time to collision directly measured in the scene is obtained by expressing the known distance of the object from the viewing camera in centimeters. The indeterminacy in the first, second, third, and fourth columns is the standard error multiplied by a factor S^{2} (with S = 2σ, where σ is the standard deviation of the final smoothing Gaussian filter), which takes into account that only the estimates separated by S along both the vertical and the horizontal axes can be thought of as roughly independent.

Table 3

Estimates of the Magnitude of the Angular Velocity for the Rotation of Fig. 5^{a}

Results are computed through Eqs. (4.1) and Eqs. (2.3) and (3.10) with respect to the values directly measured in the scene and for different values of the standard deviation of the final smoothing Gaussian filter (rightmost column). The angular velocity is in degrees per frame, and the temporal distance between consecutive frames is taken as unitary. For the indeterminacy in the first, second, third, and fourth columns, see the footnote of Table 2.

Tables (3)

Table 1

Estimates of the Horizontal Component of Image Motion for Translation of the Random-Dot Pattern of Fig. 2A^{a}

The first column reports the true displacement over the image plane (in pixels/frame), and the second column reports the standard deviation σ_{n} of the added Gaussian noise (noise-free sequences correspond to the case of a standard deviation equal to zero). The third, fourth, fifth, sixth, and seventh columns show the estimated component of image motion that was obtained by averaging the optical flow as computed by Eqs. (2.3) and (3.9), Eqs. (2.3) and (3.10), Eqs. (2.3) and (3.11), Eqs. (2.3) and (3.12), and Eqs. (4.1), respectively, through the procedures described in this paper. The average is computed over a square of 30 × 30 pixels centered in the moving square to minimize boundary effects. Similar results have been obtained for the vertical component. The indeterminacy in the third, fourth, fifth, sixth, and seventh columns is the standard error multiplied by a factor S^{2} (with S = 2σ, where σ = 3 is the standard deviation of the spatial Gaussian filter used), which takes into account the principle that only the estimates that are separated by S = 6 along both the vertical and the horizontal axes can be thought of as roughly independent.

Table 2

Estimates of the Time to Collision for the Translation of Fig. 3^{a}

Results are computed through Eqs. (4.1) and Eqs. (2.3) and (3.9) with respect to the values directly measured in the scene and for different values of the standard deviation of the final smoothing Gaussian filter (rightmost column). The raw optical flows do not depend on the final smoothing step. The time to collision is expressed in t_{d}, where t_{d} is the temporal displacement between consecutive frames and is taken as unitary. As the spatial displacement of the viewed object between consecutive frames is 1 cm, the time to collision directly measured in the scene is obtained by expressing the known distance of the object from the viewing camera in centimeters. The indeterminacy in the first, second, third, and fourth columns is the standard error multiplied by a factor S^{2} (with S = 2σ, where σ is the standard deviation of the final smoothing Gaussian filter), which takes into account that only the estimates separated by S along both the vertical and the horizontal axes can be thought of as roughly independent.

Table 3

Estimates of the Magnitude of the Angular Velocity for the Rotation of Fig. 5^{a}

Results are computed through Eqs. (4.1) and Eqs. (2.3) and (3.10) with respect to the values directly measured in the scene and for different values of the standard deviation of the final smoothing Gaussian filter (rightmost column). The angular velocity is in degrees per frame, and the temporal distance between consecutive frames is taken as unitary. For the indeterminacy in the first, second, third, and fourth columns, see the footnote of Table 2.