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Y , i = j. This is because the “background” scene can be considered to be an object itself. In this case, the image model becomes I(y) = √ 2 1 ˜2 e−t /(2ΔT ) r(y + νj (y, t))dt 2πΔT˜ ∀ y ∈ Ωj . 40) In order to complete the model we need to determine the velocities νj , which are a function of the shape and motion of objects in the scene. To do so consider again, for simplicity, the case of a scene made of a single object with depth map s. 41) where × denotes the cross-product 10 of two vectors in R3 .

14) identiﬁes an inﬁnite set of components that, when combined, give rise to the radiance r. Typically we can sort αk in decreasing order, and truncate beyond a certain index. When the sum is actually ﬁnite, we can give a name to the maximum index in the sum. 7. ρ , with αk = 0, k = 0, . . , ρ, such that ρ αk θk (x). ∞ , we say that the distribution r is band-limited if the corresponding degree of resolution ρ < ∞. Similarly, we can deﬁne a degree of resolution for the kernel h. 8. Let hvs ∈ L2 (R2 ×R2 ).

30) with initial conditions u(y, 0) = δ(x − y), and δ(·) is Dirac’s delta. The function G satisﬁes the equation u(x, ˙ t) = c u(x, t) u(x, 0) = r(x) u(y, t) = G(y, x, t)u(x, 0)dx = G(y, x, t)r(x)dx. 30). This explains why we so cherish Gaussians as point-spread functions, beyond the discussion leading to this section. Gaussians establish the equivalence between 5 Notational mea culpa: we use u for the solution of the isotropic heat equation as is customary in the PDE literature. Alas, we have already used u to describe the geometry of the lens.