Defocus Blur of a Catadioptric Sensor

  Two factors combine to cause defocus blur in catadioptric sensors, that is in addition to the normal causes present in conventional dioptric systems, such as diffraction and lens aberrations. They are (1) the finite size of the lens aperture, and (2) the curvature of the mirror. To analyze how these two factors cause defocus blur, we first consider a fixed point in the world and a fixed point in the lens. We then find the point on the mirror which reflects a ray of light from the world point through the lens point. Next, we compute where on the image plane this mirror point is imaged. By considering the locus of imaged mirror points as the lens point varies, we can compute the area of the image plane onto which a fixed world point is imaged. In Section 4.1, we derive the constraints on the mirror point at which the light is reflected, and show how it can be projected onto the image plane. Afterwards, in Section 4.2, we present numerical results for the hyperboloid mirror. Generalization to the other mirrors described in Section 2 is straightforward.

  

Figure 9: The geometry used to analyze the defocus blur. We work in the 3-D cartesian frame $(\mathbf{v},\hat\mathbf{x},\hat\mathbf{y},\hat\mathbf{z})$where $\hat\mathbf{x}$ and $\hat\mathbf{y}$ are orthogonal unit vectors in the plane z=0. In addition to the assumptions of Section 3, we also assume that the effective pinhole is located at the center of the lens and that the lens has a circular aperture. If a ray of light from the world point $\mathbf{w} = \frac{l}{\Vert \mathbf{m} \Vert} (x, y, z)$ is reflected at the mirror point $\mathbf{m}_{1} = (x_{1}, y_{1}, z_{1})$ and then passes through the lens point $\mathbf{l} = (d \cdot \cos \lambda, d \cdot \sin \lambda, c)$,there are three constraints on $\mathbf{m}_{1}$: (1) it must lie on the mirror, (2) the angle of incidence must equal the angle of reflection, and (3) the normal $\mathbf{n}$ to the mirror at $\mathbf{m}_{1}$, and the two vectors $\mathbf{l}-\mathbf{m}_{1}$ and $\mathbf{w}-\mathbf{m}_{1}$ must be coplanar.