2 Full Defocus Map Using sparse defocus map at edge locations, the full defocus map is recovered by an edge-aware interpolation method [8]. We use the MatLab software provided by Zhuo and Sim [16] and substitute their sparse defocus map with ours. Edge Width Estimation for Defocus Map from a Single Image 4 19 Results and Discussion We demonstrate the effectiveness of the proposed method by comparing our full defocus maps with results provided by [14] and [16]. The proposed edge width estimation method works well on images with a relatively low amount of noise, which is usually the case with defocused images.

Fig. 1. Edge model In our model we use the following definitions of the Gaussian filter and ideal step edge function Gσ (x) = √ x2 1 e− 2σ2 , 2πσ H(x) = 1, 0, x ≥ 0, x < 0. 2 σ x) σ ∀σ > 0, σ > 0. (1) Estimation of Edge Width For the estimation of edge width we use the unsharp masking approach. Let Uσ,α [Eσ0 ](x) be the result of unsharp masking applied to the edge Eσ0 (x): U σ,α [Eσ0 ](x) = (1 + α)Eσ0 (x) − αEσ0 ∗ Gσ = = (1 + α)Eσ0 (x) − αE√σ2 +σ2 (x). 0 (2) Using (1) and supposing σ = σ0 = σ1 , (2) holds U σ1 ,α [Eσ1 ](x) = (1 + α)Eσ1 (x) − αE√2σ1 (x) = √ 2σ2 σ2 √ = (1 + α)Eσ1 (x) − αE 2σ2 ( √ x) = Uσ2 ,α [Eσ2 ]( x).

An illustration of DHSF is presented in Fig. 3(a) and Fig. 2(c). 2 Estimating the Direction of Second Order Statistics Such as Ridges and Valleys By convolving the image patch with the DHSF (by the technique of rotating the images, as explained above), for each pixel in the image patch we obtain a pixel signal which captures a directional second order information around the pixel. The idea is to extract the directions at which these second order statistics such as ridges and valleys occur and to construct a descriptor from this information.