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� � MAS.131/MAS.531 Separating Transparent Layers in Images Abstract This paper presents an algorithm to separate different transparent layers present in images of a 3-D scene. The criterion of minimum mutual information is presented and evaluated. We find that high frequency details are faithfully recovered while the recovery of low frequency details needs improvement. INTRODUCTION Typically, when we image a 3-D scene using a camera, different 2-D layers of the scene that are at different distances from the camera are superimposed. For example, consider the images shown in Figure 1. The two images belong to the same scene containing a chair and a file cabinet. However, the images are captured by focussing at different depths. The surface that is in focus is sharp and the surface that is out of focus is blurred. (a) The file cabinet is in focus (b) The chair is in focus Fig. 1. An example illustrating superposition of transparent layers in an Image. Such images are often encountered in microscopy while studying the anatomy of various organisms. Different slices of cells or tissues are captured by focussing at different depths. In order to study any particular 2-D slice from such images, it is important to separate the effect of other 2-D layers that are super-imposed in the images. THE MODEL Let a 2-D layer f1(x, y) interfere with a 2-D layer f2(x, y). We consider the slices ga(x, y) and gb(x, y), in which either layer f1(x, y) or layer f2(x, y) respectively, is in focus. The other layer is blurred. Modeling the effect of blur as a convolution with blur kernals, we have, ga(x, y) = � f2(x, y) ⊗ h2a(x, y)� .f1(x, y) (1) gb(x, y) = f1(x, y) ⊗ h1b(x, y) .f2(x, y) where ‘⊗’ denotes convolution and ‘.’ denotes multiplication. The above model does, in fact describes the images shown in Figure 1. For example, consider a region in Figure 1(b) where the chair has holes. In that case, we are able to ‘see through’ a blurred version of the file cabinet. The regions where there are no holes in the chair in Figure 1(b), we do not ‘see through’ any version of the file cabinet. And hence, Anonymous MIT student� � � � the ‘multiply after convolution’ model. We assume that the blur-kernals are space-invariant for constant depth objects. Moreover, if the system is telecentric, then h1b(x, y) = h2a(x, y) = h(x, y). PRIOR ART Separation of different layers in images has been studied before. In [1], [2], the authors propose the use of Independent Component Analysis [3], [4] to separate reflections from Images that are caputered with a polarizer. Using local features to perceive transparency in images has been proposed in [5], [6]. The closest to our work is [7], [8], where the authors propose separation of transparent layers using a different model for superposition. The novelty of our work lies in the formulation of the transparent layer interference as a mutliplication of images as opposed to the standard approach of summation. OUR APPROACH The two equations in eq (1) have four unknowns, namely f1(x, y), f2(x, y), h2a(x, y) and h1b(x, y). And hence, this is a highly under-determined system. To solve the above system of equations, we make the assumption that h1b(x, y) = h2a(x, y) = h(x, y) (This is a reasonable assumption especially for telecentric images [7], [8]). Under this assumption, the resulting model is given by, ga(x, y) = � f2(x, y) ⊗ h(x, y)� .f1(x, y) (2) gb(x, y) = f1(x, y) ⊗ h(x, y) .f2(x, y) We are still left with three unknowns and two equations. Here, we leverage the idea of minimum mutual information that was proposed in [7], [8]. The underlying intuition is that if fˆ 1(x, y) and fˆ 2(x, y) are the solutions to the equations (2), then fˆ 1(x, y) and fˆ 2(x, y) have the least mutual information I(fˆ 1(x, y), fˆ 2(x, y)) among all possible solutions to equations (2). We use this criterion to find the optimal h(x, y) that yields minimum mutual information between the recovered images fˆ 1(x, y) and fˆ 2(x, y). Even given h(x, y), finding the solution to the equations (2) is challenging. Our algorithm for separating f1(x, y) and f2(x, y) is given below: 1) Assume a blur-kernal h(x, y) and solve for fˆ 1(x, y) and fˆ 2(x, y) using equations (2). This is not straightforward from the equations themselves. We find fˆ 1(x, y) and fˆ 2(x, y) iteratively from equations (2) as follows: a) Initialize fˆ 1(x, y) = 1 and fˆ 2(x, y) = 1. b) Iteratively update fˆ 1(x, y) and fˆ 1(x, y) as fˆ 1(x, y) = � ga(x, y) � fˆ 2(x, y) ⊗ h(x, y) fˆ 2(x, y) = � gb(x, y) � fˆ 1(x, y) ⊗ h(x, y) c) Return fˆ 1(x, y) and fˆ 2(x, y) after ten iterations. 2) Calculate the normalized minimum mutual information � � I fˆ 1(x, y), fˆ 2(x, y) In fˆ 1(x, y), fˆ 2(x, y) =[H � fˆ 1(x, y) � + H � fˆ 2(x, y) � ]/2� � where � ��� � � � H fˆ 1(x, y) = − P fˆ 1(x, y) log P fˆ 1(x, y) , fˆ1(x,y) � ��� � P fˆ 1(x, y), fˆ 2(x, y) I fˆ 1(x, y), fˆ 2(x, y) = − P fˆ 1(x, y), fˆ 2(x, y) log P � fˆ 1(x, y) � P � fˆ 2(x, y) � fˆ1(x,y), fˆ2(x,y) 3) Change the blur kernal h(x, y) and return to step 1. In our problem, we assume that the blur-kernels are Gaussian and are parameterized by their standard deviation σ. We iterate over the steps 1 − 3 mentioned above by changing σ. The final estimates of the separated layers are the ones that result in the least normalized mutual-information across all the runs. For more details, refer to the MATLAB code in the appendix. EXPERIMENTAL EVALUATION We generated multi-layered images according to equations (2) by combining two sharp images as shown in Figure 4. We used σ = 0.7 for the images. The calculated normalized mutual information is shown in the Figure 2. We observe that the algorithm correctly estimates σ to be 0.7. The recovered images are shown in Figure 4. We find that the image recovery preserves the high-frequency information but performes poorly in the low-frequency components. Shown in Figure 3 are the x-gradients for a


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MIT MAS 531 - Separating Transparent Layers in Images

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