92 research outputs found

    Optimising Spatial and Tonal Data for PDE-based Inpainting

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    Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods

    Dithering by Differences of Convex Functions

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    Motivated by a recent halftoning method which is based on electrostatic principles, we analyse a halftoning framework where one minimizes a functional consisting of the difference of two convex functions (DC). One of them describes attracting forces caused by the image gray values, the other one enforces repulsion between points. In one dimension, the minimizers of our functional can be computed analytically and have the following desired properties: the points are pairwise distinct, lie within the image frame and can be placed at grid points. In the two-dimensional setting, we prove some useful properties of our functional like its coercivity and suggest to compute a minimizer by a forward-backward splitting algorithm. We show that the sequence produced by such an algorithm converges to a critical point of our functional. Furthermore, we suggest to compute the special sums occurring in each iteration step by a fast summation technique based on the fast Fourier transform at non-equispaced knots which requires only Ο(m log(m)) arithmetic operations for m points. Finally, we present numerical results showing the excellent performance of our DC dithering method

    A projection method on measures sets

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    We consider the problem of projecting a probability measure π on a set MN of Radon measures. The projection is defined as a solution of the following variational problem: inf ”∈M N h (” − π) 2 2 , where h ∈ L 2 (℩) is a kernel, ℩ ⊂ R d and denotes the convolution operator. To motivate and illustrate our study, we show that this problem arises naturally in various practical image rendering problems such as stippling (representing an image with N dots) or continuous line drawing (representing an image with a continuous line). We provide a necessary and sufficient condition on the sequence (MN) N ∈N that ensures weak convergence of the projections (” * N) N ∈N to π. We then provide a numerical algorithm to solve a discretized version of the problem and show several illustrations related to computer-assisted synthesis of artistic paintings/drawings

    A projection algorithm on measures sets

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    We consider the problem of projecting a probability measure π\pi on a set MN\mathcal{M}_N of Radon measures. The projection is defined as a solution of the following variational problem:\begin{equation*}\inf_{\mu\in \mathcal{M}_N} \|h\star (\mu - \pi)\|_2^2,\end{equation*}where h∈L2(Ω)h\in L^2(\Omega) is a kernel, Ω⊂Rd\Omega\subset \R^d and ⋆\star denotes the convolution operator.To motivate and illustrate our study, we show that this problem arises naturally in various practical image rendering problems such as stippling (representing an image with NN dots) or continuous line drawing (representing an image with a continuous line).We provide a necessary and sufficient condition on the sequence (MN)N∈N(\mathcal{M}_N)_{N\in \N} that ensures weak convergence of the projections (ÎŒN∗)N∈N(\mu^*_N)_{N\in \N} to π\pi.We then provide a numerical algorithm to solve a discretized version of the problem and show several illustrations related to computer-assisted synthesis of artistic paintings/drawings

    Digital Color Imaging

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    This paper surveys current technology and research in the area of digital color imaging. In order to establish the background and lay down terminology, fundamental concepts of color perception and measurement are first presented us-ing vector-space notation and terminology. Present-day color recording and reproduction systems are reviewed along with the common mathematical models used for representing these devices. Algorithms for processing color images for display and communication are surveyed, and a forecast of research trends is attempted. An extensive bibliography is provided

    The development of the toner density sensor for closed-loop feedback laser printer calibration

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    A new infrared (IR) sensor was developed for application in closed-loop feedback printer calibration as it relates to monochrome (black toner only) laser printers. The toner density IR sensor (TDS) was introduced in the early 1980’s; however, due to cost and limitation of technologies at the time, implementation was not accomplished until within the past decade. Existing IR sensor designs do not discuss/address: ‱ EMI (electromagnetic interference) effects on the sensor due to EP (electrophotography) components ‱ Design considerations for environmental conditions ‱ Sensor response time as it affects printer process speed The toner density sensor (TDS) implemented in the Lexmark E series printer reduces these problems and eliminates the use of the current traditional “open-loop” (meaning feedback are parameters not directly affecting print darkness such as page count, toner level, etc.) calibration process where print darkness is adjusted using previously calculated and stored EP process parameters. The historical process does not have the ability to capture cartridge component variation and environmental changes which affect print darkness variation. The TDS captures real time data which is used to calculate EP process parameters for the adjustment of print darkness; as a result, greatly reducing variations uncontrolled by historical printer calibration. Specifically, the first and primary purpose of this research is to reduce print darkness variation using the TDS. The second goal is to mitigate the TDS EMI implementation issue for reliable data accuracy

    Towards the Control of Electrophotographic-based 3-Dimensional Printing: Image-Based Sensing and Modeling of Surface Defects

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    Electro-Photography (EP) has been used for decades for fast, cheap, and reliable printing in offices and homes around the world. It has been shown that extending the use of EP for 3D printing is feasible; multiple layered prints are already commercially available (color laser printers) but only for a very limited number of layers. Many of the advantages of laser printing make EP 3D printing desirable including: speed, reliability, selective coloring, ability to print a thermoplastic, possibilities for multi-material printing, ability to print materials not amenable to liquid ink formulations. However, many challenges remain before EP-based 3D printing can be commercially viable. A limiting factor in using the same system architecture as a traditional laser printer is that as the thickness of the part increases, material deposition becomes more difficult with each layer since the increased thickness reduces the field strength. Different system configurations have been proposed where the layer is printed on intermediate stations and are subsequently transferred to the work piece. Layer registration and uniform transfer from the intermediate station become crucial factors in this architecture. At the Print Research and Imaging Systems Modeling (PRISM) Lab preliminary tests have confirmed the feasibility of using EP for Additive Manufacturing (AM). However, similar issues were encountered to those reported in literature as the number of layers increased, resulting in non-uniform brittle 3D structures. The defects were present but not obvious at each layer, and as the part built up, the defects add up and became more obvious. The process, as in many printers, did not include a control system for the ultimate system output (print), and the actuation method (electrostatic charge) is not entirely well characterized or sensed to be used in a control system. This research intends to help the development of a model and an image-based sensing system that can be used for control of material deposition defects for an EP 3D printing process. This research leverages from the expertise at RIT and the Rochester area in Printing, Electrophotography, Rapid Prototyping, Control, and Imaging Sciences
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