Abstract : Partial differential equation (PDE) method shows better performance than traditional image processing methods, and some new ideas have never been considered in traditional image processing, such as affine invariant feature extraction, image structure and texture decomposition, etc. This method aims to establish the mathematical model of a partial differential equation, and then make the image change according to the PDE, and finally achieve the desired effect. PDE models are mathematically robust and also provide insights in developing new algorithms. Fusion of AI/ML methods and PDE models makes it even more effective.
Abdul Halim (King Abdullah University of Science and Technology)
B.V. Rathish Kumar (IIT Kanpur)
Abstract : In this talk, I will present a fourth-order PDE model with a multi-well potential function for grayscale image inpainting. Convexity splitting in time and Fourier spectral method in space has been used to derive an unconditional scheme on time. The stable scheme is both consistent and convergent. Also, I present the fractional variant of the time discretized scheme by replacing the Laplacian with its fractional counterpart. Numerical results for some standard test images will be presented and will be compared with the results of other existing models in the literature. To quantify the quality of the recovered image, we calculate the image quality metric such as PSNR, SNR, and SSIM.
[01962] Fractional Calculus Based Approach for Retinal Blood Vessel Segmentation
Format : Online Talk on Zoom
Author(s) :
Rajesh Kumar Pandey (Indian Institute of Technology (BHU) Varanasi)
Varun Makkar (Indian Institute of Technology (BHU) Varanasi)
Rathish Kumar Venkatesulu Bayya (Indian Institute of Technology Kanpur)
Abstract : We will discuss a new fractional filter and an algorithm for retinal blood vessel segmentation. The proposed fractional filter is designed with the help of a weighted fractional derivative and an exponential weight factor. Firstly, the image is denoised using developed fractional. Then, multi-scale line detector is used to compute the line responses at multiple lengths using a punctured window of fixed dimensions. The final response is the computed as the arithematic mean of all responses at different scales and the underlying image intensity. This enhances the retinal blood vessels and suppresses rest of the background. Finally, hysteresis thresholding is applied to obtain the segmented vessels. Experiments are performed on two well-studied evaluation databases named STARE and DRIVE and the simulation results are discussed.
[02860] Game theoretic Approach for Image segmentation and Image restoration by using Fractional PDE
Format : Online Talk on Zoom
Author(s) :
Kedarnath Buda (Indian Institute of Technology Kanpur, India)
Rathish Kumar Venkatesulu Bayya (Indian Institute of Technology Kanpur)
Abstract : In this study, a game theoretic algorithm through which a noisy
image is both restored and segmented simultaneously is proposed. We define
a two cost functions for two players of a Game. One player is Image
restoration and another one is Image segmentation. Both of them are further
constrained by fractional PDEs.The establish the Nash equilibrium for this
game, which is an ideal strategy for deciding the amount of image
restoration and image segmentation that can be done through the
optimization of the bi-objective cost functions. We then numerically
compute this with some real images.
[02862] Higher Order PDE Model for Effective Image Denoising
Format : Talk at Waseda University
Author(s) :
Rathish Kumar Venkatesulu Bayya (Indian Institute of Technology Kanpur)
Abdul Halim (KAUST)
Abstract : In this talk we will introduce a new higher order nonlinear PDE model for image denoising and demonstrate its ability to denoise without any staircase effect as routinely noticed with lower order PDE models. We use convexity splitting based Fourier Spectral scheme for the computation of the denoised version of a given noisy image. Fourier spectral method is both accurate and faster than many standard approaches. Performance of the model and the method will be discussed based on the benchmark test images and the related computational metrics.
[02870] A Framework for Motion Estimation with Physics-Based Constraints in Image Sequences
Format : Online Talk on Zoom
Author(s) :
Hirak Doshi (Doctoral Research Scholar)
Uday Kiran Nori (Associate Professor)
Abstract : Motion estimation using variational models has been a central topic in mathematical image processing for many years. The Horn and Schunck model's variational approach to optical flow motion estimation is a seminal work that has been studied in-depth to develop different variational models for motion estimation. However, the Horn and Schunck model's constancy assumption cannot reflect the reality of actual motion, as deformation effects of fluid, illumination variations, perspective changes, poor contrast, etc., directly affect the important motion parameters. Therefore, physics-dependent motion estimation algorithms have been extensively investigated in the literature.
In this paper, we propose a generic framework that captures physics-based constraints for motion estimation as perceived at the smallest intensity level (pixel) of an image sequence. These constraints are introduced as non-conservative terms that capture the loss of particles at the pixel-level, in the minimizing energy functional. We demonstrate our framework with two physics-based constraints, the continuity constraint for fluid motion and the harmonic constraint for capturing rotation in the images. Furthermore, we theoretically justify the effectiveness of our model through the techniques of Augmented Lagrangian and maximal monotone operators.
We establish the mathematical well-posedness of the associated PDE in the Hilbert space setting. For the linear case, we perform a decoupling of the associated PDE into diffusion equations on the curl and divergence of the flow field through a diagonalization with the Cauchy-Riemann operator. This decoupling process suggests that our approach preserves the spatial characteristics of the divergence and the vorticities of the flow field.
We adapt the first-order primal-dual Chambolle-Pock algorithm to obtain the minimization of our variational problem. We demonstrate the robustness of our approach through velocity plots and use the Average Angular Error (AAE) and End-Point Error (EPE) as performance metrics. We test our algorithm on several relevant datasets and show good results. In particular, for the Middlebury dataset, we show that our algorithm outperforms some of the state-of-the-art Horn and Schunk based flow models.
Moreover, for the fluid motion estimation case, a primal-dual implementation of our two-phase refinement model has a faster convergence rate of $O(1/N)$ compared to the $O(1/\sqrt{N})$ convergence rate of a direct primal-dual implementation of the Liu-Shen continuity-based model, where $N$ is the number of iterations. Although we do not have a theoretical proof for this observed efficiency, we provide substantial empirical evidence.
[03393] On the convergence analysis of DNN for vorticity stream function formulation and application
Format : Online Talk on Zoom
Author(s) :
Rajendra Kumar (Student)
Rathish Kumar Venkatesulu Bayya (Indian Institute of Technology Kanpur)
Ming-Chih Lai (National Yang Ming Chiao Tung University, Taiwan)
Abstract : The physics-informed neural network is a completely mesh-free method for partial differential equations. In this paper, We introduce a Physics-informed neural network for the two dimensions Navier-stokes equation in the vorticity-stream function form with boundary conditions. We estimate the error of the physics-informed neural network for vorticity stream function formulation and theoretically establish the convergence of the computational procedure. In Deep neural network representation imposing the boundary condition is one of the main issues. We successfully incorporate periodic boundary conditions in the vorticity stream function formulation which is known for its difficulty in training the model. We have successfully applied PINNS on applications such as the Double shear layer and Taylor vortex problem
