Recently, a number of iterative learning methods have been introduced to improve generalization. These typically rely on training for longer periods of time in exchange for improved generalization. LLF (later-layer-forgetting) is a state-of-the-art method in this category. It strengthens learning in early layers by periodically re-initializing the last few layers of the network. Our principal innovation in this work is to use Simulated annealing in EArly Layers (SEAL) of the network in place of re-initialization of later layers. Essentially, later layers go through the normal gradient descent process, while the early layers go through short stints of gradient ascent followed by gradient descent. Extensive experiments on the popular Tiny-ImageNet dataset benchmark and a series of transfer learning and few-shot learning tasks show that we outperform LLF by a significant margin. We further show that, compared to normal training, LLF features, although improving on the target task, degrade the transfer learning performance across all datasets we explored. In comparison, our method outperforms LLF across the same target datasets by a large margin. We also show that the prediction depth of our method is significantly lower than that of LLF and normal training, indicating on average better prediction performance.11The code to reproduce our results is publicly available at: https://github.com/amiiir-sarfi/SEAL

In Continual learning (CL) balancing effective adaptation while combating catastrophic forgetting is a central challenge. Many of the recent best-performing methods utilize various forms of prior task data, e.g. a replay buffer, to tackle the catastrophic forgetting problem. Having access to previous task data can be restrictive in many real-world scenarios, for example when task data is sensitive or proprietary. To overcome the necessity of using previous tasks’ data, in this work, we start with strong representation learning methods that have been shown to be less prone to forgetting. We propose a holistic approach to jointly learn the representation and class prototypes while maintaining the relevance of old class prototypes and their embedded similarities. Specifically, samples are mapped to an embedding space where the representations are learned using a supervised contrastive loss. Class prototypes are evolved continually in the same latent space, enabling learning and prediction at any point. To continually adapt the prototypes without keeping any prior task data, we propose a novel distillation loss that constrains class prototypes to maintain relative similarities as compared to new task data. This method yields state-of-the-art performance in the task-incremental setting, outperforming methods relying on large amounts of data, and provides strong performance in the class-incremental setting without using any stored data points.

Continual Learning (CL) research typically focuses on tackling the phenomenon of catastrophic forgetting in neural networks. Catastrophic forgetting is associated with an abrupt loss of knowledge previously learned by a model when the task, or more broadly the data distribution, being trained on changes. In supervised learning problems this forgetting, resulting from a change in the model's representation, is typically measured or observed by evaluating the decrease in old task performance. However, a model's representation can change without losing knowledge about prior tasks. In this work we consider the concept of representation forgetting, observed by using the difference in performance of an optimal linear classifier before and after a new task is introduced. Using this tool we revisit a number of standard continual learning benchmarks and observe that, through this lens, model representations trained without any explicit control for forgetting often experience small representation forgetting and can sometimes be comparable to methods which explicitly control for forgetting, especially in longer task sequences. We also show that representation forgetting can lead to new insights on the effect of model capacity and loss function used in continual learning. Based on our results, we show that a simple yet competitive approach is to learn representations continually with standard supervised contrastive learning while constructing prototypes of class samples when queried on old samples.11The code to reproduce our results is publicly available at: https://github.com/rezazzr/Probing-Representation-Forgetting

Recommendation methods fall into three major categories, content based filtering, collaborative filtering and deep learning based. Information about products and the preferences of earlier users are used in an unsupervised manner to create models which help make personalized recommendations to a specific new user. The more information we provide to these methods, the more likely it is that they yield better recommendations. Deep learning based methods are relatively recent, and are generally more robust to noise and missing information. This is because deep learning models can be trained even when some of the information records have partial information. Knowledge graphs represent the current trend in recording information in the form of relations between entities, and can provide any available information about products and users. This information is used to train the recommendation model. In this work, we present a new generic recommender systems framework, that integrates knowledge graphs into the recommendation pipeline. We describe its design and implementation, and then show through experiments, how such a framework can be specialized, taking the domain of movies as an example, and the resulting improvements in recommendations made possible by using all the information obtained using knowledge graphs. Our framework, to be made publicly available, supports different knowledge graph representation formats, and facilitates format conversion, merging and information extraction needed for training recommendation models.

We develop a high resolution face texture generation system which uses artist provided appearance controls as the conditions for a generative network. Artists are able to control various elements in the generated textures, such as the skin, eye, lip, and hair color. This is made possible by reparameterizing our dataset to the same UV mapping, allowing us to utilize image-to-image translation networks. Although our dataset is limited in size, only 126 samples in total, our system is still able to generate realistic face textures which strongly adhere to the input appearance attribute conditions because of our training augmentation methods. Once our system has generated the face texture, it is ready to be used in a modern game production environment. Thanks to our novel SuperResolution and material property recovery methods, our generated face textures are 4K resolution and have the associated material property maps required for raytraced rendering.

Point cloud neighborhoods are unstructured and often lacking in fine details, particularly when the original surface is sparsely sampled. This has motivated the development of methods for reconstructing these fine geometric features before the point cloud is converted into a mesh, usually by some form of upsampling of the point cloud. We present a novel data-driven approach to reconstructing fine details of the underlying surfaces of point clouds at the local neighborhood level, along with normals and locations of edges. This is achieved by an innovative application of recent advances in domain translation using GANs. We "translate" local neighborhoods between two domains: point cloud neighborhoods and triangular mesh neighborhoods. This allows us to obtain some of the benefits of meshes at training time, while still dealing with point clouds at the time of evaluation. By resampling the translated neighborhood, we can obtain a denser point cloud equipped with normals that allows the underlying surface to be easily reconstructed as a mesh. Our reconstructed meshes preserve fine details of the original surface better than the state of the art in point cloud upsampling techniques, even at different input resolutions. In addition, the trained GAN can generalize to operate on low resolution point clouds even without being explicitly trained on low-resolution data. We also give an example demonstrating that the same domain translation approach we use for reconstructing local neighborhood geometry can also be used to estimate a scalar field at the newly generated points, thus reducing the need for expensive recomputation of the scalar field on the dense point cloud.

Today when the number of computing systems participating in IoT is growing exponentially, the task to keep these systems secure becomes critical. One challenge is regular upgrade of IoT systems firmware to keep them up-to-date with already discovered and exploitable vulnerabilities. The other challenge is to protect the rest of the IoT ecosystem from already infected IoT devices. In this project we developed a network proxy testbed setup to be placed in-between an IoT system and the Internet - called PiNalyzer. PiNalyzer is designed to be trained to classify ingress/egress traffic and take actions to protect both: the IoT system behind it from malicious traffic from the outside; and the the rest of the network from the IoT system itself if it were compromised.

Manually creating realistic, digital human heads is a difficult and time-consuming task for artists. While 3D scanners and photogrammetry allow for quick and automatic reconstruction of heads, finding an actor who fits specific character appearance descriptions can be difficult. Moreover, modern open-world videogames feature several thousands of characters that cannot realistically all be cast and scanned. Therefore, researchers are investigating generative models to create heads fitting a specific character appearance description. While current methods are able to generate believable head shapes quite well, generating a corresponding high-resolution and high-quality texture which respects the character’s appearance description is not possible using current state of the art methods.

We propose an optimization-driven approach for automated, physics-based pattern design for tight-fitting clothing. Designing such clothing poses particular challenges since large nonlinear deformations, tight contact between cloth and body, and body deformations have to be accounted for. To address these challenges, we develop a computational model based on an embedding of the two-dimensional cloth mesh in the surface of the three-dimensional body mesh. Our Lagrangian-on-Lagrangian approach eliminates contact handling while coupling cloth and body. Building on this model, we develop a physics-driven optimization method based on sensitivity analysis that automatically computes optimal patterns according to design objectives encoding body shape, pressure distribution, seam traction, and other criteria. We demonstrate our approach by generating personalized patterns for various body shapes and a diverse set of garments with complex pattern layouts.

Processing motion capture data from optical markers for use in computer animations presents numerous technical challenges. Artifacts caused by noise, marker swaps, and marker occlusions often require manual intervention of a professionally trained marker tracking artist that spends large amounts of time and effort fixing these issues. Existing automatic solutions that attempt to fix marker data lack robustness due to either failing to properly detect and fix marker paths, or generating solutions that are challenging to integrate within current animation pipelines. In this paper, we present a method that robustly identifies invalid marker paths, removes the associated segments and generates new kinematically correct paths. We start by comparing the kinematic solutions generated by commercial software against the one generated by the state-of-the-art methods, using this information to determine which animation keyframes are invalid. Subsequently, we regenerate marker paths from the neural network based method [Holden 2018] and use a sophisticated marker filling algorithm to combine them with the original marker paths at sections where we detect the original data to be invalid. Our method outperforms alternatives by generating solutions that are both closer to the ground truth and more robust, allowing for manual intervention if required.

We complement the last three editions of the course at SIGGRAPH Asia (2015, 2016, 2018) and SIGGRAPH (2017) to make it more of a hands-on nature and include OpenISS. We explore a rapid prototyping of interactive graphical applications for stage and beyond using Jitter/Max and Processing with OpenGL, shaders, and featuring connectivity with various devices. Such rapid prototyping environment is ideal for entertainment computing, as well as for artists and live performances using real-time interactive graphics. We share the expertise we developed in connecting the real-time graphics with on-stage performance with the Illimitable Space System (ISS) v2 and its OpenISS core framework for creative near-realtime broadcasting, and the use of AI and HCI techniques in art.

Most financial institutions mainly use numerical statistics to detect anomalous (malpractice) activity. The textual content in financial records however contains precious information which to date has not been effectively used for detection of anomalous behaviors by users because these are often unintelligible, cluttered with abbreviations, numbers and symbols, which makes it difficult to build a framework system that can coherently understand and draw conclusions. Rule-based techniques have been proposed but such systems are easy to elude, as they are difficult to generalize and do not scale up. The work presented in this paper differs from previous work in that we exclusively base anomalous activities on text (excluding numerical values) in financial records and treat this as a classification problem for a deep learning network. We propose four solutions using deep learning techniques on textual data to distinguish between normal with anomalous behaviors of the users. The results of our experiments convincingly show that use of the textual content in financial records yields greater accuracy in anomalous behavior detection. They also suggest that deep learning is a viable and effective solution for real time anomaly detection by financial institutions.

Shape provides significant discriminating power in time series matching of visual or geometric data as required in many important applications in graphics and vision. The well established dynamic time warping (DTW) algorithm and its variants do this matching by determining a non-linear time mapping to minimise euclidean distances between corresponding time-warped points. However the shape of curves is not considered. In this paper, we present a new shape-aware algorithm which uses time and shape correspondence (TSC) at increasing levels of detail to define a similarity measure with an norm to aggregate the results, making it robust to noise and missing data. The norm is implicitly regularised using a shape-based error. Through extensive experiments we empirically show that our algorithm outperforms existing state of the art algorithms, works more effectively with high dimensional data, and handles noise and missing data better. We demonstrate its versatile applicability and comparative performance using a large in-house created gait data base, an action data base from Microsoft, exercise action data from a local company, a large public time series data base from University of California, Riverside and hand movement in quaternion stream data format.

We describe our experience of managing different types of artifacts between multidisciplinary teams of computer scientists and software engineers with computation and design artists while designing, developing, and deploying Illimitable Space System v2 (ISSv2) in real production environments. The artifacts include design documentation, source code, hardware and production equipment inventory, stage data, SCM and issue tracking data, git repository, and multimedia assets. These types of projects are challenging due to the nature of interdisciplinary teams and their working habits. We show how we manage the data and the teams to have enable successful public productions.

Projecting stereoscopic content onto large general outdoor surfaces, say building facades, presents many challenges to be overcome, particularly when using red-cyan anaglyph stereo representation, so that as accurate as possible colour and depth perception can still be achieved. In this paper, we address the challenges relating to long-range projection mapping of stereoscopic content in outdoor areas and present a complete framework for the automatic adjustment of the content to compensate for any adverse projection surface behaviour. We formulate the problem of modeling the projection surface into one of simultaneous recovery of shape and appearance. Our system is composed of two standard fixed cameras, a long range fixed projector, and a roving video camera for multi-view capture. The overall computational framework comprises of four modules: calibration of a long-range vision system using the structure from motion technique, dense 3D reconstruction of projection surface from calibrated camera images, modeling the light behaviour of the projection surface using roving camera images and, iterative adjustment of the stereoscopic content. In addition to cleverly adapting some of the established computer vision techniques, the system design we present is distinct from previous work. The proposed framework has been tested in real-world applications with two non-trivial user experience studies and the results reported show considerable improvements in the quality of 3D depth and colour perceived by human participants.

Abstract We present anefficient algorithm for determining an aesthetically pleasing shape boundary connecting all the points in a given unorganized set of 2D points, with no other information than point coordinates. By posing shape construction as a minimisation problem which follows the Gestalt laws, our desired shape is non‐intersecting, interpolates all points and minimizes a criterion related to these laws. The basis for our algorithm is an initial graph, an extension of the Euclidean minimum spanning tree but with no leaf nodes, called as the minimum boundary complex . and can be expressed similarly by parametrizing a topological constraint. A close approximation of , termed can be computed fast using a greedy algorithm. is then transformed into a closed interpolating boundary in two steps to satisfy ’s topological and minimization requirements. Computing exactly is an NP (Non‐Polynomial)‐hard problem, whereas is computed in linearithmic time. We present many examples showing considerable improvement over previous techniques, especially for shapes with sharp corners. Source code is available online.

ABSTRACT In this paper, we propose a new method to automatically extract keyframes from animation sequences. Our method can be applied equally to both skeletal and mesh animations. It uses animation saliency computed on the original data to help select the group of keyframes that can reconstruct the input animation with less perception error. For computational efficiency, we perform nonlinear dimension reduction using locally linear embedding and then carry out the optimal search in much lower‐dimensional space. With this approach, reconstruction of the animation from the extracted keyframes shows much better results as compared with earlier approaches. Copyright © 2012 John Wiley & Sons, Ltd.

We describe a system for generating coherent movies from a collection of unedited videos. The generation process is guided by one or more input keyframes, which determine the content of the generated video. The basic mechanism involves similarity analysis using the histogram intersection function. The function is applied to spatial pyramid histograms computed on the video frames in the collection using Dense SIFT features. A two-directional greedy path finding algorithm is used to select and arrange frames from the collection while maintaining visual similarity, coherence, and continuity. Our system demonstrates promising results on large video collections and is a first step towards increased automation in non-linear video editing.

There is an increasing interest in computer-based visualizations to provide insights and ease understandings of complex information. However, many of the proposed visualization tools/techniques seldom find real use in practice. The main reason is that they do not adequately address the issue of easing comprehensibility of underlying information. Through a case study with two static software visualization tools - SA4J (structural analysis for Java) and Creole, we investigate the comprehension support of these tools with the help of a set of comprehension criteria that assess how efficiently and effectively users are able to grasp the underlying design intent along with applied interaction mechanisms.

Fuzzy clustering is a popular method for modeling web usage data, and a number of techniques have been proposed. Performance of such techniques has been demonstrated through experiments using datasets which are often limited in the size and/or variety. This is mainly due to the difficulty in acquiring large real data, and also to the huge amount of time and effort required in performing experiments. We investigate ways to ensure dependability of such results and their analyses. For this we consider three issues. First we need to ensure that the clustering quality indices used for comparing different techniques are not biased towards any parameter specific to any of them. Second, more ground truth is provided by measuring the quality through an application of the usage model than through the clustering quality index alone. Third, given the limited data sets and experiments, use of statistical significance testing can provide more confidence in that the results obtained are not by mere chance. We present our approach for dependable performance analysis using some well-known fuzzy clustering techniques along with prediction quality used as the application specific metric.

In this paper we address the problem of distributing rendering computations for real-time display of very large 3D models using a graphics cluster. With a programmable graphics processing unit (GPU) in each node, rendering computations are increasingly carried out in two phases using two separate GPU programs: a vertex shader program for vertex (geometry) processing and a fragment shader program for pixel (color) processing. With fragment shader programs becoming more and more time consuming for increased realism and special visual effects, distributing load solely based on geometry as is done in most contemporary systems can cause significant load imbalance. There is often only a weak correlation between geometry and pixel data distribution, due to multiple factors such as occlusion of objects behind, by objects in front. Clearly, load balancing for geometry processing or pixel processing alone is not optimal. In this paper, we present a novel in-frame two-phase load-balancing technique that distributes data first for geometry and then for pixel processing. The technique is implemented on a graphics cluster and experimental results demonstrate considerable improvements in rendering performance.

Visual comprehension is the characteristic that deals with how efficiently and effectively users are able to grasp the underlying design intent along with the interactions to explore the visually represented information. To assess comprehension i.e. to measure this seemingly immeasurable factor of visualization systems, we are proposing a set of criteria based on a detailed analysis of information flow from the raw data to the cognition of information in human mind. Our comprehension criteria are adapted from the pioneering work of two eminent researchers - Donald A. Nortnan and Aaron Marcus, who have investigated the issues of human perception and cognition, and visual effectiveness respectively. These proposed criteria are refined by experts' opinion in order to compose a minimal evaluation set that is then applied to a bioinformatics visualization study tool to show the efficacy of criteria in assessing comprehension in a more quantitative manner.

The mechanism of protein folding remains largely a mystery in molecular biology, despite the enormous effort from many groups in the past decades. Currently, the protein folding mechanism is often characterized by calculating the free energy landscape versus various reaction coordinates such as the fraction of native contacts, the radius of gyration and so on. In this paper, we present an integrated approach towards understanding the folding process via visual analysis of patterns of these reaction coordinates. The three disparate processes (1) protein folding simulation, (2) pattern elicitation and (3) visualization of patterns, work in tandem. Thus as the protein folds, the changing landscape in the pattern space can be viewed via the visualization tool, PROTERAN, a program we developed for this purpose. We first present an incremental (on-line) trie-based pattern discovery algorithm to elicit the patterns and then describe the terrain metaphor based visualization tool. Using two example small proteins, a beta-hairpin and a designed protein Trp-cage, we next demonstrate that this combined pattern discovery and visualization approach extracts crucial information about protein folding intermediates and mechanism.

We present a novel approach to visualize Web usage patterns by closely coupling the visual rendering process to the data mining technique. In the first step we use relational fuzzy subtractive clustering as the mining technique to perform fuzzy clustering on Web usage sessions. In the second step, we use conventional metric multidimensional scaling to obtain an initial positional configuration in 3D space for the cluster centers, and then apply a modified Sammon mapping technique to further optimize the 3D positions. In the last step, we use the dominant membership values to assign positions to all the other sessions in the given dataset. This is computationally very efficient and at the same time retains the fidelity of the interrelationships much better. We have developed a running prototype of the proposed approach and have demonstrated the utility through experiments using several datasets, including a fairly large Web usage dataset of about 100,000 log records

A number of approaches which use model-based collaborative filtering (CF) for scalability in building recommendation systems in Web personalization have poor accuracy due to the fact that Web usage data is often sparse and noisy. Clustering, mining association rules, and sequence pattern discovery have been used to determine the access behavior model. Making use of some of the characteristics of the modeling process can provide significant improvements to recommendation effectiveness. In an earlier work, we introduced a fuzzy hybrid CF technique which inherits the advantages of both memory-based and model-based CF. In this paper, using relational fuzzy subtractive clustering as the first level modeling and then mining association rules within individual clusters, we propose a two level model-based technique, which is scalable and is an enhancement over association rule based recommender systems. Our results from comprehensive experiments using a large real life Web usage data and performance comparisons with memory-based and model-based approaches help substantiate this claim.

Abstract The task of determining common tangent lines to a pair (or more) of parametric curves has important applications in draughting systems, enveloping polygon computation, binpacking and compaction problems, and a host of other areas. In the paper, an efficient and robust algorithm to detect all common tangent lines between a pair of planar parametric curves has been presented. The algorithm uses a geometric search on the curves, and, by quickly rejecting large portions that are not likely to have a common tangent, it rapidly zeros in on the solution. The algorithm has been implemented on an IBM-compatible PC, and it works fast enough for realtime interactive use.

In this paper we introduce the potential equation that along with the rendering equation forms an adjoint system of equations and provides a mathematical frame work for all known approaches to illumination computation based on geometric optics. The potential equation is more natural for illumination computations that simulate light propagation starting from the light sources, such as progressive radiosity and particle tracing. Using the mathematical handles provided by this framework and the random-walk solution model, we present a number of importance sampling schemes for improving the computation of flux estimation. Of particular significance is the use of approximately computed potential for directing a majority of the random walks through regions of importance in the environment, thus reducing the variance in the estimates of luminous flux in these regions. Finally, results from a simple implementation are presented to demonstrate the high-efficiency improvements made possible by the use of these techniques.

Abstract An equation adjoint to the luminance equation for describing the global illumination can be formulated using the notion of a surface potential to illuminate the region of interest. This adjoint equation which we shall call as the potential equation, is fundamental to the adjoint radiosity equation used to devise the importance driven radiosity algorithm. In this paper we first briefly derive the adjoint system of integral equations and then show that the adjoint linear equations used in the above algorithm are basically discrete formulations of the same. We also show that the importance entity of the linear equations is basically the potential function integrated over a patch. Further we prove that the linear operators in the two equations are indeed transposes of each other.

Abstract This paper discusses techniques for the computation of global illumination in environments with a participating medium using a Monte Carlo simulation of the particle model of light. Efficient algorithms and data structures for tracking the particles inside the volume have been developed. The necessary equation for computing the illumination along any given direction has been derived for rendering a scene with a participating medium. A major issue in any Monte Carlo simulation is the uncertainty in the final simulation results. Various steps of the algorithm have been analysed to identify major sources of uncertainty. To reduce the uncertainty, suitable modifications to the simulation algorithm have been suggested using variance reduction methods of forced collision, absorption suppression and particle divergence. Some sample scenes showing the results of applying these methods are also included.

Abstract This paper presents a new view‐independent, energy equilibrium method for determining the light distributed in a complex 3D environment consisting of surfaces with general reflectance properties. The method does not depend on discretization of directions or discretization of surfaces to differential elements. Hence, it is a significant improvement over the earlier complete view‐independent method which is computationally intractable for complex environments or the hybrid methods which include an extended view‐dependent ray tracing second pass. The new method is based on an efficient data structure of order O(N2) called the spherical cover. The spherical cover elegantly captures the complex multidimensional directional nature of light distributed over surfaces. Subdivision techniques based on range estimation of various parameters using interval‐arithmetic‐like methods are next described for efficiently computing the spherical cover for a given 3D environment. Using the spherical cover, light is progressively propagated through the environment until energy equilibrium is reached. Complexity analysis of the propagation step is carried out to show that the method is computationally tractable. The paper also includes a comprehensive review of earlier rendering techniques viewed from the point of view of capturing the multidimensional nature of light distribution over surfaces.

Abstract The fill area primitive of GKS (Graphical Kernel System)1 is one of the more powerful features which differentiates it from earlier device independent graphics software and systems. Its specification is extremely general in the form of a closed boundary, possibly self‐intersecting, and whose interior can be filled in a variety of styles. However a complete implementation of this primitive is very complex. It is difficult to find a single graphics workstation incorporating this primitive in hardware or firmware. Most GKS implementations will have to include software for simulating the appearance of this primitive on the commonly available displays and hard‐copy graphics devices. Correct and efficient algorithms are necessary for developing this software. Because of the generality many of the existing algorithms are not directly applicable. In this paper we describe: 1. a new algorithm for clipping a fill area polygon, using what we have named as the Bridge Technique. 2. implementation of a plane sweep algorithm, by Nievergelt and Preparata,2 for solid filling and hatching, particularly applicable to vector devices. 3. extension of the plane sweep algorithm for filling with any given pattern on raster as well as vector devices. The algorithms have been designed to work for all special cases as well. In fact they have been implemented having in mind the fill area set primitive of GKS‐3D extension.3 All these algorithms have been very successfully implemented in a commercially available GKS implementation, namely indoGKS.

The brush-trajectory method, a very natural scheme for describing two-dimensional shapes used in graphic arts and typesetting applications, has been used in only a few systems largely ¢,wing to the computational complexity involved in transforming such descriptions into raster bit maps.This paper addresses the problem.For some specific brushes and trajectories we derive algebraic solutions for describing the resulting outlines.The result of dynamic transformations on the brush as it moves along the trajectory is also studied.A special closed, smooth, convex brush defined by a Jburth-order parametric equation is introduced to describe more complex shapes.An algorithmic solution to determining the outlines for an unconstrained brush is then presented.Finally, we present some ideas on a canonical brush and its use in solving the inverse problem, that is, determininl, the brushtrajectory description from given outlines.

Abstract This paper presents the development of a suite of programs for the processing of parametrically defined curves in two and three dimensions. The programs are presented in Dijkstra's non‐deterministic guarded command notation. The development of the proof of correctness for the programs is also described. Three typical curve processing tasks, such as drawing, curve path following and intersection detection, are described. The algorithms used in these programs are all based on the “divide and conquer (subdivision)” paradigm. Deviation from linearity and Euclidean bounds are two curve shape properties that are used in many of the curve processing tasks. We present fast methods of computing these properties from the algebraic representation of the curve. In particular the paper considers the cubic and the rational quadratic forms of representation.

One important requirement of a graphic design system is to be able to draw a visually pleasing curve passing through a sequence of discrete points in the Euclidean plane. These points are normally provided as input by the graphic designer or the graphic artist. Another important requirement is to be able to provide a modification technique for these curves which is simple and has control parameters which are conceptually natural to the designer or artist. This paper first reviews the curve definition as used in the METAFONT alphabet design system, discusses the drawbacks of the METAFONT curve form and then suggests a number of mathematical formulations for such curves and also the parameters for modification of these curves, which have been built into a graphic design system called Palatine Palatino has been designed and implemented at the National Centre for Software Development and Computing Techniques of the Tata Institute of Fundamental Research in Bombay, India.