A robust method for scale independent detection of curvature-based criticalities and intersections in line drawings

Highlights

• The method detects curvature based criticalities and intersections efficiently.

• The method is effective for the noisy and disconnected line drawings.

• The method is effective for line drawings with a variety of brush characteristics.

• The method is utilized to represent shapes in a design grammar interpreter.

• Features in varying scales can be detected in digital drawings.

Abstract

A novel two-phase iterative method is proposed to identify curvature based criticalities and intersections in line drawings. The method is based on evaluating a field obtained from the image via diffusion using adaptively changing ellipse-shaped analysis windows. The deviation of the average field strength values within the analysis windows from that of an external reference field is used as a metric to evaluate the criticality of a region. The external reference field is computed from an image of a straight line. The experimental results depict that the method is effective in detecting curvature based criticalities and intersections, even for noisy and disconnected drawings as well as drawings drawn with a variety of brush characteristics, such as glass, ocean, ripple, scatter, stamp, strokes effects. Our method can be employed as a part of a design grammar interpreter.

Introduction

Decomposing a whole into meaningful parts is a challenging and ill-posed problem, which many researchers have been investigating. Partitioning simplifies a form and provides a better insight into it by making its components, which are considerably simpler than the whole, explicit. A quite common approach to simplifying shapes is partitioning of shape boundaries, which are simple closed curves. In contrast to shape partitioning procedures where the parts are planar regions and the boundaries are simple closed curves, the curve partitioning procedures attempt to break a curve into a set of open curves whose boundaries are pairs of points. These points are commonly referred as critical points and, in accordance with psychophysical observations [1], they typically coincide with the locations where curve segments intersect or curvature is extremal. Therefore, the majority of the methods that deal with curve partitioning focus on developing a good approximation to curvature. In the continuous setting, the curvature at a point has an exact definition computed by analyzing the curve in an arbitrarily small interval around the point. However, for digital curves, curvature can only be estimated. There are direct methods of assigning a significance measure at each point p_i on the curve by analyzing an ordered set of neighboring points $\{p_{i-k},p_{i-k+1},\dots ,p_i,,p_{i+k-1},p_{i+k}\}$, where k determines the extent of the region of support. Note that unlike the continuous case, a region of support cannot be made arbitrarily small. Thus, one has to choose a finite size region of support, i.e., aperture. An extensive survey of direct curvature estimation methods is given in [2]. These methods typically take constant user-defined aperture. Unfortunately, using a constant predefined aperture prevents perception of features at different scales. There are guidelines in the literature that provide means of determining sizes of local regions of support by considering local curve characteristics. Many methods use relaxed indirect definitions of curvature, e.g., [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. In general relaxed definitions perform better [26], [27]. For example, relating criticality to how much a fragment of the drawing around the point deviates from a linear arrangement of points is a classic and successful idea, which is supported by psychophysical observations: Attneave [1] observed that the set of points that best represent a shape is taken from the regions where the bounding contour is most different from a straight line. This observation has been realized in both older and recent works on curve partitioning via critical points, e.g., [3], [4], [5], [6], [7], [14], [16]. The method in [4] is based on suppressing collinear points. In [19], directional flow changes in terms of chain code directions along the contour by following the 8-connected contour in clock-wise direction are used as a criticality measure. In [25], [28], bending value is defined as the difference in x and y coordinates of successive points. In [5] left and right support arms for each contour point are calculated by maximizing an objective function which results in the longest line segment with minimal deviation from the contour points. The frequency that a point appears as an end point for a support arm is used as a measure of criticality. In these methods the curve is assumed to be an ordered collection of points.

Analyzing a curve at the right scale is very important. Note that in [2], it is experimentally observed that critical point detection relies heavily on the accurate determination of the local region of support rather than the accuracy of discrete curvature estimations. Examine Fig. 1a. The contour fragments inside each window look similar to a straight line and it is difficult to differentiate them from each other. However, when the window size is increased as in Fig. 1b, the fragments inside some of the windows start to deviate from a straight line. The contour remains the same, yet how it breaks up is affected by the aperture. A constant predefined aperture cannot capture all the features that exist in various scales.

The multi-scale paradigm which has penetrated to almost every aspect of shape analysis provides a viable alternative. It makes avoiding a priori commitment to a fixed scale possible. Rosin refers to analyzing curves at their natural scale [29]. The so-called scale space methods generate a family of curves parameterized with a scale parameter and the analysis can be performed at all scales [29], [30], [31], [32], [20]. However, it generally remains unclear whether the scale refers to the local region of support (which we call aperture) or blurring of the digital curve as in maximal blurred segments in [3].

In this paper, we introduce a novel method for detecting critical points from bitmap drawings that are composed of unordered collection of pixels. We have developed our critical point detection scheme as a support to a recent design grammar interpreter [33]. It is robust against various brush types and noisy drawings that are used extensively in design grammar frameworks. A big majority of the previous methods assume that the curve is described by an ordered collection of points. This typically makes sense if the curve is a good quality boundary shape. However, due to the nature of our motivating application [33], we cannot afford to assume that the curve is a boundary shape and/or the points on it can be ordered. Our input is an iconic image which depicts a drawing with several intersections and even with possible brush effects. The method is a multi-pass (iterative) evaluation scheme, where each pass comprises a two-phase analysis of the parts of the drawing. In the first phase, candidate critical regions are determined. In the second phase, these candidate regions are evaluated in order to estimate the scale of the analysis that will be used for each candidate region at the next pass or iteration. The process stops when iterations converge. That is each critical region finds its natural scale. In both phases, a field obtained from the drawing by a diffusion process is utilized to assess the criticality of regions relative to the chosen reference field.

Our method differs from the previous ones in several ways: first of all, in our approach, concepts that define a straight-line remain implicit all the time. Rather than measuring bending or collinearity, we measure deviation from a reference “drawing” that is provided externally. Because this reference drawing happens to be straight line, the deviation correlates with curvature. Nevertheless, this brings significant advantages from a computational point of view: through the introduction of straight line reference as an external drawing, we eliminate a need to develop discrete counterparts of the continuous concepts of points being on a line, curvature, bending, etc. Furthermore, because both the analyzed drawing and the external reference are objects of the same type, i.e., images, it becomes easier to compare them. They can also be compared in a scale space. Indeed, that is what we do.

We propose a very robust way of computational handling of the scale issue, by splitting the scale concept into two. The first one is related to the determination of the size of the local region of support, i.e., aperture. The second one, on the other hand, is related to smoothing of the curve fragments. Both scales are treated as continuous and disappear at the end of calculations. For the aperture, we do not fix the size. Starting from the largest possible size, we iteratively reduce the aperture till convergence. The rate of aperture reduction varies from region to region, making the region of support a local and dynamic property computed at each pass for each candidate region. This is an important source of robustness. Furthermore, we achieve a multi-scale representation of curve fragments by embedding the entire drawing in a single field via a transform we employ from [34]. Then looking through an aperture, successive iso-intensity contours mimic smoothed forms of the curve fragment inside. This means that the aperture is not the sole mechanism to control the scale. This second scale notion is important in handling noisy drawings and brush effects. Our approach can be imagined as an improvement over the maximal blurred segments in a recent work [3]. The maximal blurred segments are points in a rectangular window. In a sense, a maximal blurred segment is a thicker curve, hence, has a reduced sensitivity to perturbations. In our case each successive iso-intensity contour define a smoothed curve. But more importantly, their collection defines a field, from which we can compute aggregate quantities for robust comparison.

To the best of our knowledge, our method is the only critical point detection method that integrates two separate notions of scale, one being related to the local region of support and the other one to the blurring of the curve.

The rest of the paper is organized as follows. In Section 2 the field representation is explained. Then the following section is devoted to the method. The implementation details are provided in Section 4 along with discussions on illustrative experimental results. Finally, in Section 5 we present a recent design grammar interpreter as our motivating application.

The research has been completed as a part of Ph.D. thesis [35].