An important step in the visual processing hierarchy is putting together fragments of features into coherent objects and inferring the spatial relationship between them. The feature fragments can be based on color, orientation, texture, etc. Grouping Wagemans et al. (2012a, b) refers to the mechanism by which the feature fragments are put together to form perceptual objects. Such objects in the real world may be isolated, fully occluding one another or partially occluding, depending on the observer’s viewpoint. In the context of partially occluding objects, Figure-ground organization (FGO) refers to determining which side of an occlusion boundary is the occluder, closer to the observer, referred to as figure and which side is the occluded, far away from the observer, termed as ground.
Gestalt psychologists have identified a variety of cues that mediate the process of FGO Koffka (1935)
. Based on the spatial extent of information integration, these cues can be classified into local and global cues. Global cues such as symmetryBahnsen (1928), surroundedness Palmer (1999), and size Fowlkes et al. (2007) of regions integrate information over a large spatial extent to determine figure/ground relationship between objects. Local cues, on the other hand, achieve the same by analysis of only a small neighborhood near the boundary of an object. Hence, they are particularly attractive from a computational standpoint. Some examples of local cues are T-junctions Heitger et al. (1992) and shading Huggins et al. (2001), including extremal edges Palmer and Ghose (2008); Ramenahalli et al. (2011).
The neural mechanism by which FGO is achieved in the visual cortex is an active area of research, referred to as Border Ownership (BO) coding. The contour fragments forming an object’s boundary are detected by Simple and Complex cells in the area V1 of primate visual cortex with their highly localized, retinotopically organized receptive fields. Cells in area V2, which receive input from V1 Complex cells, were found to code for BO by preferentially firing at a higher rate when the figural object was located on the preferred side of the BO coding neuron at its preferred orientation, irrespective of local contrastZhou et al. (2000). Recently, Williford and von der Heydt (2016), remarkably show for the first time, that V2 neurons maintain the same BO preference properties even for objects in complex natural scenes.
Many computational models Craft et al. (2007); Roelfsema et al. (2002); Zhaoping (2005) have been proposed to explain the neural mechanism by which FGO or BO coding is achieved in the visual cortex. Based on the connection mechanism, those models can be classified as feed-forward, feedback Mihalas et al. (2011); Hu et al. (2019) or lateral interaction models Zhaoping (2005). In this work, we present a neurally motivated, feed-forward computational model of FGO incorporating both local and global cues. While we do not attempt to exactly mimic the neural processing at every step, we attempt to keep it as biologically motivated as possible.
The FGO model we develop has three independent feature channels, Color, Intensity and Orientation. The main computational construct of the model is a BO computation mechanism that embodies Gestalt principles of convexity, surroundedness and parallelism, which is identical to all feature channels. In addition, we introduce many additional modifications to make it suitable for performing FGO and to incorporate local cues, as detailed in Section 3. The model, applicable to any natural image, is tested on the widely used BSDS figure/ground dataset. First, we show that even the model with only global cues, devoid of any local cues achieves good results on the BSDS figure/ground dataset. Let us call this the Reference model, against which we compare the performance of models with added local cues.
We add two local cues to the reference model, Spectral Anisotropy Ramenahalli et al. (2014) and T-Junctions. The motivation behind adding local cues is their relatively low computational cost compared to global cues. Spectral Anisotropy (SA) was shown to be a valid cue for FGO Ramenahalli et al. (2014, 2011, 2012) in predicting which side of an occlusion boundary is figure and which the background. Moreover, SA can be computed efficiently in a biologically plausible (See Section 4.1) manner using convolutions, making it an attractive candidate. T-Junctions are commonly viewed as one of the strongest cues of occlusion and their computation can be explained on the basis of end-stopped cells Heitger et al. (1992); Heitger and von der Heydt (1993); Hansen and Neumann (2002). This is the biological motivation to incorporate them into the model.
We have only a few FGO cues, specifically two local cues in our model. Both local cues influence the Orientation channel only as the properties they capture are more closely related to this feature. Certainly, many more local cues and global cues would be needed for best performance in real world images. But, here our primary motivation is to develop a common computational framework and investigate how these local and global cues can be incorporated into a model of FGO. Second, our purpose is to verify whether local cues can co-exist along with the global cues. If so, how useful are these local cues? Can they lead to a statistically significant improvement in the model’s performance when added alone? Finally, are these local cues mutually facilitatory leading to even further improvement, when added together? For these purposes, the minimalistic model with few global cues and even fewer local cues added to only one of the three feature channels provides an excellent analysis framework. Our goal is to study, from first principles, the effect of local and global cues in FGO, not necessarily to build a model with best performance. However, we compare the performance of our model with state of the art models of FGO, which are not biologically motivated, and show that our model performs competitively.
2 Related Work
FGO has been an active area of research in Psychology since nearly a century Rubin (1921) ago. The Gestalt principles of FGO and grouping such as common fate, symmetry, good continuation, similarity etc were formulated by Max Wertheimer Wertheimer (1923) along with Kurt Koffka Koffka (1935) and many others. Excellent reviews about the Gestalt principles of FGO and grouping can be found in Wagemans et al. (2012a, b). It is an active area of research in neuroscience Lamme (1995); Super and Lamme (2007); Zhou et al. (2000); Williford and von der Heydt (2014)
and computer visionRen et al. (2006); Hoiem et al. (2011); Teo et al. (2015) as well. We limit our literature review to computational models only. Even though the terms “FGO”, “BO” or “grouping” are not used in many publications we reviewed, the common goal in all of them is related to inferring depth ordering of objects.
A local shapeme based model employing Conditional Random Fields (CRF) to enforce global consistency at T-junctions was proposed in Ren et al. (2006). Hoiem et al. Hoiem et al. (2007, 2011) used a variety of local region, boundary, Gestalt and depth based cues in a CRF model to enforce consistency between boundary and surface labels. An optimization framework Amer et al. (2010) to obtain a 2.1D sketch by constraining the “hat” of the T-junction to be figure and “stem” to be ground was proposed, which uses human labeled contours and T-junctions. In an extension Amer et al. (2015), a reformulated optimization over regions, instead of pixels, was proposed. By using various cues such as, curve and junction potentials, convexity, lower-region, fold/cut and parallelism, Leichter and Lindenbaum (2009) train a CRF model to enforce global consistency. In a series of papers Palou and Salembier (2013, 2012, 2011) show how image segmentation and depth ordering (FGO) can be performed using only low-level cues. Their model uses Binary Partition Trees (BPT) Salembier and Garrido (2000) for hierarchically representing regions of an image, performs depth ordering by iteratively pruning the branches of BPT enforcing constraints based on T-junctions and other depth related cues. In a recent work Teo et al. (2015)
, which uses Structured Random Forests (SRF) for boundary detection, simultaneous boundary detection and figure-ground labeling is performed. They use shape cues, extremal edge basis functionsRamenahalli et al. (2011), closure, image torque Nishigaki et al. (2012) etc to train the SRFs.
Yu et al. (2001) present a hierarchical Markov Random Field (MRF) model incorporating rules for continuity of depth on surfaces, discontinuity at edges between surfaces and local cues such as T- and L-junctions. The model learns from a couple examples and effectively does depth segregation, thereby FGO. In Baek and Sajda (2005)
, a neurally plausible model integrating multiple figure-ground cues using belief propagation in Bayesian networks with leaky integrate and fire neurons was proposed. A simultaneous segmentation and figure-ground labeling algorithm was reported inMaire (2010) which uses Angular Embedding Yu (2009) to influence segmentation cues from figure-ground cues and vice-versa. Similar attempts with primary goal of segmenting images and labeling object classes using figure-ground cues can be seen in Ion et al. (2011, 2014).
Differentiation/Integration for Surface Completion (DISC) model Kogo et al. (2010) was proposed in which BO is computed by detecting local occlusion cues such as T- and L- junctions and comparing non-junction border locations with junction locations for BO consistency with the local cues. A Bayesian belief network based model was proposed Froyen et al. (2010) in which local cues (curvature and T-junctions) interact with medial axis or skeleton of the shape to determine BO.
In one of the early attempts Kienker et al. (1986), a two layer network with connections between “computational units” within and across layers is proposed. These units integrate bottom-up edge input with top-down attention input to realize FGO. Grossberg and Mingolla (1985); Grossberg (1994) propose that a reciprocal interaction between a Boundary Contour System (BCS) extracting edges and a Feature Contour System (FCS) extracting surfaces achieves not only FGO, but also 3D perception. A model of contour grouping and FGO was proposed in Heitger and von der Heydt (1993) central to which is a “grouping” mechanism. The model not only generates figure-ground labels, but also simulates the perception of illusory contours. Another influential model was proposed in Sajda and Finkel (1995) with feedback and feed-forward connections having 8 different computational modules to obtain representations of contours, surfaces and depth. Roelfsema et al. (2002); Jehee et al. (2007) propose multilayer feedback networks resembling the neural connection pattern in the visual cortex to perform BO assignment through feedback from higher areas. Li Zhaoping et al. Li (2003, 2000) propose a model of FGO based on V1 mechanisms. The model consists of orientation selective V1 neurons which influence surrounding neurons through mono-synaptic excitatory and di-synaptic suppressive connections. The excitatory lateral connections mimic colinear excitation Kapadia et al. (1995) and cross-orientation facilitation Slllito et al. (1995), while inhibitory connections model the iso-orientation suppression Knierim and Van Essen (1992). In a related model Zhaoping (2005), neurons in V2 having properties of convexity preference, good continuation and proximity was presented. A BO coding model which detects curvatures, L-Junctions and sends proportional signals to a BO layer was proposed by Kikuchi and Akashi (2001), where BO signals are propagated along the contour for two sides of BO.
The model proposed by Craft et al. (2007) consists of edge selective cells, BO cells and multi-scale grouping (G) cells. The G cells send excitatory feedback to those BO cells that are co-circular and point to the center of the annular G cell receptive field. The model incorporates Gestalt principles of convexity, proximity and closure. But, it is a feedback model tested only on simple geometric shapes, not real-world natural images. Several models Mihalas et al. (2011); Russell et al. (2014); Molin et al. (2013); Hu and Niebur (2017) with similar computational mechanisms have been proposed to explain various phenomena related to FGO, saliency, spatial attention, etc. A model akin to Craft et al. (2007) was proposed in Layton et al. (2012), where in addition to G cells the model consists of region cells at multiple scales. In a feedback model Domijan and Šetić (2008) based on the interaction between dorsal and ventral streams, surfaces which are of smaller size, greater contrast, convex, closed, having higher spatial frequency are preferentially determined as figures. The model also accounts for figure-ground cues such as lower region and top-bottom polarity. In a series of papers Sakai et al. (2012); Nishimura and Sakai (2004, 2005) Sakai and colleagues formulate a BO model in which localized, asymmetric surround modulation is used to detect contrast imbalance, which then leads to FGO. Russell et al. (2014) propose a feed-forward model with Grouping and Border Ownership cells to study proto-object based saliency. Though our model is inspired by this work, the goal of Russell et al. (2014) model is to explain the formation of proto-objects Rensink (2000) and saliency prediction, not Figure-Ground Organization. Another related model is proposed by Hu et al. (2019), which is a recurrent model with feedback connections, devoid of any local cues. To the best of our knowledge our’s is the first feed-forward model of FGO with both local and global cues. Also, this the the first such model tested on real-world images of the BSDS300 figure-ground dataset commonly used as a benchmark for FGO in natural images.
3 Model Description
The model consists of three independent features channels, Color, Intensity and Orientation. The features are computed at multiple scales to achieve scale invariance. Orientation selective V1 Simple and Complex cells Adelson and Bergen (1985) are excited by edge fragments of objects within their receptive field (Figure 1). Let us denote the contrast invariant response of a Complex cell at location by , where is the preferred orientation of the cell and is the spatial frequency. As the spatial frequency (see Table 2 for all parameters of the model) is same of all edge responsive cells in our model, except when explicitly stated otherwise (Section 4.1), we omit this variable for the most part. Each active Complex cell, activates a pair of BO cells, one with a BO preference direction, (a counter-clockwise rotation with respect to ) denoted as , and the other with BO preference, denoted as . When we talk about the BO response related to a specific figure/ground cue, be it local or global, a subscript is added to the right of the variable. For example, would be used to denote the BO response related to T-Junctions. Likewise, when specifying scale is necessary, it is denoted by superscript, . For example, denotes Complex cell response for orientation at location, and scale, . On the other hand, when we need to explicitly specify the feature we talk about, a subscript is added to the left of the variable. For example, represents the BO response for the Color feature channel. When a specific BO direction, feature, cue, scale or a location is not important, we just refer to them as, cells, cells, etc. Same applies in all such situations.
Without the influence of any local or global cues, the responses of both BO cells at a location will be equal, hence the figure direction at that location is arbitrary. The center-surround cells, denoted as cells, bring about global scene context integration by modulating the cell activity. The cells (Figure 1) extract light objects on dark background, while cells code for dark objects on light background. Without the influence of local cues, this architecture embodies the Gestalt properties of convexity, surroundedness and parallelism (global cues).
The local cues (see Section 4 for computational details of local cues) modulate cell activity additionally. Similar to cells, a pair of Spectral Anisotropy cells exist for the two opposite BO preference directions at each location, which capture local texture and shading gradients (see Section 4.1 for SA computation) on the two sides of the border. Let us denote by the cell capturing Spectral Anisotropy for BO direction, likewise for the opposite BO direction. The T-Junction cells (see Section 4.2 for computational details) also come in pairs, for the two opposite BO directions. Similar to cells, hold the T-Junction cue information for the two antagonistic BO directions, . Both these type of cells excite cells of the same BO direction and inhibit the opposite BO direction cells. For example, excites and inhibits .
The influence of cells, cells and cells on cells is controlled by a set of weights (not shown in Figure 1). Local cues are active in the Orientation channel only. The interplay of all these cues leads to the emergence of figure/ground relations strongly biased for one of the two BO directions at each location. The network architecture depicted in Figure 1 is the same computational construct that is applied at every scale, for every feature and orientation. The successive stages of model computation are explained in the following subsections.
3.1 Computation of feature channels
We consider Color, Intensity and Orientation as three independent feature channels in our model, the computation of each is described in the following sections.
3.1.1 Intensity channel
The input image consists of Red (), Blue () and Green () color channels. The intensity channel, is computed as average of the three channels, . As with all other feature channels, a multi-resolution image pyramid is constructed from the intensity channel (Section 3.2). The multi-resolution analysis allows incorporation of scale invariance into the model.
3.1.2 Color opponency channels
The color channels are first normalized by dividing each , or value by . From the normalized , , channels, four color channels, Red (), Green (), Blue () and Yellow () are computed as,
In Eq 4, the symbol, denotes absolute value.
The four opponent color channels, , , and are computed as,
3.1.3 Orientation channel
The Orientation channel is computed using the canonical model of visual cortex Adelson and Bergen (1985)
, where quadrature phase, orientation selective, Gabor kernels are used to model the V1 simple cells. The responses of Simple cells are non-linearly combined to obtain the contrast invariant, orientation selective response of the Complex cell. Mathematically, the receptive fields of even and odd symmetric Simple cells can be modeled as the cosine and sine components of a complex Gabor function - a sinusoidal carrier multiplied by a Gaussian envelope. The RF of a Simple Even cell,is given by,
where, and are the rotated coordinates,
is the standard deviation of the Gaussian envelope,is the spatial aspect ratio (controlling how elongated or circular the filter profile is), is the spatial frequency of the cell and is the preferred orientation of the simple cell. Similarly, the receptive field of a Simple Odd cell is defined as,
Simple even and odd cells responses, respectively denoted and are computed by correlating the intensity image, with the respective RF profiles. The Complex cell response, is calculated as,
Eight orientations in the range, , at intervals of are used.
3.2 Multiscale pyramid decomposition
Let us denote a feature map, be it Orientation (), Color (, , etc) or Intensity feature map, at image resolution by a common variable, . The next scale feature map, is computed by downsampling . The downsampling factor can be either
(half-octave) or 2 (full octave). Bi-linear interpolation is used to compute values in the down-sampled feature map,, which is the same interpolation scheme used in all cases of up/down sampling. Similarly, any feature map of a lower scale, is computed by downsampling the higher scale feature map, by the appropriate downsampling factor. As the numerical value of increases, the resolution of the map at that level in the pyramid decreases. The feature pyramids thus obtained are used to compute BO pyramids explained the next section.
In addition to the multiscale pyramids of independent feature channels, we compute the multiscale local cue pyramids for SA and T-Junctions as well. To denote the local cue map at a specific scale, as with feature pyramids, the scale parameter is used. For example, denotes the Spectral Anisotropy feature map for border ownership direction at scale, . Similarly T-Junction pyramids at different scales for BO directions are denoted by . The local cue pyramids are computed by successively downsampling the local cue maps at native resolution, and (see Section 4 for their computation details).
3.3 Border Ownership pyramid computation
The operations performed on any of the features ( or ) or the sub-type of features like is the same. BO responses are computed by modulating by the activity of center-surround feature differences on either sides of the border. Each feature map, , is correlated with the center-surround filters to get center-surround () difference feature pyramids. Two types of center-surround filters, (ON-center) and are defined as,
where are the standard deviations of the outer and inner Gaussian kernels respectively.
The center-surround dark pyramid, is obtained by correlating the feature maps, with the filter followed by half-wave rectification,
which detects weak/dark features surrounded by strong/light ones. In Eq 14, the symbol, denotes 2D correlation MATLAB (9 30). Similarly, to detect strong features surrounded by weak background, a pyramid is computed as,
The pyramid computation is performed this way for all feature channels except for the Orientation channel. For the Orientation feature channel, feature contrasts are not typically symmetric as in the case of other features, but oriented at a specific angle. Hence, the and filter kernels in Equations 14 and 15 are replaced by even symmetric Gabor filters, (ON-center) and (OFF-center) of opposite polarity respectively. But, in this case, different set of parameter values are used. Instead of , and used in Section 3.1.3, here we use , and respectively. The parameter values are modified in this case such that the width of the center lobe of the even Gabor filters (ON and OFF-center) matches the zero crossing diameter of the and filter kernels in Equations 14 and 15. As a result, the ON-center Gabor kernel detects bright oriented edges in a dark background, instead of symmetric feature discontinuities detected by . Similarly, the OFF-center Gabor filter detects activity of dark edges on bright backgrounds.
An important step in BO computation is normalization of the center-surround feature pyramids, and . Let be used to denote the normalization operation, which is same as the normalization used in Itti et al. (1998), but done after rescaling and pyramids to have the same range, . Similarly the local cue pyramids, and are also normalized using the same method and in the same range, . In the same way, pyramids are also normalized. This normalization step enables comparison of different features and local cues on the same scale, hence the combination of feature and local cue pyramids.
Since, we compute BO on the normalized light and dark CS pyramids, and separately and combine them at a later stage, let us denote, the corresponding BO pyramids by and respectively. We explain the BO pyramid computation for and which have a BO preference direction of . Computation of and is analogous.
Let denote the kernel responsible for mapping the object activity from normalized and
pyramids to the object edges, which is implemented with von Mises distribution. von Mises distribution is a normal distribution on a circleWeisstein (9 30). The un-normalized von Mises distribution, is defined as Russell et al. (2014),
where pixels is the radius of the circle on which the von Mises distribution is defined, is the angle at which the normal distribution is concentrated Weisstein (9 30) on the circle (also called mean direction), and is the modified Bessel function of the first kind. The distribution is then normalized as,
is computed analogously.
The BO pyramid, for light objects on dark background capturing the BO activity for direction is computed as,
Similarly, the BO pyramid for direction for a dark object on light background is obtained by correlating normalized maps with and summing the responses for all scales greater than the scale, at which BO map is being computed as,
where, is the synaptic weight for the inhibitory signal from the feature map of opposite contrast polarity. The symbol, is used to denote pixel-wise addition of responses from all scales greater than , by first up-sampling the response to the scale at which is being computed. The other two pyramids, and for the opposite BO direction are computed analogously.
With the BO pyramids related to dark and light pyramids already computed, we turn our attention to the computation of the local cue related BO pyramids. The local cue pyramids at different scales, and are constructed, as explained in Sections 4.1 and 4.2, by successively down-sampling the local cue maps computed at native image resolution. Both local cues excite cells of the same BO direction and inhibit the opposite BO direction cells.
The BO pyramid for BO direction related to the local cue, SA denoted as, is computed as,
where we can see the SA cell () having same BO preference as the BO cell, has an excitatory effect on the BO cell, but has an inhibitory effect. The synaptic weight, remains unchanged as in Eqs 18 and 19. The BO pyramid, related to SA, for opposite BO direction is computed in the same way.
The BO pyramid related to T-Junctions for the BO direction, is computed as,
The corresponding T-Junction pyramid for the opposite BO direction, , denoted as is computed analogously.
The combined BO pyramid for direction, is computed by summing global and local cue specific BO pyramids as,
where , and are weights such that , that control the contribution of , and cues to the BO response at that location respectively. By setting the weights to 0 or 1, we can study the effect of individual cue on BO response. It should be noted that the local cues are active only for the Orientation channel, so for the other channels, and will be set to zero, by default. In the absence of local cues, combination of light and dark BO pyramids (first term in Eq 22) results in contrast polarity invariant BO response. The corresponding BO pyramid for opposite BO preference, is computed as in Eq 22 by summing the light, dark and local cue BO pyramids of opposite BO preference.
Since the BO responses, , are computed for each orientation, there will be multiple BO responses active at a given pixel location. But the boundary between figure and ground can only belong to the figure side, i.e. there can only be one winning BO response for a given location. So, the winning BO response, denoted as is computed as,
where is the orientation for which absolute difference between antagonistic pair of BO responses is maximum over all orientations. This gives the edge orientation at that location. So, the winning BO pyramid, has non-zero response at a location only if the difference between the corresponding pair of BO responses for is non-negative. The winning BO pyramid, for the opposite direction is computed analogously.
Upto this point, the computation for all feature channels is identical. Now, if we denote the feature specific winning BO pyramid for direction for the Color channel by , Intensity feature channel by and Orientation feature channel by , then the final BO map, for BO direction is computed by linearly combining the up-sampled feature specific BO maps across scales as,
where represents pixel-wise addition of feature specific BO responses across scales after up-sampling each map to native resolution of the image. Similarly, is computed for BO direction. As we can see in Eq 24, the contribution of every feature channel to the final BO map is the same, i.e. , feature combination is equally weighted. Ten spatial scales () are used. All parameters of the model are summarized in Table 2. In the end, we get 16 BO maps at image resolution, 8 each for and BO directions respectively.
4 Adding local cues
Both local cues, SA and T-Junctions are computed at the native resolution of the images, but they influence BO cells of all scales as described in Eqs. 20, 21. In other words, the cues are computed once based on the analysis local image neighborhood, but their effect is not local111Should the effect of local cues also be local? See Section 7 for related discussion.
4.1 Computation of Spectral Anisotropy
Spectral Anisotropy, a local cue for FGO, that captures intensity and texture gradients very close to object boundaries, is computed by pooling Complex cell responses of various spatial frequencies from small image regions on either sides of the boundary (Figure 2). This computation is neurally/biologically plausible.
SA at any location, in the image, for a specific orientation, and for the BO direction, is computed for one side of the border (side determined by the BO direction, ) as,
where . The Complex cell response, , is computed as explained in Section 3.1.3, but with a different set of parameters, , , instead of , and respectively. The values of , , and other relevant parameters are listed in Table 1. Filter size is equal to and is the perpendicular distance between the point, at which SA is being computed and the center of the Gabor filters. The centers of even and odd symmetric Gabor filters, hence the Complex cell are all located at , from where the complex cell responses are pooled to compute SA. The term, determines the number of lobes in the Gabor filters. It is 2 or 4 for even symmetric Simple cells and 3 or 5 for odd symmetric Simple cells. The location from which Complex cell responses are pooled, is computed as,
Similarly, SA at the same location, , but for the opposite side of border at the same orientation, is computed as,
So, for every location there will be two SA cells capturing the spatial intensity and texture gradients on the two sides abutting the border. It has to be noted that the major axis orientation of the Gabor filters is the same as the local border orientation, . This is because we want to capture the variation of spectral power in a direction orthogonal to the object boundary, which is captured by the Complex cells with their orientation parallel to the object boundary. This biologically plausible computation of SA with Complex cells responses captures the anisotropic distribution of high frequency spectral power on figure side we observed in Ramenahalli et al. (2014). The SA maps thus obtained are decomposed into multiscale pyramids, , where superscript, denotes scale, by successive downsampling, which are used to compute the cue specific BO pyramids as explained in Section 3.3, Equation 20.
4.2 Detecting of T-Junctions
The object edges and the regions bound by those edges called “segments” are obtained using the gPb+ucm+OWT image segmentation algorithm Arbelaez et al. (2011), referred to as the gPb algorithm in other parts of this work. Image segmentation, partitioning of an image into disjoint regions, is considered as a pre-processing step occurring prior to FGO. The edges obtained using the gPb algorithm are represented as a Contour Map as shown in Figure 3 (B). The corresponding Segmentation Map is shown in Figure 3 (C). The Contour Map has uniquely numbered pieces of contours that appear to meet at a junction location. The Segmentation Map contains uniquely numbered disjoint regions bound by the contours. The Contour Map and Segmentation Maps are just a convenient way of representing the edge information. Only the locations at which exactly 3 distinct contours meet in the Contour Map (Figure 3 (B)) and correspondingly the locations at which exactly 3 distinct segments meet in the Segmentation Map (Figure 3 (C)) are considered for T-Junction determination. Such locations can be easily determined from the Segmentation and Contour maps.
As shown in Figure 3E and 3F, at each junction location we have three regions, , and and contours, , and meeting. At each such junction, a circular mask of pixels is applied and the corresponding small patches of the segmentation map and contour map are used for further analysis. We determine the contours forming the “hat” of the T-Junction (foreground) and the corresponding figure direction in two different ways: (1) based on the area of regions meeting at junction location within the small circular disk around junction; (2) based on the angle between contours meeting at the junction location. Finally, only those junctions locations for which figure direction, as determined based on both methods, is matching are introduced into the FGO model as T-Junction local cues. Matching based on two different methods improves the overall accuracy in correctly identifying the “hat” (foreground) and “stem” (background) of T-Junctions, in effect the correct figure direction.
The local neighborhood of T-Junction influence is set to be a circular region of radius 15 pixels. All the border pixels near the junction location within a radius of 15 pixels that belong to the “hat” of the T-Junction are set to +1 for the appropriate BO direction. Remember that for each orientation, we will have two T-Junction maps, one for the BO preference direction, denoted as and the other for the opposite BO preference, denoted as . A pixel in is set to +1 if the direction of figure, as determined by both methods (Sections 4.2.1 and 4.2.1) is , i.e. “stem” of the T-Junction is in the direction. Similarly, computed. The T-Junction maps thus obtained are decomposed into multiscale pyramids, , where superscript, denotes scale, by successive downsampling, which are used to compute the cue specific BO pyramids as explained in Section 3.3, Equation 21.
4.2.1 Area based T-Junction determination
Let , and be the three regions at a junction location (Figure 3E). After extracting the circular region around the junction by applying a circular mask of radius, 6 pixels, we count the number of pixels belonging to each of the regions, . The region, having the largest pixel count is determined as the figure region. In Figure 3E, is the region with largest pixel count, hence determined as the foreground. The contours abutting the figure region, as determined by pixel count, which are and (Figure 3F), form the “hat” of the T-Junction. Contour forms the “stem” of the T-Junction, which belongs to the background.
The local orientation at each contour location is known. Vectors of length pixels, normal to the local orientation are drawn at each “hat” contour location within the pixel neighborhood. If the normal vector intersects the figure region, , as determined based on region area, the edge/contour location is given a value of +1 in the T-Junction map for the appropriate BO direction, which can be or . This is done for every pixel in the edge/contour map within a neighborhood of 15 pixel radius around the T-Junction location for those contours ( and ) that form the “hat” of the T-Junction. For example, in Figures 3E and 3F, if the local orientation of and is roughly 0, then the end point of normal vector in the direction intersects with the figure region, , as determined based on the segment area. So, the T-Junction map for BO preference direction is set to +1 within the circular neighborhood of pixels. The T-Junction map for BO direction will be zero.
4.2.2 Angle based T-Junction determination
In this method, as in Section 4.2.1, a small circular patch of radius, 7 pixels is extracted from the contour map around T-junction location. Pixels belonging to each contour, meeting at the junction are labeled with a distinct number, so for each contour, we track the first 7 pixels starting from the junction location. Since the starting point for each contour, is the same, the total angle at junction location is . For each contour, we define a vector (red arrows in Figure 3F) from the junction location to the last tracked point on the contour. We then compute the angle between the vectors corresponding to contours. The contours between which angle is the largest form the “hat” of the T-junction. For example, in Figure 3F, is the angle between and , which is also the largest of the three angles, , and . So, in the angle based T-junction computation also, and are determined to form the “hat” of the T-junction. The figure direction at every pixel of the “hat” contours is determined as in Section 4.2.1.
Among all the potential T-Junctions determined using the angle based method, potential Y-Junctions and Arrow junctions are discarded based on the angle formed by the contours at junction location. If the largest angle is greater than , such junctions are discarded. Since the largest angle greater than is typically seen in the case of Arrow-junctions, we do not include them in the computation. Arrow junctions appear in a scene when the corner of a 3D structure is seen from outside. In the same way, if each angle at a junction location is within , such junctions are discarded as those are most likely Y-Junctions. Y-Junctions appear in a scene when a 3D corner is viewed from inside the object, for example, corner of a room viewed from inside the room. Rest of the T-Junctions are included in our computation. Angle based filtering of potential Arrow or Y-Junctions was not considered in previous methods Palou and Salembier (2011, 2013).
T-Junctions and their figure directions are determined using both Segment Area based and Contour Angle based methods and the T-Junctions are incorporated into the model only in those cases, where both methods give matching figure direction, which makes T-Junction determination more accurate.
Accurately determining the figure side of a T-junction from a small neighborhood of 6-7 pixel radius is quite challenging because, within that small neighborhood we generally do not have any information indicative of figure/ground relations, other than contour angle and segment area. Even though key point detection is a well studied area, hence locating a T-Junction is not problematic, deciding which of the three regions is the foreground based on information from a small neighborhood is extremely challenging. So, when locally determining figure side of a T-junction, segment area and contour angle were found to be the most exploitable properties.
5 Data and methods
The figure-ground dataset, a subset of BSDS 300 dataset, consists of 200 images of size pixels, where each image has two ground truth figure-ground labels Ren et al. (2006) and corresponding boundary maps. For each image, the two sets of figure-ground labels labels are annotated by users other than those who outlined the boundary maps. The figure-ground boundary consists of figure side of the boundary marked by +1 and the ground side boundary by -1.
The figure-ground classification accuracy (FGCA) for an image we report is the percentage of the total number of boundary pixels in the ground truth figure/ground label map for which a correct figure/ground classification decision is made by the model described in Section 3. Even though the model computes BO response at every location where cells are active, the BO responses are compared only at those locations for which ground truth figure/ground labels exist.
Whenever the two ground truth label maps differ for the same image, average of the FGCA for both ground truth label maps is reported. Since different figure-ground labelers interpret figure and ground sides differently depending on the context, such differences arise, as a result, the self-consistency between figure-ground labelings between the two sets of ground truth annotations is 88%, which is the maximum achievable FGCA for the dataset. At each pixel, the direction of figure, as determined by the model can be correct or wrong. So, the average FGCA for the entire dataset, at chance is 50%, assuming figure/ground relations at neighboring pixels are independent. This assumption is consistent with previously reported results Ren et al. (2006), where same assumption was made. The complete details of the figure-ground dataset can be found in Martin et al. (2001); Ren et al. (2006); Fowlkes et al. (2007).
The entire BSDS figure/ground dataset consisting of 200 images is randomly split into training set of 100 images and test set of 100 images. Parameters of the model are tuned for the training dataset and the optimal values of parameters found for the training set are used to evaluate the FGCA of the test set of images. The average FGCA that we report for the entire test set is the average of FGCAs of all 100 images in the test set.
To remind the readers, the model with only global cues of convexity, surroundedness and parallelism, without any local cues is referred to as the Reference model. As explained in Section 3, local cues, SA and T-Junctions are added to the Orientation feature channel of the reference model. As we have previously described in Section 3.3, by setting and in Eq 22, the model with local cues can be reduced to the reference model. Similarly, by switching the weights for each local cue to zero, the effect of the other local cue on FGO can be studied. As explained in Section 3.3, the winning BO pyramids are up-sampled to image resolution and summed across scales and feature channels (Eq 24) for each BO direction to get the response magnitude for that BO direction. The BO information derived this way is compared against the ground-truth from BSDS figure/ground dataset.
First, we wanted to quantify the performance of the reference model, which is devoid of both local cues, in terms of FGCA. With and , the overall FGCA for 100 test images was 58.44% (standard deviation = 0.1146). With only global cues, the 58.44% FGCA we achieved is 16.88% above chance level (50%). Hence, we can conclude that the global Gestalt properties of convexity, surroundedness and parallelism, which the reference model embodies, are important properties that are useful in FGO. The parameters used in the reference model computation are listed in Table 2. Unless stated otherwise explicitly, those parameters in Table 2 remain unchanged for the remaining set of results that we are going to discuss. Only the parameters specifically related to the addition of local cues are separately tuned and will be explicitly reported.
6.1 Effect of adding Spectral Anisotropy
As explained in Section 4.1, Spectral Anisotropy was computed at the native resolution of the image by pooling Complex cell responses at many scales for each orientation. For each orientation, , two SA maps, and are created for respective antagonistic BO directions with respect to . The SA maps are then decomposed into multiscale pyramids by successively downsampling. The SA pyramids are then incorporated into the model as explained in Eq.20 and Eq.22. In this case, parameters and are tuned for the training dataset and is set to 0.
The parameter tuning procedure we use here is the same for other cases as well. We use multi-resolution grid search for parameter tuning with the condition that the sum of tuned parameters should be 1. In this case, the condition was . We stop refining the resolution of the grid when the variation in FGCA upto second decimal point is zero, i.e. , only small changes are seen from third digit onward, after the decimal point.
The optimal parameters were found to be, and
for the training dataset. With these optimal parameter values, the FGCA for the test set was 62.69% (std. dev = 0.1204), which is a 7.3% improvement in the model’s performance after adding the local cue, Spectral Anisotropy, compared to the reference model’s FGCA of 58.44%. To verify if the improvement in FGCA that we see is statistically significant, we performed an unpaired sample, right tailed t-test (Table3
), where the null hypothesis was that the means of FGCAs of the reference model and the model with SA are equal. The alternate hypothesis was that the mean FGCA of the model with SA is higher than that of the reference model. The significance level,was chosen. For other results (Sections 6.2, 6.3) as well, we do the same type of test, where the reference model’s FGCA is compared with that of modified model’s FGCA having different local cues. Hereafter, we refer to them as statistical tests.
Statistical tests show that the mean FGCA of the model with SA is significantly higher than that of the reference model (). This demonstrates SA is a useful cue and can be successfully incorporated into the reference model, adding which results in statistically significant improvement in the model’s performance. This, and all other results are summarized in Table 3 for the test dataset.
6.2 Effect of adding T-Junctions
As described in Section 4.2, T-Junctions are computed at image resolution using the segmentation map and edge map obtained using the gPb Arbelaez et al. (2011) algorithm. Each of the T-Junction maps for the 16 different BO directions is successively downsampled to create multiscale T-Junction pyramids. The T-Junction pyramids are incorporated into the model as explained in Eq.21 and Eq.22 and by setting . The other two parameters, and are tuned on the training dataset. With optimal parameter values, and (and ), the FGCA for the test set was found to be 59.48% (std. dev. = 0.1127). Compared to the reference model’s FGCA of 58.44%, we see that adding T-Junctions improves the model’s performance in terms of FGCA by 1.78%. Based on the statistical tests (Table 3), we find that the improvement in FGCA that we see is indeed statistically significant.
6.3 Effect of adding both Spectral Anisotropy and T-Junctions
SA is computed as explained in Section 4.1, T-Junctions are computed as explained in Section 4.2, where T-Junctions are derived from automatically extracted edges using the gPb algorithm. Both cues are added to the Reference model according to Eq 22. The parameters , and are tuned simultaneously on the training dataset using multiresolution grid search as before, with the constraint, . The optimal values of the parameters were found to be, , and . All other parameters remained unchanged as shown in Table 2. The FGCA of the combined model with both local cues, Spectral Anisotropy and T-Junctions was 63.57% (std. dev = 0.1179) for the test dataset, which is higher that the FGCAs we obtained for the individual cues when they were added separately. We see an improvement in FGCA of 8.78% compared to that of the reference model with no local cues. As before, an unpaired sample, right tailed t-test comparing the reference model’s figure/ground decisions and the combined model’s figure/ground decisions with both SA and T-Junctions showed statistically significant improvement (Table 3).
In addition to comparing the performance of the model with both local cues with the Reference model, we also compared the performance of the model with both local cues (Ref model + SA + T-Junctions) to the model with only one (Ref model + SA) local cue. Unpaired sample right-tailed t-tests were used again with a significance level of 0.05. In this case the the null hypothesis is that adding T-Junctions to the Reference Model with SA does not lead to statistically significant improvement in FGCA. The alternate hypothesis is that adding T-Junctions leads to statistically significant improvement in FGCA when compared to the FGCA of Reference (global cues only) + SA model. Tests show adding T-Junctions to the Reference + SA model leads to a statistically significant improvement (p = ).
In summary, we show that both SA and T-Junctions are useful local cues of FGO, which produce statistically significant improvement in FGCA when added alone. When both cues are simultaneously present, they lead to even higher improvement in FGCA of the model indicating the cues are mutually facilitatory. An improvement of with only a few local and global cues at a minimal computational cost (see A for computational cost analysis) is truly impressive. Figures 4 and 5 show FGO results for some example images from the test dataset when both SA and T-Junctions are added.
Next, we compare the performance of our model with state of the art methods for which all steps are fully automated (see Table 4). Here, we are comparing the FGCA of the model with both local cues with other methods that are not neurally inspired, instead are learning based and trained on thousands of images. Our model performs better than that of Maire (2010) even when it has only a few local and global FGO cues and not specifically tuned for best performance. The performance of our model is competitive with the state of the art models given our constraints discussed above, but leaves room for improvement. The performance of the model can be substantially improved by making several simple modifications and adding more local and global cues as discussed in Section 9.
|M. Maire etal, ECCV, 2010 Maire (2010)||62%|
|X. Ren etal, ECCV, 2006 Ren et al. (2006)||68.9%|
|P. Salembier etal, IEEE TIP, 2013 Palou and Salembier (2013)||71.3%|
|CL. Teo, etal, CVPR 2015 Teo et al. (2015)||74.7%|
|D. Hoiem etal, ICCV 2007 Hoiem et al. (2011)||79%|
We see improvement in FGCA of the combined model with both local cues. This improvement, from only two local cues added to one of the three feature channels is truly impressive. Moreover, the three feature channels (Color, Intensity and Orientation) were weighted equally. Better results can be achieved if we tune the weights for individual feature channels. But, since our objective here was to study how to integrate local and global cues and measure the relative importance of local cues in FGO, feature specific weight tuning was not done, but we consider to do this in future (See Section 9 for future work). Moreover, it is important to note that FGCA of the model with both local cues is always higher than the FGCAs of models with individual local cues. This suggests the local cues are mutually facilitatory, which is further validated by the fact that we see statistically significant improvement in FGCA when T-Junctions are added as an additional cue to the Reference model having SA as one of the local cues.
We introduce a few novel methods in our work. First, demonstrating that Spectral Anisotropy can be computed with Simple and Complex cells found in area V1 is a novel contribution. The significance of this computation is that it demonstrates SA can be computed in low level visual areas, even in the striate cortex and it does not require specialized cells to detect these shading/texture gradients. Only a specific arrangement of Complex cells of various spatial frequencies on each side of the border is sufficient. These cues, first mathematically shown to be useful by Huggins et al. (2001), were psychophysically validated by Palmer and Ghose (2008). We showed these patterns are abundantly found in natural images Ramenahalli et al. (2011) and can be efficiently computed using 1D FFTs Ramenahalli et al. (2014). Now, we show that these cues can be computed in a biologically plausible manner, using Complex cells found commonly in striate cortex.
Next, in the detection of T-Junctions, we filter out Y-Junctions and Arrow junctions using the angle property of these junction types. Since Y-Junctions and Arrow junctions are not occlusion cues, ideally those should not be considered as T-Junctions, hence we device a method to remove such junctions. To the best of our knowledge, previous methods Palou and Salembier (2011, 2012, 2013) that use T-Junctions as FGO cues have not looked closely at this issue, which we consider novel in our approach. Also, we explicitly compute the local figure/ground relations at a T-Junction local based on local information, which is new. And, the way we organize the local cue computation such that the same computational routine can be used for incorporation of both cues into the model is noteworthy. With this, the implementation is made more efficient, allowing easy parallelization using Graphics Processing Units (GPU) and other hardware. Moreover, the combination of features and local cues is done at a late stage (Eq 22), which allows independent and parallel computation of features and local cues, which again makes the model computationally more efficient, allowing parallelization.
Even though we see FGCA improvement when T-junctions are added, it is a relatively small, but statistically significant, improvement compared to that adding SA. Since T-Junctions are generally regarded as strong cues of occlusion, this small, statistically significant improvement may seem counter-intuitive. But it is important to note that T-Junctions are extremely sparse, can be computed only at a few locations where exactly 3 different regions partially occlude each other, whereas SA can be computed at every border location of an object. Given the sparsity of T-junctions, they can still be considered stronger FGO cues compared to SA. The presence of “inverted” T-Junctions Palou and Salembier (2013, 2012, 2011), could also be the reason for diminished effect of T-Junctions. From a computational cost perspective (A), even though the cost is , given their sparsity (typically 3-10 T-Junctions per image), adding them as a local cue is justified.
Even though it is commonly assumed that T-Junctions are unambiguous cues of occlusion, no systematic, data-driven analysis of the utility of T-Junctions as a classic Gestalt cue was available until now. Moreover, there are few instances where researchers argue from the opposite perspective. Tse and Albert (1998) argue that high level surface and volume analysis takes place first, and only after such an analysis, a T-Junction is interpreted to be an occlusion cue. As a result, we may not consciously notice the prevalence of “inverted” T-Junctions.
The traditional view that T-Junctions are unambiguous cues of occlusion has also been challenged by psychophysics experiments of McDermott (2004), where they find that making occlusion decisions from a small aperture, typically a few pixels wide, in real images is hard for humans. Some studies also suggest junctions in general, hence T-Junctions, can be cues for image segmentation, but not for occlusion reasoning van der Helm (2011). These previous works and our own results do not support the generally held view that T-Junctions are the most unambiguous occlusion cues. But, these cues are useful and produce statistically significant improvement in FGCA. This is an important contribution of our work.
While comparing the performance of our model with existing methods, as noted in Section 6.3, we need to keep in mind some important differences. First, our model is not trained on image features, hence generalization to any other dataset does not require additional training. Second, our model is neurally inspired, built to provide a general framework for incorporating and studying local and global Gestalt cues, not specifically optimized for best accuracy. Moreover, we use only a handful of cues, yet perform better than some existing models (Maire (2010) in Table 4). While Maire (2010) uses 64 different shapemes, descriptors of local shape derived from object edges, Ren et al. (2006) incorporates empirical frequencies of 4 different junction types derived from training data, in addition to shapemes in a Conditional Random Field based model. Also, Ren et al. (2006) compare figure/ground relations with the ground-truth only at a partial set of locations where their edge detection algorithm finds a matching edge with the ground-truth. It is not clear what percentage of edges match with the ground-truth. Palou and Salembier (2013) use 8 color based cues, in addition to T-Junctions and local contour convexity in their model. The other two (Teo et al. (2015); Hoiem et al. (2011) ) models use a much larger number of cues to achieve FGO. Moreover, the models we are comparing with are neither strictly Gestalt cue based nor neurally motivated. To the best of our knowledge, there are no comparable neurally inspired, feed-forward, fully automated models that are tested on the BSDS figure-ground dataset. The model proposed by Sakai et al. (2012) is tested on BSDS FG database, but it requires human drawn contours.
The method we use to report FGCA can be very different from the methods of other models listed in Table 4. We report the average of all pixels for all 100 test images, which can be considerably lower than computing the FGCA image by image and then averaging the FGCA of all images. It is not clear from other methods in Table 4, how the FGCA numbers were reported. Moreover, the exact split of the dataset into train and test set also has an effect. For some splits, the FGCA can be higher. The methods reported in Table 4 may not have used the same test/train split as it is not reported in previous methods. So, instead of comparing with existing methods in terms of absolute FGCA, a more appropriate way to look at our results would be from the perspective of relative improvement after adding each cue. From this perspective, we do see statistically significant improvement with the addition of each local cue. Moreover, our motivation in this study was always to quantify the utility of local and global cues and build a general framework to incorporate and study the effect of multiple local and global cues.
Lastly, we investigate if the influence of local cues should be strictly local or global. In our model, even though the local cues, SA and T-Junctions, are computed based on the analysis of a strictly local neighborhood around the object boundary, they modulate the activity of cells at all scales, i.e. , their influence is global in nature. Should the influence of local cues be also local? To answer this question, we added local cues only at the top 2 layers of the model, tuned the optimal parameters, , and accordingly and recomputed FGCA. We found that with local cue influence at only the top two layers, the FGCA we obtained was lower than having them at all scales (See B for details). This confirms the influence of local cues should not be local, even though their computation should be strictly local to reduce the computational cost, which is the case in our model.
We develop a biologically motivated, feed-forward computational model of FGO with local and global cues. Spectral Anisotropy and T-Junctions are the local cues newly introduced into the model, which only influence the Orientation channel among the three feature channels. First, we show that even the reference model, with only a few global cues, convexity, surroundedness and parallelism, completely devoid of any local cues performs significantly better than chance level (50%) achieving a FGCA of 58.44% on the BSDS figure-ground dataset. Each local cue, when added alone leads to statistically significant improvement in the overall FGCA, compared to the reference model devoid of local cues, indicating their usefulness as independent local cues of FGO. The model with both SA and the T-Junctions achieves an 8.77% improvement in terms of FGCA compared to that of the model without any local cues. Moreover, the FGCA of the model with both local cues is always higher than that of the models with individual local cues, indicating the mutually facilitatory nature of local cues. In conclusion, SA and T-Junctions are useful, mutually beneficiary local cues and lead to statistically significant improvement in the FGCA of the feed forward, biologically motivated FGO model, either when added alone or together.
As we show in A, the computational complexity of adding both local cues is relatively low, yielding improvement in model’s performance. Given that the feature channel weights are un-optimized, model consists of only a few global and local cues, local cues added to only one of three feature channels and the model is not optimized for best FGCA222See Chapter 9 for a discussion on how FGCA of the model can be improved even with existing local cues., the performance of the model is highly impressive.
9 Future Work
In future, we intend to improve the FGCA of the model by tuning the inhibitory weight, for each feature and each local cue (Eqs. 18 – 21) and tuning feature specific weights in Eq 24. In addition, increasing the number of scales, having cells and cells of multiple radii can all lead to better FGCA. cells cells of multiple radii would capture the convexity and surroundedness cues better. Also, the model’s figure-ground response is computed by modulating the activity of cells, which are computed using Gabor filter kernels. The response of cells may not always exactly coincide with human drawn boundaries in the ground-truth, with which we compare the model’s response to calculate FGCA. Hence, averaging the BO response in a small pixel neighborhood and then comparing that with the ground-truth FG labels could yield improved FGCA. In future, we would like to explore these ideas in order to improve FGCA. Moreover, color based cues Troscianko et al. (1991); Zaidi and Li (2006), global cues such as symmetry Ardila et al. (2011) and medial axis Ardila et al. (2012) can be incorporated to improve the FGCA and make the model more robust.
In the biologically plausible SA computation 4.1, we used the Complex cell responses in all our computation. It would be interesting to see if similar or better FGCA can be achieved with Simple Even or Odd cells alone. In that case, the cost of computing SA would reduce by more than half. This would make the overall FGO model computation even more efficient. Also, in Section 4.1, filter size increment was in steps of 2 pixels. Having finer filter size resolutions (for example, instead of ) will be considered to improve the FGCA even more.
From a computational cost perspective, image segmentation using the gPb Arbelaez et al. (2011)
algorithm is the most expensive step in the FGO model with local cues. In order to decrease the computational cost, more efficient image segmentation algorithms should be explored. One efficient algorithm with similar performance as gPb (F-score,Arbelaez et al. (2011) = 0.70 vs. F-score, Leordeanu et al. (2014) = 0.69 on BSDS 500 dataset) by Leordeanu et al. (2014) is a good candidate. Replacing gPb Arbelaez et al. (2011) algorithm with the algorithm by Leordeanu et al. (2014) for image segmentation, hence T-Junction computation, can substantially reduce the computational overload, while achieving similar performance. Other recent methods with better image segmentation performance can also be considered. We would also consider parallelization of the model using GPUs and FPGAs in future.
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Appendix A Computational complexity of adding local cues
The most computationally intensive part of SA computation is the correlations involved in Eq 25, which has a computational complexity of when implemented in Fourier domain, where and are the number of rows and columns in the image.
The computationally intensive part of T-junction computation is the gPb Arbelaez et al. (2011) based image segmentation. We utilize this algorithm as is, hence we will not delve into exact estimation of computational complexity for this step. Once the contours and segmentation maps are obtained using gPb algorithm, the computation of each T-Junction using both methods described in Section 4.2.1 and Section 4.2.2 involves multiplying the edge maps, segmentation maps with masks of appropriate sizes, counting and tracking pixels, computing angles, etc, which roughly translates into a computational complexity of for both methods, where pixels for Segment Area based T-Junction computation (Section 4.2.1) and pixels for Contour Angle based T-Junction computation (Section 4.2.2). Typically
T-Junctions are found in an image. So, once edges/segmentation map is computed, since only few T-Junctions are typically present in images and the size of mask is not very large, subsequent computation is not very time consuming. With appropriate modifications, it should be possible to reduce the computational complexity of T-Junction determination even further, which is not optimized at the moment.
Appendix B Local cues influencing only top 2 layers
Should the influence of local cues also be strictly local? Local cues, by definition, should be computed based on the analysis of a small patch of an image to determine figure-ground relations. This is what makes them computationally more efficient. But, should their influence also be local? There is no a priori reason why their influence should be strictly local. To verify whether there is higher benefit in adding them locally only at the top layer (i.e. , at native image resolution only), we added them only at the top layer. For SA it resulted in a noticeable, but very small improvement. For T-Junctions, the change was barely noticeable. This could be due to extremely small size of von Mises filter kernels that we use ( pixels) in comparison with the images size ( pixels). So, we added the local cues to the top two layers. For each local cue added separately, the optimal parameters of the model were recomputed and those parameters were used to compute the FGCA. The versions of the model with local cues only at the top 2 layers did not give rise to better FGCA than what we saw earlier with the cues added at all scales. The results are summarized in Table 5.
|Model||k = 2||k = 10|
|Ref + SA||62.42%||62.69%|