iccv iccv2013 iccv2013-208 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Fan Wang, Qixing Huang, Leonidas J. Guibas
Abstract: Joint segmentation of image sets has great importance for object recognition, image classification, and image retrieval. In this paper, we aim to jointly segment a set of images starting from a small number of labeled images or none at all. To allow the images to share segmentation information with each other, we build a network that contains segmented as well as unsegmented images, and extract functional maps between connected image pairs based on image appearance features. These functional maps act as general property transporters between the images and, in particular, are used to transfer segmentations. We define and operate in a reduced functional space optimized so that the functional maps approximately satisfy cycle-consistency under composition in the network. A joint optimization framework is proposed to simultaneously generate all segmentation functions over the images so that they both align with local segmentation cues in each particular image, and agree with each other under network transportation. This formulation allows us to extract segmentations even with no training data, but can also exploit such data when available. The collective effect of the joint processing using functional maps leads to accurate information sharing among images and yields superior segmentation results, as shown on the iCoseg, MSRC, and PASCAL data sets.
Reference: text
sentIndex sentText sentNum sentScore
1 edu Abstract Joint segmentation of image sets has great importance for object recognition, image classification, and image retrieval. [sent-6, score-0.217]
2 In this paper, we aim to jointly segment a set of images starting from a small number of labeled images or none at all. [sent-7, score-0.212]
3 To allow the images to share segmentation information with each other, we build a network that contains segmented as well as unsegmented images, and extract functional maps between connected image pairs based on image appearance features. [sent-8, score-1.237]
4 These functional maps act as general property transporters between the images and, in particular, are used to transfer segmentations. [sent-9, score-0.931]
5 We define and operate in a reduced functional space optimized so that the functional maps approximately satisfy cycle-consistency under composition in the network. [sent-10, score-1.561]
6 A joint optimization framework is proposed to simultaneously generate all segmentation functions over the images so that they both align with local segmentation cues in each particular image, and agree with each other under network transportation. [sent-11, score-0.718]
7 The collective effect of the joint processing using functional maps leads to accurate information sharing among images and yields superior segmentation results, as shown on the iCoseg, MSRC, and PASCAL data sets. [sent-13, score-1.152]
8 Compared with single image segmentation, co-segmentation has the potential of aggregating information from multiple images to improve the segmentation of individual images. [sent-18, score-0.265]
9 So far this has been approached by computing point-based maps between pairs of images using local descriptors such as SIFT. [sent-20, score-0.293]
10 It alleviates the imperfections of the individual maps and the difficulty of making path — choices when transferring information between images. [sent-24, score-0.278]
11 In this paper we present a novel framework, called consistent functional maps, for representing and computing consistent appearance relations among a collection of images. [sent-25, score-0.839]
12 The proposed framework modifies the functional map framework [16] to use it on pairs of images instead of shapes, and further extends it to handle multiple images under consistency constraints. [sent-26, score-0.902]
13 The basic idea of functional maps is to equip each image with a linear functional space, and represent relations between images as linear maps between these functional spaces. [sent-27, score-2.423]
14 This functional representation is powerful because image descriptors and segmentations can be considered as functions on images, and their relations can thus be encoded as linear constraints on the linear map between the two spaces. [sent-28, score-0.902]
15 In particular, with a properly chosen basis for each functional space, optimizing functional maps between images becomes optimization of the familiar transformation matrices. [sent-29, score-1.669]
16 Most importantly, for our purposes, in a network of images connected by such functional maps, the functional setting admits of an easy approach for enforcing the consistency of these maps. [sent-31, score-1.518]
17 By introducing a latent basis for the functional space associated with each image, the consistency of the functional maps is equivalent to the fact that each functional map transforms the source image latent basis to the target image latent basis. [sent-32, score-2.618]
18 This leads to a simple formulation of the cycle-consistency constraint, enabling us to compute consistent functional maps among multiple images by solving a tractable optimization problem. [sent-33, score-1.002]
19 Given the consistent functional maps computed between 884499 pairs of images, we jointly optimize segmentations of all images so that they (i) are consistent with each other when transported via the functional maps, and (ii) agree with segmentation boundary clues presented on each image (e. [sent-34, score-2.235]
20 We note that this optimization procedure is easily modified to incorporate labeled images as input, in which case we simply let the labeled images provide additional clues for segmentation. [sent-37, score-0.296]
21 Related Work Earlier work on joint segmentation mainly compared the visual features of image pairs, such as foreground color histogram [18], SIFT [14], saliency [5], and Gabor features [9]. [sent-42, score-0.333]
22 In the supervised setting, a pool of object-like candidate segmentations were generated and a random forest regressor was trained to score each pair of segmentations [24]. [sent-47, score-0.23]
23 Recently, segmentation masks were transferred from the training windows to similar windows in test images [11], and images were jointly segmented in a energy minimization framework with multiple unary potentials [12]. [sent-50, score-0.499]
24 Functional maps are related to graph matching for feature correspondences in object categorization and image matching [4, 13, 7, 23]. [sent-51, score-0.328]
25 The edges of the graph reflect the underlying spatial structure of the image, such as region proximity, and are used to guarantee the geometric consistency of nearby regions during matching. [sent-53, score-0.175]
26 However, our framework solves the graph matching problem in a functional setting, which is fundamentally different from point-wise correspondences and leads to a linear system with an easilyobtained optimal solution for each pairwise functional map. [sent-57, score-1.387]
27 These approaches are typically formulated as solving constrained optimization problems, where the objective functions encode the score of maps, and the constraints enforce the consistency of maps along cycles. [sent-59, score-0.441]
28 However, these approaches assume that correct maps are dominant in the graph so that the small number of bad maps can be identified through their participation in many bad cycles. [sent-60, score-0.496]
29 The consistency property of functional maps is also related to diffusion maps [6] and vector diffusion maps [22]. [sent-62, score-1.491]
30 Building blocks To better explain the proposed approach, we present a brief introduction to functional maps adapted from [16] for mapping meshed 3D shapes to our super-pixel setting, and formulate the cycle-consistency constraint. [sent-76, score-0.875]
31 We formulate image segmentation as computing an indicator functions on the super885500 Figure 1: Overview of the proposed framework. [sent-79, score-0.402]
32 Any segmentation Oi ∈ Pi corresponds otof a binary oinnd Kica. [sent-87, score-0.217]
33 On the o(tph)er = = ha 1n,d∀,p any Ofunction f ∈( p F)i = =ind 0u,c∀pes a segmentation Othei o=t h{epr|f ha(pn)d > a tyi} f,u given a properly chosen threshold ti. [sent-89, score-0.217]
34 To improve efficiency, we reduce the search space of segmentation indicator functions to a subspaceFi ⊂ Fi ofdimension M < K foreachimage tIoi, spanned by a ⊂b as Fis Bi = (bi1 , · · · , biM). [sent-91, score-0.373]
35 Relations between images can be easily described as linear functional maps in the functional setting. [sent-97, score-1.529]
36 Specifically, a functional map from Fi to Fj is given by a pmecatirfiicxa Xij a∈ f RnMcti×onMa,l mwhapere f Xij maps a function f ∈ Fi with coeffi∈cieRn t vector f to the fumncatpiosn a af f? [sent-98, score-0.882]
37 In the functional setting, the cycleconsistency constraint can be described as the fact that a transported function along any loop should be identical to the original function. [sent-104, score-0.779]
38 Suppose we are given a connected directed graph G that connects some pairs of images in I. [sent-105, score-0.185]
39 L Fet C denote the space of all cycles in wG,i tthhe end gthee ( cycle consistency ecnoontset trahient s can eb oef d aellsc cryibceleds as X∀(iIki 00,·I·i 1,X·i 1·i2 ,XIi k0)i1 ∈f C =, f f ∈ Fi0. [sent-108, score-0.267]
40 This latent space is expected ,to· ·in· c,ylude functions that are consistent across multiple images, e. [sent-112, score-0.25]
41 The first stage computes a reduced functional space on each image. [sent-125, score-0.71]
42 The second stage optimizes consistent functional maps between pairs of images. [sent-126, score-0.984]
43 The objective function combines a term that quantifies the quality ofpair-wise functional maps, and another term that enforces the consistency among all functional maps. [sent-127, score-1.431]
44 Given these consistent functional maps, the final stage generates the segmentations by jointly optimizing segmentation functions that (i) align with the segmentation clues on each image and (ii) are consistent with neighboring image segmentations after transportation by functional maps. [sent-128, score-2.344]
45 Reduced Functional Spaces We choose Fi as the eigen-space spanned by the first M eigenvectors of the normalized graph Laplacian Li ∈ RK×K, motivated by some practical success of using thes∈e eigenvectors or combinations of them as segmentation indicator functions [20]. [sent-131, score-0.632]
46 An important distinction of the proposed approach from previous methods is that we do not commit to any segmentation indicator function at this stage. [sent-132, score-0.274]
47 Instead, these are jointly selected later over all input images using optimized functional maps. [sent-133, score-0.786]
48 The segmentation functions can be approximated well in the reduced eigen-space. [sent-134, score-0.365]
49 2 shows that when M = 8855 11 Approximation of (a) with 30 basis functions (18. [sent-136, score-0.172]
50 2% error); (c) Binary version of (b) by thresholding (3% error); (d) The gray lines are reconstruction errors of typical segmentation functions; the red line shows the case in (b) and (c); the blue line is the average of all cases. [sent-137, score-0.217]
51 30, the normalized error between the original segmentation function and its projection to the reduced space is usually less than 20%. [sent-138, score-0.295]
52 e2r) w, othrdes, a optimizing othre i segmentation cfeudnc ttioo n3s% i. [sent-142, score-0.254]
53 1), and how to compute consistent functional maps faogre se (a§ch5. [sent-146, score-0.924]
54 Similarity graph We compute a sparse similarity graph G for the input image ccoomllepcuttieon a I s,p arsned only compute f Gun fctoiron tahle maps bimetawgeeen c pairs oofn images specified by Gute. [sent-152, score-0.441]
55 Aligning image features When computing the functional map Xij from image Ii to image Ij, it is natural to enforce that Xij agrees with the features computed from the images. [sent-164, score-0.671]
56 In the functional setting, this is equivalent to the constraint that Xijdi ≈ dj, where di and dj are corresponding descriptor fun≈ctio dns on the two images represented in the reduced functional space. [sent-165, score-1.457]
57 Figure 3: Visualization of some probe functions that are put in correspondence by the functional map. [sent-178, score-0.763]
58 4 shows an example of functional map with and without the regularization term — the one with regularization is closer to a diagonal matrix, meaning that eigenvectors are transported only to their counterparts with similar frequencies. [sent-206, score-0.84]
59 2, we formulate the cycleconsistency constraint of functional maps by introducing a latent basis Yi for each Fi, and force each functional map Xij to transformfo rY eia icnhto F Yj . [sent-210, score-1.775]
60 We choose m = 20 for 885522 (a)(b) Figure 4: The functional map (a) with and (b) without commutativity regularization. [sent-213, score-0.671]
61 With this setup, we formulate the map consistency term as fcons = ? [sent-215, score-0.196]
62 We thus impose an additional constraint YTY = Im, where the latent basis matrix Y is simply (Y1T, ·· · , YNT)T. [sent-228, score-0.176]
63 4-6, we arrive at the following optimization problem for computing consistent functional maps: min ? [sent-233, score-0.798]
64 5 and a segmentation function transferred along a cycle is illustrated in Fig. [sent-242, score-0.378]
65 When the latent basis matrix Y is fixed, Xij can be optimized independently, i. [sent-247, score-0.177]
66 7 for solving the latent basis matrix Y becomes min trace(YTWY ) s. [sent-260, score-0.176]
67 × Figure 5: (a) Training image with ground truth segmentation; (b) test images; segmentation results transferred from (a) through the maps obtained by Eq. [sent-326, score-0.492]
68 7 without the consistency term in (c) and with the consistency term in (d). [sent-327, score-0.262]
69 Figure 6: Given a cycle of 3 images in (a), the segmentation function of the first image is transferred along the cycle. [sent-348, score-0.426]
70 The final function transferred back looks like the original one more in (c) when the maps are consistent than that in (b) when map consistency is not enforced. [sent-349, score-0.49]
71 Figure 7: Generated segmentation function (c) compared with normalized cut results (b). [sent-382, score-0.274]
72 Generating Consistent Segmentations Given the consistent functional maps {Xij }, the final stage voefn nth teh proposed approach jointly optimizes an approximate segmentation indicator function fi ∈ Fi for each image. [sent-386, score-1.408]
73 We then generate the final segmentation by rounding/binarizing fi into a segmentation indicator function. [sent-387, score-0.603]
74 Joint optimization of segmentation functions To optimize the coefficient vectors fi of segmentation functions fi = Bifi, we minimize an objective function which consists of a map consistency term fmap and a segmentation term fseg. [sent-390, score-1.431]
75 The map consistency term fmap ensures the segmentation functions are consistent with the optimized functional maps: fmap = ? [sent-391, score-1.489]
76 (11) The segmentation term sums the alignment score between each segmentation function and segmentation clues provided on each image. [sent-396, score-0.763]
77 For labeled images, we re-define Li as the normalized graph Laplacian of the graph that only connects super-pixels within the foreground or background segment. [sent-399, score-0.291]
78 In the same spirit as optimizing the latent basis Y in Eq. [sent-416, score-0.183]
79 The corresponding segmentation f ∈un Rction of each image Ii is then obtained by si = Bifi? [sent-433, score-0.217]
80 Rounding segmentation functions The continuous functions si generated above already delineate the objects in the images well. [sent-437, score-0.463]
81 To convert them into binary indicator functions, we simply sample 30 thresholds within the interval [min(si) , max(si)] uniformly, and choose the threshold whose corresponding segmentation has the smallest normalized cut score [20]. [sent-438, score-0.331]
82 Table 1 shows the accuracy of our unsupervised joint segmentation method, two other state-of-the-art unsupervised co-segmentation algorithms [10, 19], and a supervised method [24]. [sent-445, score-0.486]
83 Additionally, we also show the comparison of average accuracy with the segmentation transfer method [12] in Table 2. [sent-449, score-0.254]
84 The accuracy of our unsupervised joint segmentation method is shown in Fig. [sent-453, score-0.341]
85 Semi-supervised joint segmentation accuracy of our method is compared with [24] and [12] in Fig. [sent-459, score-0.258]
86 Discussion The functional maps are surprisingly effective in building natural correspondences between images. [sent-536, score-0.889]
87 The superior performance shows the effectiveness of the consistent functional maps and the joint optimization framework in generating robust results from the image network, despite the imperfections of individual maps. [sent-537, score-1.038]
88 Although exact pixel or region correspondences may exist, in the more general functional formulation cycle consistency can be easily enforced, yielding improved results. [sent-538, score-0.876]
89 Our framework is compared with another state-of-the-art segmentation transfer method [12], and the results are in Fig. [sent-545, score-0.254]
90 In the last column, we also include the performance of our unsupervised framework with all images in each class jointly segmented without any label information. [sent-548, score-0.254]
91 087695 0Nu1mbero20fTain3gSCBRImoahlvnp4oWdge0r−mFsG50 (a) Unsupervised (b) Supervised Figure 9: Segmentation accuracy as a function of (a) unlabeled images in the unsupervised setting and (b) labeled images in the supervised setting. [sent-552, score-0.354]
92 This suggests that, because the image set does not have much variation in object appearance, adding labeled images does not provide much additional information beyond the clues contained in the unlabeled images. [sent-554, score-0.21]
93 This improves accuracy in general because added images provide more segmentation cues to each other (Fig. [sent-558, score-0.265]
94 In the supervised case, we vary the number of training images selected from the “training set”, and evaluate the performance on a separate “validation set” of 90 images (Fig. [sent-560, score-0.182]
95 As the number of training images increases, the segmentation accuracy improves rapidly, then gradually saturates. [sent-563, score-0.289]
96 To further confirm this observation, we pretend that the foregrounds of the test images are known, and select training images based on similarities in foreground color histogram (Color-FG) and foreground shape histogram (Shape-FG [2]). [sent-566, score-0.27]
97 Conclusion We have proposed a framework for joint image segmentation, in which functional between images are jointly esti885555 classNL[12]FMaps-sFMaps-uns cpcflaoa csnwese3 3N0 07[8 13140. [sent-569, score-0.796]
98 Using the obtained functional maps, segmentation functions of all images are jointly optimized so that they are consistent under functional transport and well-aligned with each image’s own segmentation cues. [sent-592, score-2.067]
99 We believe that this approach, focussing on establishing transport mechanisms for image properties in a network setting and then using global analysis tools over the entire network, can be beneficial to other vision problems on joint analysis of image collections. [sent-597, score-0.202]
100 Functional maps: A flexible representation of maps between shapes. [sent-703, score-0.211]
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