raster-to-vector conversion. Each tool follows a different strategy leading to different results, but in any case when applied to digital images representing real ...
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT)
REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < case good performances but also several perceptual drawbacks when applied to digital pictures acquired by consumer devices. The paper is structured as follows. Sections II and III introduce some details of triangulation by using respectively the DDT and WBT approaches. The next section briefly reviews the heuristics used to drive the final polygonalization. Section V is devoted to the description of the Watershed approach. Section VI gives a brief description of some related commercial tools. Section VII reports some details about the experimental results together with some comparisons with other techniques. A final Section closes the paper tracking also directions for future works. II. SVGENIE A triangulation is a partition of a two-dimensional plane into a triangles set. The triangulation techniques, named Data Dependent Triangulation (DDT), try to optimize the approximation of a particular function, taking also advantage from the information obtained from the co-domain. The optimum triangulation research, by considering a fixed criterion, is a not well posed problem; however good results are obtained by considering locally optimal criterions. A triangulation is called “Optimal” if it is impossible to obtain a further improvement by swapping any edges. The algorithm introduced by Lawson ([11]) is an iterative technique able to find a locally optimal triangulation. This approach inspects all the inner edges and exchanges those that reduce the total cost of the triangulation; the algorithm iterates the process until the total cost is still unchanged. Another approach, based on Lawson’s method and developed by Yu, Morse and Sederberg ([23]), has been used as starting point to develop the proposed approach: 1. Every pixel of the image is split into two triangles by using the diagonal with the lowest cost; 2. The obtained triangulation is furthermore adapted to the image data by executing VWHS in an iterative process, which is stopped when a locally optimal triangulation is reached; 3. If the polygon is formed by adjacent triangles is convex, the following look-ahead step is achieved: a. The diagonals are swapped if the swapping introduces a reduction of the edges cost sum; b. Otherwise, for each edge ( of the polygon, with L if the exchange of ( with its opposite
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Fig. 1. Example of look-ahead " #%$ Initial configuration, " &'$ Exchange of E1 with the opposite diagonal E’1. " ($ Simultaneous exchange of E1 with E’1 and E2 with E’2.
2
diagonal ( ¶, and the simultaneous swapping of the edge ( with its opposite diagonal ( ¶ decrease the costs of the edges of the polygon and the costs of the edges ( ,…, ( of the adjacent triangles, then both exchanges are achieved (see Figure 1). To estimate the triangulation, the optimality criterion is fixed using the following cost function:
&RVW ( ( ) 31 32 31 32
(1)
where E is a common edge of two triangles and 3 D , E are the gradient of the planes 3 , that are the interpolating linear functions of the triangles. A few iterations are needed to guarantee an effective coupling between original edges distribution and related triangle vertexes map. In our case, to further speed-up the overall process we used the following approximation: FRVW H D E D E D D E E
(2)
that is cost effective. Such cost function is similar to the simplified cost function used in ([7]) to describe the roughness of triangulated terrain (Sobolev seminorm). Further investigation will be devoted to compare the cost function (2) with alternative approaches as described in([17],[23]). Therefore the global triangulation cost is summarized in: cost (7 )
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E1 D 2 E 2 D1 D 2 E1 E 2 (3)
The DDT approach aims to minimize the global triangulation cost. Three/Four iterations are usually needed to minimize the triangulation cost. Actually, after the fourth iteration the triangulation cost changes slightly introducing less improvement. III. SVGWAVE The Wavelet Based Triangulation algorithm (WBT) generates a triangulation which approximates an image, taking advantage from the characteristics given by the multiresolution analysis carried out by the discrete wavelet transformation. The wavelet transformation permits to extract the frequencies coefficients from a signal, supplying a decomposition of the signal with a hierarchical set of approximations and proper details. Due to the wavelet locality property, the decomposition coefficients provide information about the inner details of restricted regions of the image. Once the wavelet decomposition coefficients until the level O , where L «Oand O is the total number of iterations, have been obtained (e.g. using a standard bi-orthogonal model) the successive steps to generate the triangulation are: x Create a initial triangulation using predefined structures
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < and the wavelet coefficient of each level O (L «/); x For each level in the range between the first O and the final level O) iterate Q time the following steps: o Select the triangles with a higher energy detail; o Refine the selected triangles; o Adapt all the triangles, finding the good trade-off between the current error and the Delaunay’ s criterion; o Adapt all the vertices with a neighborhood low energy detail; o Decimate the shorter edges; o Decimate the unnecessary vertices; o Repeat the triangle adaptation step. The image is then subdivided into a regular region grid, taking into account the current resolution and the initial level. Each predefined model represents the better way to achieve a triangulation of a simple region, due to the presence of horizontal, vertical, diagonal or no predominant discontinuities. By using the wavelet coefficients it is possible to discriminate among the aforementioned cases building region by region the initial triangulation of the image. To improve the reconstruction quality of the image, maintaining the complexity of the triangulation, local operators are then applied. At each level the resolution of the grid is increased, and each triangle belongs to several regions (the triangle covers a region if the center of the region belongs to the triangle). The refining operation increases the number of triangles into the highly texturized regions and reduces the number of triangles into the less texturized regions. In order to achieve the refinement, the triangles containing regions with a energy detail higher than a fixed threshold,
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(2)
are subdivided. The subdivision of the triangle 7 is achieved considering the direction of the possible discontinuity: the region with the greatest energy is considered if its center belongs to the area of 7, that means it do not belongs to the boundaries; 7is then subdivided into three smaller triangles, as shown in Figure 2a. If the center of the region belongs to the boundary of the triangle 7, adjacent to the triangles 7¶, the subdivision is accomplished jointly to 7¶, in function of the common edge to which the center belongs (Figure 2b). Two triangles are joinable if the direction of the discontinuity, for both the triangles, forms a convex angle with the common edge, and this angle is higher than 0 1 S DUFFRV G 3 > 2 1 S 01 S @ , where
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Fig. 2. Examples of Triangle subdivisions. (a) Subdivision into three triangles, (b) Joined Subdivision for center belonging on boundaries.
The first adjustment operator performs exchange among the edges by using the Delaunay’ s criterion of optimality and a suitable error measure. The Delaunay’ s criterion estimates as locally optimal those triangles wherein the minimum angle present is also the maximum angle possible. As measure of the inner triangle error the Mean Square Error (MSE) is used. The MSE is calculated on each pixel of the image whose center belongs to the considered triangle. These two classes of goodness are then averaged through a weighted mean by using the parameter . For each triangle 7, considering the polygons formed with every adjacent triangles of 7, all the possible exchanges of the diagonals are evaluated and the combination with the lower cost is performed (see Fig. 1). The second adjustment operator aims to move the vertices by using the region centers as attraction points. If a vertex belongs to an area without details, it will be moved, in a fixed neighborhood, to the region center with the maximum rate between the energy detail and the square distance between the vertex itself and the center. Using this technique the vertices will be condensed into the area of the image with larger discontinuities, thus enhancing the characterization of the edges (Figure 3). F F
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Fig. 3 - Displacement GH'IJLK'M GONQPRITS9GU V GH WYX IZR[\'GON%]'M ^_U Ga`9b N%]dcaG]9V GU belonging to the red area has not enough detail energy to satisfy the displacement constraints; also in the blue area the only center with such caN%]fe7V U7Igb ]9V ehb eiV \'Gj`%U GaG]LN%]'G%W%X k9ZlmGU V GH
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Other used local operators are the edge removal operator and the vertex removal operator. An edge can be removed if it is shorter than a fixed threshold value. If an edge must be removed its vertices are removed and replaced by the middle point (Figure 4); if it is impossible to use the middle point, one of the vertices of the removed edge is used.
is a fixed coupling parameter. If both the single triangle
subdivision and the coupled triangles subdivision could not be achieved, the simple subdivision using the barycentre of the triangle is accomplished.
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Fig. 4 Example of edge removal. (a) The edge is shorter than a fixed minimum threshold; then the vertex is fixed as a new vertex for E. (b) Structure after the removal of E and substitution of the old vertices and with.
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < To achieve a further reduction of the triangulation complexity, it is possible to find and remove vertices with redundancies. By considering the intensity values of the vertices as the third spatial coordinate, each triangle is set up with a particular orientation given by the normal to its surface. If the triangulation is consistent each normal to the triangles will have outgoing directions. A vertex belonging to a set of triangles with similar orientation, in function of a fixed criterion, is redundant, therefore it can be removed. The used criterion is: Given a vertex Y which is associated to the triplet (F 1 , F 2 , F 3 ) of intensity values, it can be removed if
Yp | (
(Y p , Y) then M 1,...,3 Fr q F q d W q ,
1.
2.
(3)
where W 1 , W 2 and W 3 are the thresholds for each component of the triplet. Once the vertex has been removed a new triangulation of the polygon, formed by the union of the triangles connected to the removed vertex, must be generated (Figure 5).
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4. Fig. 5 Example of vertex removal. (a) The red vertex is a candidate to removal because of its similitude to the connected vertices. (b) Structure after vertex rejection and new triangulation.
Each adjustment and removal operation must be controlled before the execution due to possibility of lacks into the final triangulation.
The polygonalization can also be obtained by incremental fusion steps using different increasing thresholds, generating different levels of approximation: starting from zero, the values of w are increased at each fusion step until the threshold value x reaches a fixed maximum approximation target ; the difference between the values of x , into two successive iterations, is fixed by the parameter VWHS. At the same time the parameter VWHSy fixes the respective increments of the thresholds z and { ; this parameter is estimated using the following, empirically estimated, function:
Both the SVGenie and SVGWave approaches require a polygonalization step. Major details can be found in ([3], [5]). The polygonalization algorithm, starting from the derived triangulation, inspects every triangle looking for similarity with one of the adjacent polygon or triangle to achieve the fusion. The measure of similarity is achieved in terms of intensity values into a specific color space (i.e REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < polygonalization; if 71 do not belong to any polygon but 7 belongs to the polygon3 and 71 is similar to 3 then 71 is combined to 3; if 7 and 7 belong to two different polygons 3 and 3 and those polygons are similar, then 31 and 32 are combined together and 32 is removed from the polygonalization.
x x
V. SWATERG SwaterG ([4]) is a raster to vector approach based on two major steps: partitioning and contouring. The partitioning step uses watershed decomposition that has been largely used in Computer Vision both for image segmentation and classification ([22]). The algorithm returns a partition of the input image into a number (typically large) of sub-regions called “basins” with the property that each one of them is almost homogenous. First, minimal values that represent the deepest point in the “catchment basin” are computed. This is obtained evaluating the gradient of the image. For each RBG-component of the image, let
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,
,
2
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,
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position code are updated and search continues until the path returns to the first pixel (see Figure 6). VI. SVG TOOLS Some commercial and free tools have been developed for raster-to-vector conversion. Each tool follows a different strategy leading to different results, but in any case when applied to digital images representing real scenes (e.g. picture) the final results show some drawbacks. After several experiments, the best trade-off between quality and size, with respect to our case study, has been obtained using the following software: Vector Eye [21], KVec [10] and Autotrace [2]. In the following some related details for each of them are briefly reported. $ 9HFWRU(\H Vector Eye generates an SVG file using the following steps: segmentation, simplification, contour extraction, curve Starting Point
be the gradient of the layer, where
6 * , and *
,
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operator and 6[ and 6\ are Sobel Filters). In order to merge each computed level, we experimented with several metrics choosing /YQ as the most suitable in terms of efficiency and final results. Values of gradient are then normalized into the range [0,255]:
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The various level of gradient quantization can significantly reduce the well known over-segmentation problem thus obtaining a lower total number of basins. Let H be the histogram of the gradient, list of pixels are processed starting from minimum gradient values by assigning to each point in the current list a basin in the following manner: for z = 0 to #number_of_gradient_levels for each pixel p in H(z) Let m be the number of the basins in the 8-neighborhood of the pixel p, then p can be marked as •a new basin (m = 0) •as an edge (m > 1) •aggregated to an existing basin (m=1)
After the segmentation step, a chain code representation ([8]) of each basin is computed following its perimeter with a deterministic automata. Chain codes are used to represent regions by a sequence of segments with a given direction and size. To build chain code for each basin a deterministic automata has been implemented. Starting from the top-left point, the procedure looks for the next pixel of the perimeter in the neighbourhood following a clockwise direction. Both automata’ s state and vector description with the relative
&KDLQFRGH&& 69* SDWKG ´0[\KOYOYKO KY]³VW\OH ´«´!
Fig.6. Vectorial representation of basins.
approximation. Initially, the algorithm produces a draft segmentation using two threshold values: ORZBFRORUBWKUHVKROG and KLBFRORUBWKUHVKROG. These values are used to establish the minimum color difference between adjacent regions: x ORZBFRORUBWKUHVKROG (an integer value in [0, 440]) is used for regions with low details; x KLBFRORUBWKUHVKROG (an integer value greater than ORZBFRORUBWKUHVKROG and lower than 440) is the difference threshold between details. The segmentation step simplifies and prunes regions with low details having an overall area lower than an integer value PLQLPXPBDUHD. In the third step regions contours are extracted, using line segments. Finally, the overall contours are approximated through Bezier cubic curves. The parameters used are: x UHOBFXUYHILW and DEVBFXUYHILW establish a maximum error (in pixel) between contours and the approximating curve; UHOBFXUYHILW depends on the curve length, DEVBFXUYHILW is curve independent; x OLQHDUBVLPSOLI\, is used to transform curves into lines, limiting the maximum allowed error. Overall results using VectorEye are good: images appear sharp even if not very detailed, colors maintain a good quality and the PSNR value is high. Also in term of ESS (bit per pixel)
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < are on the average. % .9HF The first step performed by KVec is color quantization: the colors number is reduced using the parameter TXDQWL]H. Subsequently the image is segmented using a technique able to find connected pixel regions of similar color (with a difference lower than the value of the parameter JDSILOO). Regions containing a number of pixels lower than a threshold value JULW are pruned. The next step performs the contour tracking allowing to manage regions with a width lower than the threshold value OZLGWK as lines of suitable thickness. Contours are then approximated using Bezier curves. It is also possible to perform a sub-sampling (parameter VXEVDPSOLQJ) in order to reduce the final file size. Results are not very good: files size are too high, images are blurred, with a bad “ blocked” effect and colors are too much different from the original image. Numerically PSNR value is low. & $XWRWUDFH Autotrace performs two preliminary operations: color reduction and GHVSHFNOLQJ. Color reduction is obtained using FRORUFRXQW (an integer value in [1, 256]). 'HVSHFNOLQJ step removes tightened areas, using GHVSHFNOHWLJKWQHVV (a real value in [0, 8]). After the contour tracking, an iterative process is executed in order to approximate contours using splines (parameter HUURUWKUHVKROG establishes the maximum allowed error). Splines are then simplified using two different methods: x globally, using a single line when the limit imposed by OLQHUHYHUVLRQWKUHVKROG is respected; x locally, a single portion of a curve is substituted by a line, using the OLQHWKUHVKROG parameter. This method lead to a “ snow” effect caused by the presence of gaps between near elements. This fact penalizes the final quality both in terms of PSNR and perceptual quality. On the other hand file size is very competitive. In the next Section the overall performances of these software will be compared with our recent solutions using a meaningful dataset. VII. EXPERIMENTAL RESULTS Each triangle/polygon can be easily remapped into a SVG representation in the following way:
where x1,y1, x2,y2, x3,y3 are the vertex coordinates (xi,yi for further vertices), and #FFFFFF is the filling colour, obtained using the aforementioned equation. The generalization to RGB images is simply obtained, replacing #FFFFFF with #RRGGBB, where RR, GG, BB are respectively the hex representation of the red, green and blue mean value inside the considered triangle/polygon. All the proposed methods have been implemented in C
6
programming language. A large set of experiments have been performed in order to verify the capabilities of the proposed techniques comparing the SVG rendering with respect to some commercial tools. In particular all parameters have been manually tuned to obtain the best overall performances. Figure 7 shows a magnified details of a digital picture vectorized with our three methods compared with the SVG rendering of VectorEye, AutoTrace and Kvec. The picture shows how our results outperform other tools in terms of perceptual quality. Similar results have been observed using a large dataset composed by images from Kodak collection ([9]), classical images, text and clip-arts. While SVGenie and SVGWave tend to work granularly, introducing some “ zipper” effect, SWaterG shows good performances, mainly for digital photography. In some cases we are not able to distinguish between the original raster photography and the final SVG file. Further subjective experiments are in progress. Quantitatively experiments concerning sizes of output files both in SVG and SVGZ together with PSNR values are reported in Figures 8, 9, 10. Results show how all the methods reach satisfactory level in terms of perceptual quality and good performances with respect to final quality and overall bit rate. The different resolution size of relative sensor with respect to the viewfinder display in mobile devices is a relevant issue. Figure 11 shows SVG rendering (using SVGenie just for lack of space) of color images acquired by using an Evaluation Kit [24](STV6500 - E01 EVK 502 VGA). The first two images (Figure 11.a and 11.b) have been captured with a QVGA format (160x120 pixels) typically used in mobile phone devices; the last image (Fig. 11.c) has been captured in CGA format (320x240 pixels) typically used in PDA devices. The SVG images have been directly rendered doubling the original resolution. VIII. CONCLUSION In this paper a comparative study about different raster to vector conversion techniques has been presented. Major details, on line demo and results can be found at the following web address: http://svg.dmi.unict.it/. Future researches will be devoted to study advanced region merging heuristics, together with the use of Bezier curves, region gradient filling and filter enhancement. ACKNOWLEDGMENT Authors would like to thank STMicroelectronics – AST Catania Lab, for support and collaboration. REFERENCES [1]
[2]
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(a)
(b)
(c)
(d)
(e)
(f)
)LJXUH: A comparison between the proposed method and other methods (scale 4:1) D SVGenie, E SVGWave, F Vector Eye G SWaterG H Autotrace, I KVec.
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