Error Free Iterative Morphological Decomposition Algorithm for Shape Representation

Problem statement: A generalized skeleton transform allows a shape to be represented as a collection of modestly overlapped octagonal shape parts. One problem with several generalized Morphological skeleton transforms is that they generate noise after decomposition. The noise rate may not be effective for ordinary images; however this effect will be more when applied on printed or handwritten characters. Approach: The present study tackled this issue by applying a noise removal algorithm after morphological decomposition. Results: The algorithm was applied on various types of decomposition images. The Present method was compared with generalized skeleton algorithm and octagon-generating decomposition algorithm. Conclusion: The error rates with original image were evaluated using various error functions. The experimental results indicated that the present decomposition algorithm produces images with good clarity when compared with other algorithms.


INTRODUCTION
SHAPE representation is an important issue in image processing and computer vision.Efficient shape representation provides the foundation for the development of efficient algorithms for many shaperelated processing tasks, such as image coding [1,2] , shape matching and object recognition [3][4][5][6][7] , contentbased video processing [8,9] and image data retrieval [10,11] .Mathematical morphology is a shapebased approach to image processing [12,13] .Basic morphological operations can be given interpretations using geometric terms of shape, size and distance.Therefore, mathematical morphology is especially suited for handling shape-related processing and operations.Mathematical morphology also has a welldeveloped mathematical structure, which facilitates the development and analysis of morphological image processing algorithms.A number of morphological shape representation schemes have been proposed [1,2,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]30] . Manyof them use the structural approach.That is, a given shape is described in terms of its simpler shape components and the relationships among the components.
The notion of skeleton or medial axis transform was first introduced by Blum [34] .Lantuejoul showed that the skeleton can be computed using morphological operations [35] .The term 'Skeleton' is often used to describe thinning algorithms that preserve homotopy but do not necessarily support exact shape reconstruction [36,37] .In this study, our focus is on building efficient structural shape representations that allow exact as well as approximate reconstructions of the input shapes.Therefore, we are following a structural and algebraic approach to shape representation.Recently new algorithms for skeletonization and thinning, for 2D images based on primitive concept approach were proposed [31][32][33] .
The 'Morphological Skeleton Transform' (MST) is a leading morphological shape representation algorithm [14] .In the MST, a given shape is represented as a union of all maximal disks contained in the shape.In general, there is much overlapping among the maximal disks.The 'Morphological Shape Decomposition' (MSD) is another important morphological shape representation scheme [15] , in which a given shape is represented as a union of certain disks contained in the shape.The overlapping among representative disks of different sizes is eliminated.A new morphological shape representation algorithm that can be viewed as a compromise between the MST and the MSD was recently proposed [23,29] .In this scheme, overlapping among representative disks of different sizes is allowed, but severe overlapping among such disks is avoided.This algorithm is called as 'Overlapped Morphological Shape Decomposition' (OMSD).The advantages of these basic algorithms include that they have simple and well-defined mathematical characterizations and they are easy and efficient to implement.This study focuses on building efficient structural shape representations that allow exact as well as approximate reconstructions of the input shapes.

MATERIALS AND METHODS
In this study, we will first introduce an algorithm for generating skeleton points which will find a special maximal octagon for each image point of a given shape.In the decomposition algorithm a given shape will be decomposed into a collection of modestly overlapping disk components.And it will also provide enough information to allow efficient collection of disk components to be selected for exact representation of the given shape.The exact representation of the given shape is achieved through a noise removal algorithm.
An algorithm for representation of skeleton points of a given image in the form of flow chart is shown in Fig. 1.In this algorithm, the skeleton points are derived by repeatedly applying erosion operation using eight structuring elements in the following order: B , B ,...B ,B ,B ,...B , B , B ,... as shown in Fig. 2 The symbols '*','_', '+' represents origin, zero and one respectively.That is these eight structuring elements will be applied in cyclic sequence.This process of representation of skeleton points is same as 'octagongenerating decomposition algorithm.
The proposed (EFMD) Error Free Morphological Decomposition algorithm, while reconstructing the image removes noise and this is given in the form of a flow chart in Fig. 3 The proposed algorithm utilizes the number of skeleton points, in their co-ordinate positions, corresponding structuring elements and noise removal filter for reconstruction of the image.The process will be repeated for the skeleton points obtained by the algorithm to generate skeleton points shown in Fig. 1.

RESULTS
To test the integrity of the noise removal decomposition algorithm nine different images are taken and they are showed in the Fig. 4. Figure 5-7 show the reconstructed images using (GST) Generalized Skeleton Transform algorithm, (OGD) Octagon-Generating Decomposition algorithm and (EFMD) Error Free Morphological Decomposition algorithm.Various error functions as stated in equation (1-7) are applied on all reconstructed images using the three algorithms.The error rate is defined as the ratio between the number of image points that are not represented and the number of points in the original shape.(2) because it is possible to relate MSE to theoretical issues related to rate/distortion curves and least-squares minimization in statistical theory more easily than with any other measures and (3) because PSNR is a logarithmic measure which correlates with the logarithmic response to image intensity of the HVS.Generally speaking, as a rule of thumb, the higher PSNR will frequently correspond to better decompression noticeably.But the present study has tested this for reconstruction of the images.

DISCUSSION
All the error functions reflect this fact between the original image f (x, y) and the reconstructed image g(x, y) .Table 1-3 show the error rates of reconstructed images with original image using GST algorithm, OGD algorithm and the EFMD algorithm, respectively.Table 1-3 average error rate of error functions on all nine different images is evaluated.It is evident that the error rate of the present method is reduced to half or even more when compared with other two algorithms.
One more interesting point is that the error rate of the proposed method is less than the other two methods for all images by using all error functions.The PSNR is high for the present method for all images.It indicates that it has high signal to noise ratio.Figure 8-10 show reconstructed images after reversing the background color of the image for generalized skeleton algorithm, octagon generating decomposition algorithm and, proposed error free decomposition algorithm respectively.Even in this case the proposed error free reconstruction algorithm has shown less noise when compared with other algorithms.The above fact is evident from the Table 4-6.When we reverse the background of the image, the shape component is not clear with GST, due to the effect of dilation on background intensity as shown in Fig. 5.By this, error rate is increased.

Table 1 :
Error calculations using different error functions after reconstruction using generalized skeleton transform algorithm

Table 3 :
Error calculations using different error functions after reconstruction using error free decomposition algorithm

Table 5 :
Error calculations using different error functions after reconstruction using octagon-generating decomposition algorithm (image and

Table 6 :
Error calculations using different error functions after reconstruction using error free decomposition algorithm.(imageand background