A Fast Statistical Method for Multilevel Thresholding in Wavelet ... - arXiv
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A Fast Statistical Method for Multilevel Thresholding in Wavelet
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Domain
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Madhur Srivastava a, Prateek Katiyar a1, Yashwant Yashu a
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a2
, Satish K. Singha3, Prasanta K. Panigrahi b*
Jaypee University of Engineering & Technology, Raghogarh, Guna – 473226, Madhya Pradesh, India
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b
Indian Institute of Science Education and Research- Kolkata, Mohanpur Campus, Mohanpur - 741252, West Bengal, India
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______________________________________________________________
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ABSTRACT
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An algorithm is proposed for the segmentation of image into multiple levels using
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mean and standard deviation in the wavelet domain. The procedure provides for
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variable size segmentation with bigger block size around the mean, and having
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smaller blocks at the ends of histogram plot of each horizontal, vertical and
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diagonal components, while for the approximation component it provides for finer
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block size around the mean, and larger blocks at the ends of histogram plot
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coefficients. It is found that the proposed algorithm has significantly less time
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complexity, achieves superior PSNR and Structural Similarity Measurement Index
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as compared to similar space domain algorithms[1]. In the process it highlights
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finer image structures not perceptible in the original image. It is worth
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emphasizing that after the segmentation only 16 (at threshold level 3) wavelet
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coefficients captures the significant variation of image.
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Keywords: Discrete Wavelet Transform; Image Segmentation; Multilevel Thresholding; Histogram;
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Mean and Standard Deviation; .
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*Corresponding author, Mob. : +91 9748918201.
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E-mail addresses: [email protected], (P.K. Panigrahi).
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___________________________________________________________________________________
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1. INTRODUCTION
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Image segmentation is the process of separating the processed or unprocessed
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data into segments so that members of each segment share some common
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characteristics and macroscopically segments are different from each other. It is
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instrumental in reducing the size of the image keeping its quality maintained
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since most of the images contain redundant informations, which can be effectively
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unglued from the image. The purpose of segmentation is to distinguish a range of
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pixels having nearby values. This can be exploited to reduce the storage space,
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increase the processing speed and simplify the manipulation. Segmentation can
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also be used for object separation. It may be useful in extracting information from
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images, which are imperceptible to human eye [2].
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Thresholding is the key process for image segmentation. As thresholded images
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have many advantages over the normal ones, it has gained popularity amongst
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researchers. Thresholding can be of two types – Bi-level and Multi-level. In Bi-level
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thresholding, two values are assigned – one below the threshold level and the
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other above it. Sezgin and Sankur [3] categorized various thresholding
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techniques, based on histogram shape, clustering, entropy and object attributes.
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Otsu’s method [4] maximizes the values of class variances to get optimal
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threshold. Sahoo et al. [5] tested Otsu’s method on real images and concluded
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that the structural similarity and smoothness of reconstructed image is better
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than other methods. Processing time of the algorithm in Otsu's method was
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reduced after modification by Liao et al. [6]. In Abutaleb’s method [7], threshold
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was calculated by using 2D entropy. Niblack’s [8] method makes use of mean and
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standard deviation to follow a local approach. Hemachander et al. [9] proposed
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binarization scheme which maintains image continuity.
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In Multilevel thresholding, different values are assigned between different ranges
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of threshold levels. Reddi et al. [10] implemented Otsu’s method recursively to
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get multilevel thresholds. Ridler and Calward algorithm [11] defines one threshold
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by taking mean or any other parameter of complete image. This process is
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recursively used for the values below the threshold value and above it separately.
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Chang [12] obtained same number of classes as the number of peaks in the
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histogram by filtered the image histogram. Huang et al. [13] used Lorentz
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information measure to create an adaptive window based thresholding technique
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for uneven lightning of gray images. Boukharouba et al. [14] used the distribution
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function of the image to get multi-threshold values by specifying the zeros of a
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curvature function. For multi-threshold selection, Kittler and Illingworth [15]
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proposed a minimum error thresholding method. Papamarkos and Gatos [16] used
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hill clustering technique to get multi-threshold values which estimate the
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histogram segments
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Comparison of various meta-heuristic techniques such as genetic algorithm,
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particle swarm optimization and differential evolution for multilevel thresholding
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is done by Hammouche et al. [17].
by
taking
the
global
minima
of
rational
functions.
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Wavelet transform has become a significant tool in the field of image processing
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in recent years [18][19]. Wavelet transform of an image gives four components of
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the image – Approximation, Horizontal, Vertical and Diagonal [20]. To match the
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matrix dimension of the original image, the coefficients of image is down sampled
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by two in both horizontal and vertical directions. To decompose image further,
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wavelet transform of approximation component is taken. This can continue till
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there is only one coefficient left in approximation part [21]. In image processing,
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Discrete Wavelet Transform (DWT) is widely used in compression, segmentation
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and multi-resolution of image [22].
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In this paper a hybrid multilevel color image segmentation algorithm has been
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proposed, using mean and standard deviation in the wavelet domain. The method
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takes into account that majority of wavelet coefficients lie near to zero and
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coefficients representing large differences are a few in number lying at the
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extreme ends of histogram. Hence, the procedure provides for variable size
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segmentation, with bigger block size around the weighted mean, and having
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smaller blocks at the ends of histogram plot of each horizontal, vertical and
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diagonal components. For the approximation coefficients, values around weighted
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mean of histogram carry more information while end values of histogram are less
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significant. Hence, in approximation components segmentation is done with finer
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block size around weight mean and larger block size at the end of the histogram
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[1]. The algorithm is based on the fact that a number of distributions tends toward
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a delta function in the limit of vanishing variance. A well-known example is normal
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distribution
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In this paper, a recently established new parameter – Structural Similarity Index
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Measurement (SSIM) [23] is used to compare the structural similarity of image
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segmented by proposed algorithm and by spatial domain algorithm with the
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original image. It uses mean, variance and correlation coefficient of images to
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relate the similarity between the images.
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In section 2, illustration of approach for new hybrid algorithm is provided followed
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by algorithm in section 3. Section 4 consists of the observations seen and results
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obtained in terms of SSIM, PSNR and Time Complexity by new algorithm. Finally,
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section 5 provides the inference of the results obtained by the new algorithm.
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2. METHODOLOGY
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Keeping in mind the fact that wavelet transform is ideally suited for study of
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images because of its multi-resolution analysis ability, we implement the above
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principle in the wavelet domain and find that the proposed algorithm is superior to
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the space domain algorithm of Arora et al [1].
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Following has to be done to implement the proposed methodology. Segregate the
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colored image IRGB into its Red(IR), Green(IG) and Blue(IB). In the proposed
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methodology different approaches have been applied for approximation and detail
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coefficients of wavelet transformed image for each IR, IG and IB. The coefficients
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are divided into blocks of variable size, using weighted mean and variance of each
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sub-band of histogram of coefficients. For approximation coefficients, finer block
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size is taken around mean while broader block size at the end of histogram.
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Whereas, in case of detail coefficients thresholding is done by having broader
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block size around mean while finer block size at the end of respective histogram.
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Take inverse wavelet transform for each thresholded I R, IG and IB component.
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Reconstruct the image by concatenating IR, IG and IB components. Following section
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provides the algorithm used.
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3. ALGORITHM
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For Vertical/Horizontal/Diagonal coefficients
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1. Input n (no. of thresholds)
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2. Input f ( Vertical/Horizontal/Diagonal coefficients matrix)
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3. a = min( f ); b = max ( f );
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4. me= weighted mean f (a to b)
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5. T1 = me ; T2 = me + 0.0001 ;
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6. Repeat steps from (a) to (h) (n-1)/2 times
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(a) m1 = weighted mean f (a to T1)
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(b) m2 = weighted mean f (T2 to b )
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(c) d1 = standard deviation f (a to T1 )
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(d) d2 = standard deviation f (T2 to b )
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(e) T11 = m1 – (k1 * d1 ); T22 = m2 +( k2 * d2 ) ;
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(f) f (T11 to T1 )= weighted mean f (T11 to T1 )
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(g) f (T2 to T22 )= weighted mean f (T2 to T22 )
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(h) T1 = T11 – 0.0001; T2 = T22 + 0.0001;
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9. f (a to T1)= weighted mean f (a to T1)
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10. f (T1 to b)= weighted mean f (T1 to b)
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11.Output f (Quantized input matrix)
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For Approximation coefficients
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1. Input n (no. of thresholds)
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2. Input f ( Approximation coefficients matrix)
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3. a = min( f ); b = max ( f );
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4. me= weighted mean f ( a to b )
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5. Repeat steps from (a) to (f) (n-1)/2 times
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(a) m = weighted mean f (a to b)
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(b) d = standard deviation f (a to b)
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(c) T1 = m – (k1 * d ); T2 = m +( k2 * d ) ;
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(d) f (a to T1)= weighted mean f (a to T1)
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(e) f (T2 to b)= weighted mean f (T2 to b)
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(f) a = T1 + 0.0001; b =T2 – 0.0001;
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6. f (a to me)= weighted mean f (a to me)
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7. f (me+1 to b)= weighted mean f (me+1 to b)
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8. Output f (Quantized input matrix)
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4. RESULTS AND OBSERVATIONS
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Experiments have been performed on various images using MATLAB 7.1 on a
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system having processing speed of 1.73 GHz and 2GB RAM. The histogram plot
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shown in Fig. 1 verifies the variable segmentation with bigger block size around
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the mean while smaller blocks at the each end of histogram plot.
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174 Fig.1. (a)
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Fig.1.(b)
Fig.1.(c)
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Fig.1. Results: (a) histogram in wavelet domain
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(b) segmentation with threshold levels seven
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(c) segmentation with threshold levels nine
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From the plots in Fig. 1(b) and (c), it can be easily seen that when threshold levels
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are increased, quantization becomes finer around the ends of histogram plot. To
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vary the block size, one can choose the values of k1 and k2 accordingly. The result
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of proposed algorithm is tested on variety of images.
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In Fig. 2, original Aerial image with segmented images in space domain and by
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proposed algorithm are shown at different thresholding levels.
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Fig.2. (b)
Fig.2. (c)
Fig.2. (d)
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Fig.2. (e)
Fig.2. (f)
Fig.2. (g)
Fig.2. Results:(a) Original Aerial Image.
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(b,c,d) Segmentation in space domain at threshold level 3,5 and 7.
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(e,f,g) Segmentation by proposed algorithm at threshold level 3,5
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and 7.
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In Fig. 3, histograms of Approximation, Horizontal, Vertical and Diagonal
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coefficients of R, G and B components of original and segmented Aerial image in
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wavelet domain is depicted.
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Fig.3. (1)
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Fig.3. (3)
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Fig.3. (7)
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Fig.3. (9)
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Fig.3. (11)
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Fig.3. (13)
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Fig.3. (15)
Fig.3. (16)
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Fig.3. (19)
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Fig.3. (21)
Fig.3. (22)
Fig.3. (23)
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Fig.3. Results:(1,9,17) Histogram of Approximation coefficients of R, G and B
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components respectively in wavelet domain of Aerial
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image.
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(2,10,18) Histogram of Approximation coefficients of R, G and B
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components respectively in wavelet domain of Aerial
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image thresholded at level 3.
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(3,11,19) Histogram of Horizontal coefficients of R, G and B
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components respectively in wavelet domain of Aerial
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image.
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(4,12,20) Histogram of Horizontal coefficients of R, G and B
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components respectively in wavelet domain of Aerial
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image thresholded at level 3.
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(5,13,21) Histogram of Vertical coefficients of R, G and B
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components respectively in wavelet domain of Aerial
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image.
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(6,14,22) Histogram of Vertical coefficients of R, G and B
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components respectively in wavelet domain of Aerial
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image thresholded at level 3.
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(7,15,23) Histogram of Diagonal coefficients of R, G and B
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components respectively in wavelet domain of Aerial
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image.
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(8,16,24) Histogram of Diagonal coefficients of R, G and B
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components respectively in wavelet domain of Aerial
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image thresholded at level 3.
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Fig. 4 shows the histogram of segmented R, G and B components of Aerial image
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in wavelet domain at thresholding levels 3, 5 and 7.
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Fig.4. (a)
Fig.4. (b)
Fig.4. (c)
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Fig.4. (d)
Fig.4. (e)
Fig.4. (f)
Fig.4. (g)
Fig.4. (h)
Fig.4. (i)
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Fig.4. Results:(a,b,c) Histogram of B, G and R components thresholded in wavelet domain at level 3 of Aerial image.
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(d,e,f) Histogram of B, G and R components thresholded in wavelet
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domain at level 5 of Aerial image.
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(g,h,i) Histogram of B, G and R components thresholded in wavelet
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domain at level 7 of Aerial image.
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Table 1: Comparison of SSIM[20] of Aerial image (512 x512, 768.1 kB) between image segmentation
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in space domain and by proposed algorithm. Threshold Level 3 5 7
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SSIM in Space Domain 0.9520 0.9506 0.9505
SSIM in Wavelet Domain 0.9666 0.9676 0.9678
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Table 2: Comparison of PSNR of Aerial image (512 x512, 768.1 kB) between image segmentation in
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space domain and by proposed algorithm. Threshold Level 3 5 7
PSNR in Space Domain (dB) 22.1656 22.0107 22.0045
PSNR in Wavelet Domain (dB) 23.1395 23.4560 23.5226
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Table 3: Comparison of Time Complexity of Aerial image (512 x512, 768.1 kB) between image
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segmentation in space domain and by proposed algorithm. Threshold Level
Time Complexity in Space Time Complexity in Wavelet Domain (sec) Domain (sec) 3.0888 1.6224 3.6192 1.7628 3.9780 1.9188
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The results of Table 1, 2 and 3 are plotted in the fig. 5. The red color graph
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represents the result of space domain algorithm while the black color represents
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the results of proposed algorithm.
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Fig.5. (a)
Fig.5. (b)
Fig.5. (c)
Fig.5. Results: (a) Comparison of PSNR of Aerial image between image segmentation in space domain and by proposed algorithm. (b) Comparison of SSIM of Aerial image between image segmentation in space domain and by proposed algorithm. (c) Comparison of Time Complexity of Aerial image between image
segmentation in space domain and by proposed algorithm.
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In Fig. 6, original Earth image with segmented images in space domain and by
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proposed algorithm are shown at different thresholding levels.
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Fig.6. (b)
Fig.6. (c)
Fig.6. (d)
Fig.6. (e)
Fig.6. (f)
Fig.6. (g)
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Fig.5. Results:(a) Original Earth Image.
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(b,c,d) Segmentation in space domain at threshold level 3,5 and 7.
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(e,f,g) Segmentation by proposed algorithm at threshold level 3,5
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and 7.
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In Fig. 7, histograms of Approximation, Horizontal, Vertical and Diagonal
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coefficients of R, G and B components of original and segmented Earth image in
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wavelet domain is depicted.
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Fig.7. (1)
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Fig.7. (3)
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Fig.7. (11)
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Fig.7. (17)
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Fig.7. (22)
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Fig.7. (23)
Fig.7. (24)
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Fig.7. Results:(1,9,17) Histogram of Approximation coefficients of R, G and B
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components respectively in wavelet domain of Earth
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image.
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(2,10,18) Histogram of Approximation coefficients of R, G and B
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components respectively in wavelet domain of Earth
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image thresholded at level 3.
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(3,11,19) Histogram of Horizontal coefficients of R, G and B
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components respectively in wavelet domain of Earth
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image.
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(4,12,20) Histogram of Horizontal coefficients of R, G and B
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components respectively in wavelet domain of Earth
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image thresholded at level 3.
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(5,13,21) Histogram of Vertical coefficients of R, G and B
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components respectively in wavelet domain of Earth
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image.
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(6,14,22) Histogram of Vertical coefficients of R, G and B
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components respectively in wavelet domain of Earth
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image thresholded at level 3.
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(7,15,23) Histogram of Diagonal coefficients of R, G and B
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components respectively in wavelet domain of Earth
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image.
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(8,16,24) Histogram of Diagonal coefficients of R, G and B
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components respectively in wavelet domain of Earth
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image thresholded at level 3.
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Fig. 8 shows the histogram of segmented R, G and B components of Aerial image
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at thresholding levels 3, 5 and 7.
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Fig.8. (a)
Fig.8. (b)
Fig.8. (c)
Fig.8. (e)
Fig.8. (f)
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Fig.8. (d)
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Fig.8. (g)
Fig.8. (h)
Fig.8. (i)
Fig.8. Results:(a,b,c) Histogram of B, G and R components thresholded in wavelet domain at level 3 of Earth image.
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(d,e,f) Histogram of B, G and R components thresholded in wavelet
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domain at level 5 of Earth image.
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(g,h,i) Histogram of B, G and R components thresholded in wavelet
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domain at level 7 of Earth image.
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Table 4: Comparison of SSIM of Earth image (512 x512, 768.1 kB) between image segmentation in
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space domain and by proposed algorithm. Threshold Level 3 5 7
SSIM in Space Domain 0.9685 0.9672 0.9674
SSIM in Wavelet Domain 0.9806 0.9796 0.9797
534 535
Table 5: Comparison of PSNR of Earth image (512 x512, 768.1 kB) between image segmentation in
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space domain and by proposed algorithm. Threshold Level 3 5 7
537 538
PSNR in Space Domain (dB) 24.1114 23.8892 23.8944
PSNR in Wavelet Domain (dB) 25.9095 26.2669 26.4056
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Table 6: Comparison of Time Complexity of Earth image (512 x512, 768.1 kB) between image
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segmentation in space domain and by proposed algorithm. Threshold Level 3 5 7
Time Complexity in Space Time Complexity in Wavelet Domain (sec) Domain (sec) 3.0264 1.6692 3.6036 1.7628 3.9312 1.8564
541 542
The results of Table 4, 5 and 6 are plotted in the fig. 9. The red color graph
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represents the result of space domain algorithm while the black color represents
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the results of proposed algorithm.
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Fig.9. (a)
Fig.9. (b)
Fig.9. (c)
Fig.9. Results: (a) Comparison of PSNR of Earth image between image segmentation in space domain and by proposed algorithm. (b) Comparison of SSIM of Earth image between image
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segmentation in space domain and by proposed algorithm.
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(c) Comparison of Time Complexity of Earth image between image
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segmentation in space domain and by proposed algorithm.
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Table 7: Comparison of PSNR, SSIM and Time Complexity of various test images between image
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segmentation in space domain and by proposed algorithm. Image Name
Threshold Level
House
3 5 7
SSIM in SSIM in PSNR in PSNR in Time Time Complexity Complexity Space Wavelet Space Wavelet in Space in Wavelet Domain Domain Domain Domain Domain Domain (dB) (dB) (sec) (sec) 0.9870 0.9875 22.5739 23.3216 0.8268 0.3276 0.9846 0.9870 22.4972 23.6323 0.8892 0.3900 0.9846 0.9872 22.5652 23.7875 0.9672 0.3744
3 5 7
0.9783 0.9776 0.9776
0.9863 0.9866 0.9874
21.8314 21.6453 21.6396
23.8069 24.1578 24.3705
2.8392 3.4164 3.6972
1.7160 1.7472 1.9188
3 5 7
0.9758 0.9755 0.9754
0.9823 0.9859 0.9866
19.6385 19.5087 19.5033
21.6214 22.4168 22.5544
2.8236 3.3852 3.6504
1.6224 1.7472 1.9032
3 5 7
0.9667 0.9646 0.9645
0.9742 0.9756 0.9759
20.4878 20.3132 20.3057
21.0555 21.6169 21.7436
3.0108 3.5568 3.9312
1.6848 1.7628 1.8564
256x256 Lenna 512x512 Pepper 512x512 Baboon 512x512 561 562
The results of Table 7 are plotted in the fig. 10. The red color graph represents the
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result of space domain algorithm while the black color represents the results of
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proposed algorithm.
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Fig.10. (a)
Fig.10. (b)
Fig.10. (c)
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Fig.10. (d)
Fig.10. (e)
Fig.10. (f)
Fig.10. (g)
Fig.10. (h)
Fig.10. (i)
Fig.10. (j)
Fig.10. (k)
Fig.10. (l)
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Fig.10. Results: (a,b,c) Comparison of PSNR, MSSIM and Time Complexity of House
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image between image segmentation in space domain and
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by proposed algorithm.
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(d,e,f) Comparison of PSNR, MSSIM and Time Complexity of Lenna
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image between image segmentation in space domain and
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by proposed algorithm.
(g,h,i) Comparison of PSNR, MSSIM and Time Complexity of Pepper
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image between image segmentation in space domain and
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by proposed algorithm. (j,k,l) Comparison of PSNR, MSSIM and Time Complexity of
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Baboon image between image segmentation in space
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domain and by proposed algorithm.
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5. CONCLUSION
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The performance of the proposed hybrid algorithm has been compared with the
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algorithm reported by Arora et al.[1]. Time taken by hybrid algorithm in wavelet
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domain is approximately half of the time taken by space domain algorithm. It can
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also be seen that segmentation done in wavelet domain gives improved PSNR
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compared to segmentation done by Arora et al. at same threshold level using
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mean and standard deviation. The number of thresholds required to reach the
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saturation PSNR is far less than thresholds required in space domain. SSIM of
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image segmented in wavelet domain is always better than the image segmented
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by using Arora et al. algorithm. Finally, more distinct regions can be observed in
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an image using wavelet domain segmentation compared to space domain
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segmentation. It is worth emphasizing that after the segmentation only 4 times
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the n (number of thresholds) + 1
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variation of image.
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[8] W. Niblack, An Introduction to Digital Image Processing, Prentice Hall, 1986 ,
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Method, IEEE Trans. Systems, Man and Cybernetics, 8 (1978) 630-632.
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C.C. Chang, L.L. Wang, A fast multilevel thresholding method based on
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lowpass and highpass filtering, Pattern Recognition Letters 18(1977), 1469-1478.
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Q. Huang, W. Gai, W. Cai, Thresholding technique with adaptive window
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selection for uneven lighting image. Pattern Recognition Letters 26(2005),801-
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808.
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S. Boukharouba, J.M. Rebordao, P.L. Wendel, An amplitude segmentation
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method based on the distribution function of an image. Comput. Vision Graphics
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Image Process. 29(1985), 47-59.
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J. Kittler, J. Illingworth, Minimum Error Thresholding, Pattern Recognition,
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Graphics Models Image Process. 56(1994), 357-370.
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[17] K. Hammouche, M. Diaf, P. Siarry, A comparative study of various meta-
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[18] Sachin P Nanavati and Prasanta K Panigrahi, 2005. Wavelets: Applications to
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New York Univ., NY, USA, 13-4(2004), 600-612.
693 694
695
A Fast Statistical Method for Multilevel Thresholding in Wavelet
2
Domain
3
Madhur Srivastava a, Prateek Katiyar a1, Yashwant Yashu a
4
a2
, Satish K. Singha3, Prasanta K. Panigrahi b*
Jaypee University of Engineering & Technology, Raghogarh, Guna – 473226, Madhya Pradesh, India
5 6
b
Indian Institute of Science Education and Research- Kolkata, Mohanpur Campus, Mohanpur - 741252, West Bengal, India
8
______________________________________________________________
9
ABSTRACT
10
An algorithm is proposed for the segmentation of image into multiple levels using
11
mean and standard deviation in the wavelet domain. The procedure provides for
12
variable size segmentation with bigger block size around the mean, and having
13
smaller blocks at the ends of histogram plot of each horizontal, vertical and
14
diagonal components, while for the approximation component it provides for finer
15
block size around the mean, and larger blocks at the ends of histogram plot
16
coefficients. It is found that the proposed algorithm has significantly less time
17
complexity, achieves superior PSNR and Structural Similarity Measurement Index
18
as compared to similar space domain algorithms[1]. In the process it highlights
19
finer image structures not perceptible in the original image. It is worth
20
emphasizing that after the segmentation only 16 (at threshold level 3) wavelet
21
coefficients captures the significant variation of image.
22
Keywords: Discrete Wavelet Transform; Image Segmentation; Multilevel Thresholding; Histogram;
23
Mean and Standard Deviation; .
24
*Corresponding author, Mob. : +91 9748918201.
25
E-mail addresses: [email protected], (P.K. Panigrahi).
26
___________________________________________________________________________________
27 28
1. INTRODUCTION
29 30
Image segmentation is the process of separating the processed or unprocessed
31
data into segments so that members of each segment share some common
32
characteristics and macroscopically segments are different from each other. It is
33
instrumental in reducing the size of the image keeping its quality maintained
34
since most of the images contain redundant informations, which can be effectively
35
unglued from the image. The purpose of segmentation is to distinguish a range of
36
pixels having nearby values. This can be exploited to reduce the storage space,
37
increase the processing speed and simplify the manipulation. Segmentation can
38
also be used for object separation. It may be useful in extracting information from
39
images, which are imperceptible to human eye [2].
40 41
Thresholding is the key process for image segmentation. As thresholded images
42
have many advantages over the normal ones, it has gained popularity amongst
43
researchers. Thresholding can be of two types – Bi-level and Multi-level. In Bi-level
44
thresholding, two values are assigned – one below the threshold level and the
45
other above it. Sezgin and Sankur [3] categorized various thresholding
46
techniques, based on histogram shape, clustering, entropy and object attributes.
47
Otsu’s method [4] maximizes the values of class variances to get optimal
48
threshold. Sahoo et al. [5] tested Otsu’s method on real images and concluded
49
that the structural similarity and smoothness of reconstructed image is better
50
than other methods. Processing time of the algorithm in Otsu's method was
51
reduced after modification by Liao et al. [6]. In Abutaleb’s method [7], threshold
52
was calculated by using 2D entropy. Niblack’s [8] method makes use of mean and
53
standard deviation to follow a local approach. Hemachander et al. [9] proposed
54
binarization scheme which maintains image continuity.
55 56
In Multilevel thresholding, different values are assigned between different ranges
57
of threshold levels. Reddi et al. [10] implemented Otsu’s method recursively to
58
get multilevel thresholds. Ridler and Calward algorithm [11] defines one threshold
59
by taking mean or any other parameter of complete image. This process is
60
recursively used for the values below the threshold value and above it separately.
61
Chang [12] obtained same number of classes as the number of peaks in the
62
histogram by filtered the image histogram. Huang et al. [13] used Lorentz
63
information measure to create an adaptive window based thresholding technique
64
for uneven lightning of gray images. Boukharouba et al. [14] used the distribution
65
function of the image to get multi-threshold values by specifying the zeros of a
66
curvature function. For multi-threshold selection, Kittler and Illingworth [15]
67
proposed a minimum error thresholding method. Papamarkos and Gatos [16] used
68
hill clustering technique to get multi-threshold values which estimate the
69
histogram segments
70
Comparison of various meta-heuristic techniques such as genetic algorithm,
71
particle swarm optimization and differential evolution for multilevel thresholding
72
is done by Hammouche et al. [17].
by
taking
the
global
minima
of
rational
functions.
73 74
Wavelet transform has become a significant tool in the field of image processing
75
in recent years [18][19]. Wavelet transform of an image gives four components of
76
the image – Approximation, Horizontal, Vertical and Diagonal [20]. To match the
77
matrix dimension of the original image, the coefficients of image is down sampled
78
by two in both horizontal and vertical directions. To decompose image further,
79
wavelet transform of approximation component is taken. This can continue till
80
there is only one coefficient left in approximation part [21]. In image processing,
81
Discrete Wavelet Transform (DWT) is widely used in compression, segmentation
82
and multi-resolution of image [22].
83 84
In this paper a hybrid multilevel color image segmentation algorithm has been
85
proposed, using mean and standard deviation in the wavelet domain. The method
86
takes into account that majority of wavelet coefficients lie near to zero and
87
coefficients representing large differences are a few in number lying at the
88
extreme ends of histogram. Hence, the procedure provides for variable size
89
segmentation, with bigger block size around the weighted mean, and having
90
smaller blocks at the ends of histogram plot of each horizontal, vertical and
91
diagonal components. For the approximation coefficients, values around weighted
92
mean of histogram carry more information while end values of histogram are less
93
significant. Hence, in approximation components segmentation is done with finer
94
block size around weight mean and larger block size at the end of the histogram
95
[1]. The algorithm is based on the fact that a number of distributions tends toward
96
a delta function in the limit of vanishing variance. A well-known example is normal
97
distribution
98 99 100
In this paper, a recently established new parameter – Structural Similarity Index
101
Measurement (SSIM) [23] is used to compare the structural similarity of image
102
segmented by proposed algorithm and by spatial domain algorithm with the
103
original image. It uses mean, variance and correlation coefficient of images to
104
relate the similarity between the images.
105 106
In section 2, illustration of approach for new hybrid algorithm is provided followed
107
by algorithm in section 3. Section 4 consists of the observations seen and results
108
obtained in terms of SSIM, PSNR and Time Complexity by new algorithm. Finally,
109
section 5 provides the inference of the results obtained by the new algorithm.
110 111
2. METHODOLOGY
112 113
Keeping in mind the fact that wavelet transform is ideally suited for study of
114
images because of its multi-resolution analysis ability, we implement the above
115
principle in the wavelet domain and find that the proposed algorithm is superior to
116
the space domain algorithm of Arora et al [1].
117 118
Following has to be done to implement the proposed methodology. Segregate the
119
colored image IRGB into its Red(IR), Green(IG) and Blue(IB). In the proposed
120
methodology different approaches have been applied for approximation and detail
121
coefficients of wavelet transformed image for each IR, IG and IB. The coefficients
122
are divided into blocks of variable size, using weighted mean and variance of each
123
sub-band of histogram of coefficients. For approximation coefficients, finer block
124
size is taken around mean while broader block size at the end of histogram.
125
Whereas, in case of detail coefficients thresholding is done by having broader
126
block size around mean while finer block size at the end of respective histogram.
127
Take inverse wavelet transform for each thresholded I R, IG and IB component.
128
Reconstruct the image by concatenating IR, IG and IB components. Following section
129
provides the algorithm used.
130 131
3. ALGORITHM
132 133
For Vertical/Horizontal/Diagonal coefficients
134
1. Input n (no. of thresholds)
135
2. Input f ( Vertical/Horizontal/Diagonal coefficients matrix)
136
3. a = min( f ); b = max ( f );
137
4. me= weighted mean f (a to b)
138
5. T1 = me ; T2 = me + 0.0001 ;
139
6. Repeat steps from (a) to (h) (n-1)/2 times
140
(a) m1 = weighted mean f (a to T1)
141
(b) m2 = weighted mean f (T2 to b )
142
(c) d1 = standard deviation f (a to T1 )
143
(d) d2 = standard deviation f (T2 to b )
144
(e) T11 = m1 – (k1 * d1 ); T22 = m2 +( k2 * d2 ) ;
145
(f) f (T11 to T1 )= weighted mean f (T11 to T1 )
146
(g) f (T2 to T22 )= weighted mean f (T2 to T22 )
147
(h) T1 = T11 – 0.0001; T2 = T22 + 0.0001;
148
9. f (a to T1)= weighted mean f (a to T1)
149
10. f (T1 to b)= weighted mean f (T1 to b)
150
11.Output f (Quantized input matrix)
151 152
For Approximation coefficients
153
1. Input n (no. of thresholds)
154
2. Input f ( Approximation coefficients matrix)
155
3. a = min( f ); b = max ( f );
156
4. me= weighted mean f ( a to b )
157
5. Repeat steps from (a) to (f) (n-1)/2 times
158
(a) m = weighted mean f (a to b)
159
(b) d = standard deviation f (a to b)
160
(c) T1 = m – (k1 * d ); T2 = m +( k2 * d ) ;
161
(d) f (a to T1)= weighted mean f (a to T1)
162
(e) f (T2 to b)= weighted mean f (T2 to b)
163
(f) a = T1 + 0.0001; b =T2 – 0.0001;
164
6. f (a to me)= weighted mean f (a to me)
165
7. f (me+1 to b)= weighted mean f (me+1 to b)
166
8. Output f (Quantized input matrix)
167 168
4. RESULTS AND OBSERVATIONS
169
Experiments have been performed on various images using MATLAB 7.1 on a
170
system having processing speed of 1.73 GHz and 2GB RAM. The histogram plot
171
shown in Fig. 1 verifies the variable segmentation with bigger block size around
172
the mean while smaller blocks at the each end of histogram plot.
173
174 Fig.1. (a)
175 176
177 178
Fig.1.(b)
Fig.1.(c)
179 180
Fig.1. Results: (a) histogram in wavelet domain
181
(b) segmentation with threshold levels seven
182
(c) segmentation with threshold levels nine
183
From the plots in Fig. 1(b) and (c), it can be easily seen that when threshold levels
184
are increased, quantization becomes finer around the ends of histogram plot. To
185
vary the block size, one can choose the values of k1 and k2 accordingly. The result
186
of proposed algorithm is tested on variety of images.
187
In Fig. 2, original Aerial image with segmented images in space domain and by
188
proposed algorithm are shown at different thresholding levels.
189 190 191 192 193 Fig.2. (a)
194 195 196 197 198 199 200
Fig.2. (b)
Fig.2. (c)
Fig.2. (d)
201 202 203 204 205 206 207
Fig.2. (e)
Fig.2. (f)
Fig.2. (g)
Fig.2. Results:(a) Original Aerial Image.
208
(b,c,d) Segmentation in space domain at threshold level 3,5 and 7.
209
(e,f,g) Segmentation by proposed algorithm at threshold level 3,5
210
and 7.
211
In Fig. 3, histograms of Approximation, Horizontal, Vertical and Diagonal
212
coefficients of R, G and B components of original and segmented Aerial image in
213
wavelet domain is depicted.
214 215 216 217 218 219
220
Fig.3. (1)
Fig.3. (2)
222 223 224 225 226 227
228
Fig.3. (3)
Fig.3. (4)
Fig.3. (5)
Fig.3. (6)
230 231 232 233 234 235
236 238 239 240 241 242 243 244
Fig.3. (7)
Fig.3. (8)
245 246 247 248 249 250
251
Fig.3. (9)
Fig.3. (10)
Fig.3. (11)
Fig.3. (12)
253 254 255 256 257 258
259 261 262 263 264 265 266 267 268 269
Fig.3. (13)
Fig.3. (14)
270 271 272 273 274 275
276
Fig.3. (15)
Fig.3. (16)
Fig.3. (17)
Fig.3. (18)
278 279 280 281 282 283
284 286 287 288 289 290 291 292 293
Fig.3. (19)
Fig.3. (20)
294 295 296 297 298 299
300
Fig.3. (21)
Fig.3. (22)
Fig.3. (23)
Fig.3. (24)
302 303 304 305 306 307
308 310
Fig.3. Results:(1,9,17) Histogram of Approximation coefficients of R, G and B
311
components respectively in wavelet domain of Aerial
312
image.
313
(2,10,18) Histogram of Approximation coefficients of R, G and B
314
components respectively in wavelet domain of Aerial
315
image thresholded at level 3.
316
(3,11,19) Histogram of Horizontal coefficients of R, G and B
317
components respectively in wavelet domain of Aerial
318
image.
319
(4,12,20) Histogram of Horizontal coefficients of R, G and B
320
components respectively in wavelet domain of Aerial
321
image thresholded at level 3.
322
(5,13,21) Histogram of Vertical coefficients of R, G and B
323
components respectively in wavelet domain of Aerial
324
image.
325
(6,14,22) Histogram of Vertical coefficients of R, G and B
326
components respectively in wavelet domain of Aerial
327
image thresholded at level 3.
328
(7,15,23) Histogram of Diagonal coefficients of R, G and B
329
components respectively in wavelet domain of Aerial
330
image.
331
(8,16,24) Histogram of Diagonal coefficients of R, G and B
332
components respectively in wavelet domain of Aerial
333
image thresholded at level 3.
334 335
Fig. 4 shows the histogram of segmented R, G and B components of Aerial image
336
in wavelet domain at thresholding levels 3, 5 and 7.
337 338
339
Fig.4. (a)
Fig.4. (b)
Fig.4. (c)
340
341
Fig.4. (d)
Fig.4. (e)
Fig.4. (f)
Fig.4. (g)
Fig.4. (h)
Fig.4. (i)
342 343
344 345
Fig.4. Results:(a,b,c) Histogram of B, G and R components thresholded in wavelet domain at level 3 of Aerial image.
346
(d,e,f) Histogram of B, G and R components thresholded in wavelet
347
domain at level 5 of Aerial image.
348
(g,h,i) Histogram of B, G and R components thresholded in wavelet
349
domain at level 7 of Aerial image.
350 351 352
Table 1: Comparison of SSIM[20] of Aerial image (512 x512, 768.1 kB) between image segmentation
353
in space domain and by proposed algorithm. Threshold Level 3 5 7
354
SSIM in Space Domain 0.9520 0.9506 0.9505
SSIM in Wavelet Domain 0.9666 0.9676 0.9678
355
Table 2: Comparison of PSNR of Aerial image (512 x512, 768.1 kB) between image segmentation in
356
space domain and by proposed algorithm. Threshold Level 3 5 7
PSNR in Space Domain (dB) 22.1656 22.0107 22.0045
PSNR in Wavelet Domain (dB) 23.1395 23.4560 23.5226
357 358
Table 3: Comparison of Time Complexity of Aerial image (512 x512, 768.1 kB) between image
359
segmentation in space domain and by proposed algorithm. Threshold Level
Time Complexity in Space Time Complexity in Wavelet Domain (sec) Domain (sec) 3.0888 1.6224 3.6192 1.7628 3.9780 1.9188
3 5 7 360
The results of Table 1, 2 and 3 are plotted in the fig. 5. The red color graph
361
represents the result of space domain algorithm while the black color represents
362
the results of proposed algorithm.
363 364 365 366 367 368 369 370 371 372
Fig.5. (a)
Fig.5. (b)
Fig.5. (c)
Fig.5. Results: (a) Comparison of PSNR of Aerial image between image segmentation in space domain and by proposed algorithm. (b) Comparison of SSIM of Aerial image between image segmentation in space domain and by proposed algorithm. (c) Comparison of Time Complexity of Aerial image between image
segmentation in space domain and by proposed algorithm.
373 374
In Fig. 6, original Earth image with segmented images in space domain and by
375
proposed algorithm are shown at different thresholding levels.
376 377 378 379 380 Fig.6. (a)
381 382 383 384 385 386 387
Fig.6. (b)
Fig.6. (c)
Fig.6. (d)
Fig.6. (e)
Fig.6. (f)
Fig.6. (g)
388 389 390 391 392 393
394
Fig.5. Results:(a) Original Earth Image.
395
(b,c,d) Segmentation in space domain at threshold level 3,5 and 7.
396
(e,f,g) Segmentation by proposed algorithm at threshold level 3,5
397
and 7.
398
In Fig. 7, histograms of Approximation, Horizontal, Vertical and Diagonal
399
coefficients of R, G and B components of original and segmented Earth image in
400
wavelet domain is depicted.
401
404 405
406
Fig.7. (1)
Fig.7. (2)
409 410 411 412 413 414
Fig.7. (3)
Fig.7. (4)
415 416 417 418 419
420
Fig.7. (5)
Fig.7. (6)
422
425 426
427
Fig.7. (7)
Fig.7. (8)
Fig.7. (9)
Fig.7. (10)
429 430 431 432 433
434
436 437 438 439 440
441
Fig.7. (11)
Fig.7. (12)
Fig.7. (13)
Fig.7. (14)
Fig.7. (15)
Fig.7. (16)
443 444 445 446 447
448 450 451 452 453 454
455
457 458 459 460 461
462
Fig.7. (17)
Fig.7. (18)
Fig.7. (19)
Fig.7. (20)
Fig.7. (21)
Fig.7. (22)
464
467 468
469 471 472 473 474 475
476
478 479 480 481 482 483
Fig.7. (23)
Fig.7. (24)
484 485
Fig.7. Results:(1,9,17) Histogram of Approximation coefficients of R, G and B
486
components respectively in wavelet domain of Earth
487
image.
488
(2,10,18) Histogram of Approximation coefficients of R, G and B
489
components respectively in wavelet domain of Earth
490
image thresholded at level 3.
491
(3,11,19) Histogram of Horizontal coefficients of R, G and B
492
components respectively in wavelet domain of Earth
493
image.
494
(4,12,20) Histogram of Horizontal coefficients of R, G and B
495
components respectively in wavelet domain of Earth
496
image thresholded at level 3.
497
(5,13,21) Histogram of Vertical coefficients of R, G and B
498
components respectively in wavelet domain of Earth
499
image.
500
(6,14,22) Histogram of Vertical coefficients of R, G and B
501
components respectively in wavelet domain of Earth
502
image thresholded at level 3.
503
(7,15,23) Histogram of Diagonal coefficients of R, G and B
504
components respectively in wavelet domain of Earth
505
image.
506
(8,16,24) Histogram of Diagonal coefficients of R, G and B
507
components respectively in wavelet domain of Earth
508
image thresholded at level 3.
509
Fig. 8 shows the histogram of segmented R, G and B components of Aerial image
510
at thresholding levels 3, 5 and 7.
511 512 513 514 515
Fig.8. (a)
Fig.8. (b)
Fig.8. (c)
Fig.8. (e)
Fig.8. (f)
516 517
519 520 521 522 523
Fig.8. (d)
524
525 526
Fig.8. (g)
Fig.8. (h)
Fig.8. (i)
Fig.8. Results:(a,b,c) Histogram of B, G and R components thresholded in wavelet domain at level 3 of Earth image.
527
(d,e,f) Histogram of B, G and R components thresholded in wavelet
528
domain at level 5 of Earth image.
529
(g,h,i) Histogram of B, G and R components thresholded in wavelet
530
domain at level 7 of Earth image.
531 532
Table 4: Comparison of SSIM of Earth image (512 x512, 768.1 kB) between image segmentation in
533
space domain and by proposed algorithm. Threshold Level 3 5 7
SSIM in Space Domain 0.9685 0.9672 0.9674
SSIM in Wavelet Domain 0.9806 0.9796 0.9797
534 535
Table 5: Comparison of PSNR of Earth image (512 x512, 768.1 kB) between image segmentation in
536
space domain and by proposed algorithm. Threshold Level 3 5 7
537 538
PSNR in Space Domain (dB) 24.1114 23.8892 23.8944
PSNR in Wavelet Domain (dB) 25.9095 26.2669 26.4056
539
Table 6: Comparison of Time Complexity of Earth image (512 x512, 768.1 kB) between image
540
segmentation in space domain and by proposed algorithm. Threshold Level 3 5 7
Time Complexity in Space Time Complexity in Wavelet Domain (sec) Domain (sec) 3.0264 1.6692 3.6036 1.7628 3.9312 1.8564
541 542
The results of Table 4, 5 and 6 are plotted in the fig. 9. The red color graph
543
represents the result of space domain algorithm while the black color represents
544
the results of proposed algorithm.
545 546 547 548
549 550 551 552
Fig.9. (a)
Fig.9. (b)
Fig.9. (c)
Fig.9. Results: (a) Comparison of PSNR of Earth image between image segmentation in space domain and by proposed algorithm. (b) Comparison of SSIM of Earth image between image
553
segmentation in space domain and by proposed algorithm.
554
(c) Comparison of Time Complexity of Earth image between image
555
segmentation in space domain and by proposed algorithm.
556 557 558
559
Table 7: Comparison of PSNR, SSIM and Time Complexity of various test images between image
560
segmentation in space domain and by proposed algorithm. Image Name
Threshold Level
House
3 5 7
SSIM in SSIM in PSNR in PSNR in Time Time Complexity Complexity Space Wavelet Space Wavelet in Space in Wavelet Domain Domain Domain Domain Domain Domain (dB) (dB) (sec) (sec) 0.9870 0.9875 22.5739 23.3216 0.8268 0.3276 0.9846 0.9870 22.4972 23.6323 0.8892 0.3900 0.9846 0.9872 22.5652 23.7875 0.9672 0.3744
3 5 7
0.9783 0.9776 0.9776
0.9863 0.9866 0.9874
21.8314 21.6453 21.6396
23.8069 24.1578 24.3705
2.8392 3.4164 3.6972
1.7160 1.7472 1.9188
3 5 7
0.9758 0.9755 0.9754
0.9823 0.9859 0.9866
19.6385 19.5087 19.5033
21.6214 22.4168 22.5544
2.8236 3.3852 3.6504
1.6224 1.7472 1.9032
3 5 7
0.9667 0.9646 0.9645
0.9742 0.9756 0.9759
20.4878 20.3132 20.3057
21.0555 21.6169 21.7436
3.0108 3.5568 3.9312
1.6848 1.7628 1.8564
256x256 Lenna 512x512 Pepper 512x512 Baboon 512x512 561 562
The results of Table 7 are plotted in the fig. 10. The red color graph represents the
563
result of space domain algorithm while the black color represents the results of
564
proposed algorithm.
565 566 567 568
569 570
Fig.10. (a)
Fig.10. (b)
Fig.10. (c)
571 572 573 574 575 576
Fig.10. (d)
Fig.10. (e)
Fig.10. (f)
Fig.10. (g)
Fig.10. (h)
Fig.10. (i)
Fig.10. (j)
Fig.10. (k)
Fig.10. (l)
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Fig.10. Results: (a,b,c) Comparison of PSNR, MSSIM and Time Complexity of House
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image between image segmentation in space domain and
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by proposed algorithm.
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(d,e,f) Comparison of PSNR, MSSIM and Time Complexity of Lenna
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image between image segmentation in space domain and
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by proposed algorithm.
(g,h,i) Comparison of PSNR, MSSIM and Time Complexity of Pepper
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image between image segmentation in space domain and
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by proposed algorithm. (j,k,l) Comparison of PSNR, MSSIM and Time Complexity of
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Baboon image between image segmentation in space
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domain and by proposed algorithm.
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5. CONCLUSION
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The performance of the proposed hybrid algorithm has been compared with the
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algorithm reported by Arora et al.[1]. Time taken by hybrid algorithm in wavelet
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domain is approximately half of the time taken by space domain algorithm. It can
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also be seen that segmentation done in wavelet domain gives improved PSNR
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compared to segmentation done by Arora et al. at same threshold level using
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mean and standard deviation. The number of thresholds required to reach the
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saturation PSNR is far less than thresholds required in space domain. SSIM of
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image segmented in wavelet domain is always better than the image segmented
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by using Arora et al. algorithm. Finally, more distinct regions can be observed in
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an image using wavelet domain segmentation compared to space domain
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segmentation. It is worth emphasizing that after the segmentation only 4 times
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the n (number of thresholds) + 1
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variation of image.
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