Gallstones are characterized by the precipitation of cholesterol and bile pigments, leading to the formation of stones. These stones could be symptomatic or asymptomatic depending on the area which they are found as well as their size. Obstruction in the liver could result due to several conditions, such as tumors, inflammation and gall stones. An obstruction in the liver leads to the accumulation of waste products such as bile in the blood as well as the skin and eyes therefore leading to conditions such as jaundice.
Fatty liver results in the enlargement and tenderness of the liver as well as abnormal liver function. This is commonly known to arise from excessive consumption of alcohol. When diseases such as cirrhosis and hepatitis progress in some cases, they can lead to liver cancer. However, it is more common for cancer to spread to the liver from other parts of the body.
Although liver diseases vary in description, they share common symptoms such as jaundice (yellowing of the skin and eyes), light stools, loss of appetite, diarrhoea, vomiting, nausea, dark urine, ascites, (which is the swelling of the abdomen due to the accumulation of fluid) abdominal pain and in some cases pruritus they may not be present until the disease has reached an advanced stage. However, the severity and also the type of disease can be distinguished following a range of test fuch as liver function tests. The diagrams below show the summary of hepatic failure including the features, complications and investigations as well as the coomon causes, investigations and managements for acute liver disease.
CASE SUMMARY A 39 year old male sailor who recently completed a 4 month tour of south east asia, went to his clinic with complaints of flu – like symptoms as well as mild pyrexia, nausea, vomiting, pain in his upper right quadrant and darkened urine. After reviewing his medical history, it was discovered that he had contracted several common sexually transmitted diseasease after living promiscuously but they were cured with antibiotic administration. On further examination, a mild scleral icterus was noted, indicating jaundice. A chest examination showed no abnormalities and the tenderness of the upper right quadrant was linked to the observed hepatomegaly through palpation. However, he showed no signs of splenomegaly or lymphadenopathy. The examination also revealed several tattoos and some of which were obtained during his tour of duty.
LABORATORY INVESTIGATIONS AND DIAGNOSIS Some biochemical tests were carried out on the patient’s blood and urine and the results obtained are shown below in table 2 with reference values followed by a figure 2, a chart on his urine output alongside the level of some biomarkers.
C- Reactive Protein (CRP) and Erythrocyte Sedimentation Rate (ESR): Erythrocyte sedimentation rate (ESR) is a test used in determining the presence of infections, tumors and inflammation as well as autoimmune diseases. This test is based on the analysis of acute-phase response which normally starts within days or hours of the beginning inflammation. This is because the increase in acute phase proteins is often accompanied by the increase of ESR. This occurs because the acute phase proteins lead to a clustering of red blood cells. ESR however cannot be used as an indication of the area where the inflammation is occurring. This patient has an elevated ESR of 120mm/hr which about 9 times more than the reference As ESR cannot be used as a single indicator, it is used alongside other tests such as C- reactive protein. C-reactive protein is an acute phase protein and in high levels it is used in the indication of infection and inflammation. As its levels fall in relation to the presence of inflammation, it is a better marker at monitoring inflammation. This patient shows a highly elevated CRP level (230 mg/LT) which is about 100 times the reference range (<3 mg/LT), therefore further suggesting that the patient is having an inflammatory response, probably due to an infection.
Alkaline Phosphatase: Alkaline phosphatase (ALP) is found mostly in the bile ducts therefore increased levels in the blood usually indicates an biliary obstruction, therefore hampering the transportation and delivery of bile. It is also used as a test in detecting liver and bone disease as elevated levels usually indicate liver damage or a problem with the bone cells. Therefore when used alongside other liver test such as aspartate aminotransferase (AST), bilirubin or alanine aminotransfere (ALT) a definite diagnosis of liver disease can be made. In this patient, the ALP test shows a result of 350 IU/L which is highly elevated when compared to the reference range of 20 ââ‚¬” 70 IU/L. This therefore suggests obstruction of the bile ducts which could have resulted from an inflammation of the bile duct. This is because it may lead to the narrowing of the bile duct. The most common causes of inflammation are viral and bacterial infections.
Alanine transaminase: Alanine aminotransferase ( ALT) is a test used in detecting liver injury. Alongside AST, a diagnosis of liver disease can be made with ALT as they are believed to be essential in the detection of liver injury. As ALT is found in cells of liver, during hepatocellular damage, it seeps out, into the blood. In this patient, there was a notable increase of ALT of about twice the normal expected range. Although a higher range is expected in diseases such as acute hepatitis, liver cell damage is still indicated.
Bilirubin: It is a pigment of bile formed from the breakdown of heme in red blood cells. An increase in bilirubin levels can have various causes such as hepatocellular damage and biliary obstruction. In this patient, there is an increase in the total bilirubin as well as conjugated bilirubin. The elevation of the total bilirubin shows reduced liver function as some of the symptoms experienced by the patient can be attributed to this increase, such as the yellowing of the sclera. Total bilirubin, is used as a test in used in diagnosing jaundice and in this patient, a highly elevated level of 30mg/dL compared to the reference range of 0.2-1.3 mg/dL indicates an onset of jaundice. There is also a noticeably high amount of bilirubin in the urine (conjugated hyperbilirubinemia) resulting in the darkened colour observed by the patient. This could occur due to various reasons such as infections affecting the liver. There are several infections which could lead to hyperbilirubinemia and they include include viral Epstein Barr virus, cytomegalovirus ( CMV) and viral hepatitis.
PATHOPHYSIOLOGY The symptoms experienced by this patient although they are mostly flu symptoms are accompanied with other symptoms such as scleral icterus, pain in the right upper quadrant and darkened urine. The yellowing of the patientââ‚¬â„¢s sclera is believed to be pointer to the onset of jaundice. This could have been due to the inability of the body to excrete bilirubin properly or due to a high level of hepatocellular damage. This theory is further supported by the levels of bilirubin (total bilirubin of 30mg/dL) observed in the laboratory tests carried out.The colour of his urine is also attributed to hyperbilirubinemia as increased amounts of conjugated bilirubin are excreted in the urine due to poor hepatic funtion
The hepatomegaly which was observed during palpation of the liver is believed to be due to inflammation. The pain which this patient experiences in the upper right quadrant usually indicates defects with the gall bladder and the liver as it could be caused due to an obstruction of the bile duct, or diseases such as non-alcholic fatty liver disease. hepatitis ( inflammation of the liver), cholecystitis (inflammation of the gallbladder resulting from cholelithiasis) and cancer. Other diseases which affect the right upper quadrant that are unrelated to the liver and the hepatobiliary system include pneumonia affecting the lung. In cholecystitis, the inflammation of bile duct leads to an irritatiuon of structures in the abdominal cavity and therefore leads to pain in the right upper quadrant. The nausea and vomitting experienced by this patient although it could be wrongly attributed to a condition such as food poisoning, due to the fact that the patient is a sailor and must have come in contact with some raw food especially in the region where he just briefly returned from ( sdouth east asia which is highluy endemic of salmonellosis). However food posioning resulting from infections such as salmonellosis, are known to show symptoms within 2 days of incubation. Threfore food poisoning is ruled out. However, conditions which could cause symptoms of nausea and vomiting include infectious and inflammtory conditions. Therefore, infections such as appendictis, cholecystitis ( a gall baldder infection) and viral hepatitis are suspected. The assumption of inflammation is supported by the CRP and ESR test carried out on the patient as an elevation of the tests indicate inflammation which was further confirmed during the palpation of the liver. The increased amount of the white blood cells ( lymphocytes) in the complete blood count, indicates the presence of an infection as infetcion as they are present in high numbers during infections. The presence of tatoos observed during physical examination of this patient further suggests a high susceptibility to infections, such as forms viral hepatitis (especially those known to be endemic in South East Asia) and HIV. The promiscuous life which was ascertained from his previous medical history also suggests that the patient could have contracted an infection during his travel, leading to the noticed symptoms.
These symptoms which the patient show are generally similar to a clinical stage of viral hepatitis known as the the prodromal stage . They are also common symsptoms of cholecystitis.
Further Laboratory Tests As the laboratory tests which were carried out were not enough to ascertain the exact ailment of the patient. Further Laboratory tests and clinical investigations which can be done include:
Viral serologic testing
Enzyme-linked immunosorbent assay (ELISA).
Strip immunoassay (SIA)
Reverse transcriptase polymerase chain reaction (RT/PCR). RT/PCR is a test with high sensitivity and specificity (> 98%) for diagnosing HCV infection. RT/PCR identifies genetic material of the HCV. (Centers for Disease Control and Prevention (2002) Centers for Disease Control and Prevention (2002) National Institutes of Health Consensus Development Conference Statement: Management of Hepatitis C. Retrieved June 10-12, 2002, from http://consensus.nih.gov/cons/116/Hepc091202.pdf.CDC, 2002)
Computed axial tomography (CAT), etc
The image below shows the progression from symptoms to diagnosing in a patient suspected of having viral hepatitis and also suffering from jaundice.
Treatment and Management Based on the symptoms and clinical findings, two of the suspected diseases that this patient might be suffering from are viral hepatitis and cholecytitis. The treatment for cholecstitis include
Daily stimulation of gallbladder contraction with intravenous CCK
Elective lapatoscopic cholecystectomy
Some treatments for viral hepatitis, depending on the type include:
Administration of nutritional supplements such as vitamin C
Edge Detection Methods in Digital Image Processing
The current work focuses on the study of different edge detection techniques and analysis of there relative performances. The recent advance of image processing has motivated on the various edge detection techniques. There are many ways to perform the edge detection. However the majority of different methods may be categorized into two groups, i.e. Gradient based and Laplacian based. Also we introduce stochastic gradient method which gives the better result in the presence of noise. The effectiveness of the stochastic process is demonstrated experimentally.
Key words: Edges, Salt and paper noise, stochastic process
Introduction Edge detection  is a process that detects the presence and location of edges constituted by sharp changes in color, intensity (or brightness) of an image. Since, it can be proven that the discontinuities in image brightness are likely corresponding to discontinuities in depth, discontinuities in surface orientation, changes in material properties and variations in scene illumination. In the ideal case, the result of applying an edge detector to an image may lead to a set of connected curves that indicate the boundaries of objects, the boundaries of surface markings as well curves that correspond to discontinuities in surface orientation [1, 2]. However, it is not always possible to obtain such ideal edges from real life images of moderate complexity. Edges extracted from non-trivial images are often hampered by fragmentation (i.e. edge curves are not connected), missing edge segments, as well as false edges (i.e. not corresponding to interesting phenomena in the image), which all lead to complicating the subsequent task of image interpreting .
A grey scale image can be represented by two-dimensional values of pixel in which each pixel represents the intensity. In image processing, the digitization process can be done by sampling and quantization of continuous data. The sampling process samples the intensity of the continuous-tone image, such as a monochrome, color or multi-spectrum image, at specific locations on a discrete grid. The grid defines the sampling resolution. The quantization process converts the continuous or analog values of intensity brightness into discrete data, which corresponds to the digital brightness value of each sample, ranging from black, through the gray, to white. A digitized sample is referred to as a picture element, or pixel. The digital image contains a fixed number of rows and columns of pixels. Pixels are like little tiles holding quantized values that represents the brightness at the points of the image. Pixels are parameterized by position, intensity and time. Typically, the pixels are stored in computer memory as a raster image or raster map, a two-dimensional array of small integers. Image is stored in numerical form which can be manipulated by a computer. A numerical image is divided into a matrix of pixels (picture elements).
Digital image processing allows one to enhance image features of interest while attenuating detail irrelevant to a given application, and then extract useful information about the scene from the enhanced image. Images are produced by a variety of physical devices, including still and video cameras, x-ray devices, electron microscopes, radar, and ultrasound, and used for a variety of purposes, including entertainment, medical, business (e.g. Documents), industrial, military, civil (e.g. traffic), security, and scientific. The goal in each case is for an observer, human or machine, to extract useful information about the scene being imaged.
Organization of the paper The paper is organized as follows:
In Section – 2 we briefly discuss the different types of edge detection methods and also discuss the criteria for edge detection. In Section – 3 we also consider Stochastic Gradient operator by which we can give the better result in noisy environment. Some experimental results are shown in Section – 4 along with few remarks. In Section – 5 some conclusions are drawn.
2. Motivation behind Edge Detection: The main objective of edge detecting process is to extract the accurate edge line with good orientation without changing the properties of the image. The brightness and contrast of an image is discreet function, mentioned in . They are corresponds to:
Surface orientation Discontinuities
Changes in material properties
Variations in scene brightness
In general, a group of connected curves which denotes the boundary of the image surface. If the boundary detection step is successfully completed, then the subsequent task of interpreting pixel values contents in the original image may therefore be substantially simplified. But, it is not always possible to get exact edges from the real life images. Edges extracted from non-trivial images are often hampered by fragmentation i.e. the edge curves are not connected, missing edge segments, false edges etc., which complicate the subsequent task of interpreting the image data.
2.1 Edge Detection
Edge is a part of an image that contains significant variation. The edges provide important visual information since they correspond to major physical, photometrical or geometrical variations in scene object. Physical edges are produced by variation in the reflectance, illumination, orientation, and depth of scene surfaces. Since image intensity is often proportional to scene radiance, physical edges are represented by changes in the intensity function of an image  Therefore, it should be mandatory to find out the occurrence in perpendicular to an edge.
2.2 Different Types of Edges
The common types of edges, mentioned in  are following .
A Sharp Step, as shown in Figure 1(a), is an idealization of an edge. We know that an image is always band limited, so this type of graph never occurs. A Gradual Step, as shown in Figure 1(b), is very similar to a Sharp Step, but it has been smoothed out. But the change in intensity is not as quick or sharp. A Roof, as show in Figure 1(c), is different than the first two edges. The derivative of this edge is discontinuous. A Roof can have a variety of sharpness, widths, and spatial extents. The Trough, also shown in Figure 1(d), is the inverse of a roof.
There are many methods for edge detection, but most of them can be grouped into two categories, search-based andÂ zero-crossingÂ based mentioned in . The search-based methods detect edges by first computing a measure of edge strength, usually a first-order derivative expression such as the gradient magnitude, and then searching for local directional maxima of the gradient magnitude using a computed estimate of the local orientation of the edge, usually the gradient direction. The zero-crossing based methods search for zero crossings in a second-order derivative expression computed from the image in order to find edges, usually the zero-crossings of theÂ LaplacianÂ or the zero-crossings of a non-linear differential expression. As a pre-processing step to edge detection, a smoothing stage, typically Gaussian smoothing, is almost always applied.
The main objective  of edge detection in image processing is to reduce data storage while at same time retaining its topological properties, to reduce transmission time and to facilitate the extraction of morphological outlines from the digitized image.
2.3 Criteria for Edge Detection There are large numbers of edge detection operators available, each designed to be sensitive to certain  types of edges. The Quality of edge detection can be measured from several criteria objectively. Some criteria are proposed in terms of mathematical measurement, some of them are based on application and implementation requirements. In all five cases a quantitative evaluation of performance requires use of images where the true edges are known. The criterions are Good detection, Noise sensitivity, Good localization, Orientation sensitivity, Speed and efficiency. Criteria of edge detection will helps to evaluate the performance of edge detectors. Correspondingly, different techniques have been developed to find edges based upon the above criteria, which can be classified into linear and non linear techniques.
2.4 Procedure for Edge detection
Edges characterized object boundaries. It is also useful for segmentation, registration, and identification of objects in images which are mentioned in. Edge points can be considered as pixels of abrupt gray-scale change. Therefore we can say that, it is reasonable to define edge points in binary images as black pixels with at least one white nearest neighbor, that is, pixel locations(m,n) such that u(m,n)=0 and g(m,n)=1, where
g(m,n) [u(m,n) u(mÂ±1,n)] OR [u(m,n) u(m,n Â± 1)]
Figure 4: Gradient of f(x,y) along r direction
Where denotes the logical exclusive-OR operation. For a continues image f(x,y), its derivatives can be assumed a local maximum in the direction of the edge. Therefore, one edge detection technique is to measure the gradient of f along r in a direction Î¸, i.e.
âˆ‚f/âˆ‚r = âˆ‚f/âˆ‚x * âˆ‚x/âˆ‚r âˆ‚f/âˆ‚y * âˆ‚y/âˆ‚r = fx cosÎ¸ fysinÎ¸
2.5. Various Techniques for Edge Detection
There are many ways to perform edge detection. Various edge detection algorithms have been developed in the process of finding the perfect edge detector. However, the most may be grouped into two categories, gradient and Laplacian. The gradient method detects the edges by looking for the maximum and minimum in the first derivative of the image. The Laplacian method searches for zero crossings in the second derivative of the image to find edges.
2.6 Gradient-based method
The first derivative assumes a local maximum at an edge. For a gradient image f(x, y), at location (x, y), where x and y are the row and column coordinates respectively, typically consider the two directional derivatives, mentioned in [2, 12]. The two functions that can be expressed in terms of the directional derivatives are the gradient magnitude and the gradient orientation.
The gradient magnitude is defined by
g(x,y) (âˆ†x2 âˆ†y2)1/2
âˆ†z = f(x n,y) – f(x-n,y) and âˆ†y = f(x,y n) – f(x,y-n)
Where n is a small integer usually unity.
This quantity gives the maximum rate of increase of f(x, y) per unit distance in the gradient orientation of g(x, y). The gradient orientation is also an important quantity. The gradient orientation is given by
Î˜(x,y) atan (âˆ†y/âˆ†x)
Here the angle is measured with respect to the x-axis. The direction of the edge at (x, y) is perpendicular to the direction of the gradient vector at that point. The other method of calculating the gradient is given by estimating the finite difference.
2.6.1 Robert Edge Detector
The calculation of the gradient magnitude of an image is obtained by the partial derivatives Gx and Gy at every pixel location. The simplest way to implement the first order partial derivative is by using the Roberts cross gradient operator, mentioned in .
Gx= f (i, j) – f (i 1, j 1)
Gy= f (i 1, j) – f (i,j 1)
The above partial derivatives can be implemented by approximating them to two 2×2 masks. The Roberts operator masks are:
These filters have been the shortest support, thus the position of the edges is more accurate, but the problem with the short support of the filters is its vulnerability to noise. It also produces very weak response to genuine edges unless they are very sharp.
2.6.2 Prewitt Edge detector
The Prewitt edge detector [2, 12] is a much better operator than Roberts’s operator. This operator having a 3 x 3 masks deals better with the effect of noise. An approach using the masks of size 3 x 3 is given below, the arrangement of pixels about the pixels [i, j].
The partial derivatives of the Prewitt operator are calculated as
Gx = (a6 ca5 a5) – (a4 ca1 a2)
Gy = ( a2 ca3 a4) – (a0 ca7 a6)
The constant c implies the emphasis given to pixels closer to the centre of the mask. Gx and Gy are the approximation at [i, j].
Setting c= 1, the Prewitt operator is obtained. Therefore the Prewitt masks are as follows
These masks have longer support. They differentiate in one direction and average in the other direction, so the edge detector is less vulnerable to noise.
2.6.3 Sobel Edge Detector
The Sobel operator [2, 12] is the most known among the classical methods. The Sobel edge detector applies 2D spatial gradient convolution operation on an image. It uses the convolution masks shown in to compute the gradient in two directions (i.e. row and column orientations), and then works out the pixels’gradient through g=|gr gc|. Finally, the gradient magnitude is threshold. The Sobel edge detector is very much similar to the Prewitt edge detector. The difference between the both is that the weight of the centre coefficient is 2 in the Sobel operator. The partial derivatives of the Sobel operator are calculated as
Gx = (a6 2a5 a4) – (a4 2a1 a2)
Gy = ( a2 2a3 a4) – (a4 2a7 a6)
Therefore the Sobel masks are:
Although the Prewitt masks are easier to implement than Sobel masks.
2.6.4. 4-neibougher operator
Instead of calculating edge strength at the point (r-Â½,c-Â½), it is desired to calculate it at the point (r,c),mentioned in [2, 12]. To take care of this, 3X3 mask are used as against 2X2 mask in Roberts operator. Then
d1 = g5 – g1 and d2 = g7 – g3
The corresponding masks are given by
2.6.5. Compass operator
Compass operators [11, 12] measure gradients in a selected number of directions shown in figure. An anti-clockwise circular shift of the eight boundary elements of these masks gives a 450 rotation of the gradient direction.
Let gk(m,n) denote the compass gradient in the direction Î¸k = Ï€/2 kÏ€/4, k=0,â€¦,.,7. The gradient at location (m,n) is defined as
Which can be threshold to obtain the edge map as before Only four of the preceding eight compass gradients are linearly independent. Therefore, it is possible to define four 3X3 arrays that are mutually orthogonal and span the space of these compass gradients. Compass gradients with higher angular resolution can be designed by increasing the size of the mask.
2.6.6 The Canny Edge Detector
The Canny operator, mentioned in [2, 12] is one of the most widely used edge finding algorithms. Canny proposed a method that was widely considered to be the standard edge detection algorithm in the industry. In regard to regularization explained in image smoothing, Canny saw the edge detection as an optimization problem. He considered three criteria desired for any edge detector: good detection, good localization, and only one response to a single edge. Then he developed the optimal filter by maximizing the product of two expressions corresponding to two former criteria (i.e. good detection and localization) while keeping the expression corresponding to uniqueness of the response constant and equal to a pre-defined value. The solution (i.e. optimal filter) was a rather complex exponential function, which by variations it could be well approximated by first derivative of the Gaussian function. This implies the Gaussian function as the smoothing operator followed by the first derivative operator. Canny showed that for a 1D step edge the derived optimal filter can be approximated by the first derivative of a Gaussian function with variance s as follow:
The Canny approach to edge detection is optimal for step edges corrupted by white Gaussian noise. This edge detector is assumed to be output of a filter that both reduces the noise and locates the edges. Its ‘optimality’ is related to the following performance criteria:
Good detection: Both the probability of missing real edge points and incorrectly marking non-existent edge points must be minimal.
Good localization: The distance between the actual and detected location of the edge should be minimal.
Minimal response: This criteria state that multiple responses to a single edge and ‘false’ edges due to noise must be eliminated.
After optimizing the above criteria in a certain fashion, an efficient approximation to the required operator is the first derivative of the two-dimensional Gaussian function G(x, y) applied to the original image. For example, the partial derivative with respect to x is defined as follows:
The first step of the edge detection algorithm is to convolve the image I(x, y) with a two dimensional Gaussian filter and differentiate in the direction of n.
Candidate’s edge pixels are identified as the pixels that survive a thinning process known as no maximal suppression. Any gradient value that is not a local peak should be set to zero. Each pixel in turn, forms the centre of a 3×3 neighborhood. The gradient magnitude is estimated for two locations, one on each side of the pixel in the gradient direction, by interpolation of the surrounding values. If the value of the centre pixel is larger than these of the surrounding pixels, the pixel is considered a maximal point. Otherwise, the pixel value is set to zero. The last step of the algorithm is to threshold the candidate edges in order to keep only the significant ones. Canny suggests hysterics threshold instead of a global threshold values. The high threshold is used to find “seeds” for strong edges. These seeds are grown to as long as an edge as possible, in both directions, until the edge strength falls below the low threshold value.
3. Stochastic Gradients The above stated edge detection techniques  are poorly effective when the image is noisy. Because in above stated edge detection techniques passed the images through a low pass filter. So the noise cannot remove properly. A better alternative is to design a edge extraction masks, which take into an account of the presence in noise in a control manner. The following figure shows the edge model whose transition region is one pixel wide to detect the presence of an edge at location P.
Then calculate the horizontal gradient using formula as
Here, (m,n) and =(m,n) are the optimum forward
and backword estimates of u(m,n) from noisy observation given over some finite region W of left and right half plane respectively. W can be defined as
Forward estimates along the horizontal can be defined as
Similarly, we can find (m,n) at the point P(m,n). In the same manner we can find the vertical gradient at the point P(m,n).Let it be g2 (m,n) . From g1(m,n) and g2(m,n) we calculated g(m,n) using clockwise circular shift of the eight boundary elements of masks. The mask(g1) is implemented using the following matrix and g2 , g3, g4, g5, g6, g7 and g8 derivative equations are isotropic for rotation increments of 450 respectively.
Therefore, g(m,n) as follows:
3.1 Laplacian based method:
The Laplacian based methods, mentioned in  search for zero crossings in the second derivative of the image in order to find edges, usually the zero-crossings of the Laplacian or the zero-crossings of a non-linear differential expression.
3.2 Laplacian of Gaussian
Gaussian filters are the most widely used filters in image processing and extremely useful as detectors for edge detection. It is proven that they play a significant role in biological vision particularly in human vision system. Gaussian-based edge detectors are developed based on some physiological observations and important properties of the Gaussian function that enable to perform edge analysis in the scale space.
The principle used in the Laplacian of Gaussian method is, the second derivative of a signal is zero when the magnitude of the derivative is maximum. The Laplacian of a 2-D function f(x, y) is defined as
The two partial derivative approximations for the Laplacian for a 3 x 3 region are given as
âˆ†2f = 4( a8) – ( a1 a3 a4 a7)
âˆ†2f = 8( a8) – (a0 a1 a2 a3 a4 a5 a6 a7)
The masks for implementing these two equations are as follows
The above partial derivative equations are isotropic for rotation increments of 900 and 450, respectively. Edge detection is done by convolving an image with the Laplacian at a given scale and marks the points where the result have zero value, which is called the zero-crossings. These points should be checked to ensure that the gradient magnitude is large.
4 Experimental Results We experimented with a picture “train” by passing it from many edge detection operators and the outputs are given bellow –
Original Image Sobel Output
4-neighbour output Compass Output
LOG Output Prewitt Output
Robert Output Canny output
Stochastic (noisy image) Sobel (noisy image)
4.1. Performance of Edge Detection Operation
To test the performance [11, 13] of our algorithm, we have applied it to a noisy image in sobel and stochastic operator where n0 be the number of edges pixels declared and n1 be the number of missed or new edges pixels of a noisy image. n0 is fixed for noiseless as well as noisy image, then the edge detection error can be given as follows.
In above stated example, the error for sobel operator used on noisy image with SNR 10dB is 24% whereas it is only 1.5% for stochastic operator.
5. Conclusion: We have reviewed and summarized the characteristics of different common operators. Each approach has its own advantages and drawbacks in different areas but experimental comparisons on different approaches show which approach is suitable for which image. Although, all operators are roughly equivalent in case of noiseless image, but for practical applications, we conclude that the canny edge detection operator gives the best result in edge detection in a noiseless image. But in case of noisy image, the stochastic gradient is found to be quite effective. There are still possibilities for further scope.