Kalman Predictor Based Edge Detector for Noisy Images

Pradipta Roy Prabir Kumar Biswas Binoy Das Optronics Centre, Department of E & ECE Optronics Centre, Integrated Test Range, Chandipur, India IIT, Kharagpur, India Integrated Test Range,Chandipur Email: pradiptar@yahoo.com Email: pkb@ece.iitkgp.ernet.in Email: bkdas32@yahoo.com

Abstract— Edge Detection is a primary but one of the most

essential segmentation tasks of image processing. Though numerous techniques are available for edge detection, it is hard to find a generalized version adaptive to all situations. Edge detection challenge gets stiffer in case of noisy images, because most of the derivative based edge detectors are very sensitive to noise. In this paper, we have tried to attack the edge detection problem from a different perspective. Instead of finding gradient, we run a Kalman Predictor over the image from two opposite directions of horizontal and vertical dimensions. Error between estimated and actual pixel values provides cue for edge localization, which is further processed by dual threshold to get the true edges. Proposed edge detector performs quite satisfactorily in case of noisy images and can be used for text extraction from noisy document image or medical images corrupted by artifacts.

Keywords — Edge Detection, Kalman Predictor

I. INTRODUCTION Edges are one of the fundamental and critical components

of an image from the processing point of view. Edges extracted from the images are used in different processing levels of object detection: from preliminary segmentation to higher order descriptor level. In general, edge is defined as the image region where an intensity transition happens. In other words, an edge signifies the boundary between two dissimilar regions. The magnitude of transition (sharp or gradual) or dissimilarities for consideration as an edge is subject specific, though a sharp transition in intensity profile is often considered as an occurrence of edge[1].

Different edge detection schemes were studied over the years in plenty of scholarly articles. Unfortunately, none of them can be applied to all edge detection problems without changing algorithmic parameters like threshold. Over the years many Edge Detectors were introduced by researchers. The primary and most popular category of edge detectors are derivative ( 1st and 2nd order) and gradient (magnitude and direction) based detectors. This type of edge detectors includes Roberts, Sobel, Prewitt and Laplacian Operators. All of them are represented by different spatial masks or kernels which are convolved with the image. Second category of edge detectors is combination of multiple processes to detect and localize edges. These detectors are more adaptive to various situations. The popular detectors in this category are Difference of Gaussian (DOG), Marr Hildreth Edge Detectors etc. But the most popular and efficient of them is Canny Edge Detector [2]. Actually Canny Edge Detection is a combination of four

sequential processes to detect and localize true edge points. Canny Edge detection process includes smoothing, edge detection by gradient operator, non maxima suppression and hysteresis thresholding. There are many other instances of multi-process edge detection. For example, the method proposed by Roy et. al. which is based on local thresholding and global variance eliminating need of manual threshold selection for edges [3]. There is another kind of edge detector which does not use the gradient or derivative to find the edges. Nomura et. al. proposed one of this kind of edge detection methods which employs a reaction diffusion model to solve Fitzhugh Nagumo differential equation on image grid [4]. The solution gives a pulse in the transition of intensity levels.

Kalman Filter is perhaps the most well known mathematical tool in the field of estimation theory which estimates the process variable in presence of measured noisy data [5]. Kalman Filter is best linear estimator available as it minimizes the mean square error. This tool can be used as Filter to smooth out noisy data set. It can also be used as a predictor to estimate the next sample from previous data samples with recursive updation along the data set coverage. The application of Kalman filter spans a staggeringly vast engineering domain from Navigation of flight vehicles to Global Positioning System, from tracking targets to estimation of noisy data in statistical signal processing [6]. In Computer Vision applications Kalman Filter is used in cases of tracking of objects from video frames, tracking human gestures, vehicle movement and even biological motions like hearts movement in case of real time medical image processing [7]. There are some instances of Kalman Filters used in edge detection applications. Rangarajan et al proposes an edge detection scheme which first smoothens the image using Kalman Filter and then applies directional derivatives to find the edges [8]. A line by line edge detection using estimation and edge following by Kalman Filter is proposed by Basseville et al. [9]. In a video frame or image sequences, edge motion is calculated using extended Kalman Filter by McIvor [10]. In their work, Dikici et al. integrated Kalman Filter based left ventricle tracking with a graph cut based edge detection scheme in Echocardiography [11]. In his work, Murphy proposes a Kalman Filter based edge detection scheme for acoustic images [12]. He had reinitialized the Kalman Filter each time an edge is detected. He has disregarded the process noise and its covariance in calculating estimate error covariance.

In this paper, we have presented Kalman Predictor Based Edge Detection Scheme for noisy images. We have used Kalman Predictor starting from each row and each column

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from both forward and backward direction. Then a dual threshold scheme is used to isolate the spurious noise like elements. We have initialized the Kalman Filter at the start of row or column. The initialization is not done in between, to avoid multiple responses for ramp edges, which is a major drawback of Murphy’s method. Our Method is purely edge detection method, hence other pre and posts processing operations (for example smoothing, non maxima suppression and hysteresis thresholding in case of Canny Detector) were not performed in this work. Hence, we have compared our results with conventional gradient based operators like Sobel in different noisy image cases. The performance were quite satisfactory both qualitatively and quantitative metric wise.

The organization of this paper is as follows. We have first introduced the Kalman Filter Theory for Prediction and abrupt change detection. Then proposed algorithm for finding edges from 2D image data set has been described. An exhaustive study with different noisy images is presented in the result section. Finally the paper has been concluded with discussions and future scope of work.

II. KALMAN PREDICTOR BASED EDGE DETECTOR As we are using a digital image in finite 2D grid points, we

will use discrete Kalman Filter for our edge detectors. The primary assumptions of Kalman Filter are that it operates on linear stochastic process and the noise distribution is Gaussian in nature. If we take a pixel data set xk (either in row or column wise) where k is data point index spanning from 1 to n , then a state space equation can be written as

xk = A. xk-1 + Pnk-1 (1)

where, A is the scale factor that relates the previous state variable to present state variable. For our case of first order linear system , we will take A as unity. Pn is the process noise with zero mean and Q is the covariance value. The measurement mk is related to process variable xk by another linear stochastic equation,

mk = H. xk + Mnk (2)

where, H is measurement scale factor, Mn is measurement noise with zero mean and R covariance value. With these two preliminary state equations, we will define the sequential Kalman equations which can be categorized into two classes of equations: Time update and Measurement update equations. In the Time update equations the present state variable and error covariance value are predicted and the measurement update phase these values are corrected with the help of measurement values and computed Kalman Gain. The time update equations are given by

xk- = A x k-1 (3)

Pk- = A Pk-1 AT + Q (4)

where, xk- , Pk

- are the a-priory estimate of state variable and error covariance calculated from previous iteration values. The measurement update equations start with computing Kalman Gain (Kk) and then updating the state and error covariance posteriori values with measurement and Kalman gain. These equations are given by

Kk = Pk- . HT (H Pk

- HT + R) -1 (5)

xk = xk- + Kk ( mk – H xk

- ) (6)

Pk = (1 – Kk H) Pk- (7)

Equations (3) to (7) are recursively applied to pixel data set with measurement values mk as original pixel values and xk as estimated pixel values.

A. Parameter Tuning and application of Kalman Filter for Edge Detection In our edge detection algorithm, we want to estimate the

pixel intensity values (xk) in a sequential pixel data set (mk) in a row (or column) wise operation. The property of Kalman Predictor is to follow the data pattern in a sequence. If there is an abrupt change in data pattern, Kalman predictor takes some sample steps to converge again. The difference between actual data and Kalman Predicted data at the transition point gives a measure of change in intensity level or edges. This is the basic idea for detecting edges using Kalman Filter and is shown in Figure 1. The Response of Predictor to the intensity changes is governed by parameter tuning. The response should not be too sensitive to detect noises or should not be too ideal to omit the edges.

Fig. 1. Transition detection using Kalman predictor

The State and Measurement Scale Factor (A and H) are taken as unity. This is because next and previous state data are directly related and of same kind (both are intensities). Similarly, the measurement and state variables are both pixel intensities hence H can be taken as 1.

For a proper estimation four parameters are to be adjusted manually, which is called ‘Predictor Parameter Tuning’. During all the tuning experiments, we fix all three parameters and vary single parameter at a time to judge the performances. These parameters are initial estimates of state variable x0 and

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error covariance P0. We have simulated the filter with a data sequence and found that x0 has very small impact on filter convergence. This is shown in Figure 2.

Fig. 2. Kalman predictor convergence with three

initial state values, X0 as 5, 0 and 10 Same sort of data set is used to find out the effect of

choosing P0. We can see from Figure 3 that except very low values of P0, the filter performance is independent of initial value of error covariance. This is because, very low value of P0 signifies that we are absolutely certain about initial estimate of state variable, which is a wrong assumption. We have chosen P0 as 10 for rest of our results.

Choosing the values of Q and R are the key of tuning the predictor for detecting abrupt changes of data profile. Both the parameters have effect on how the predictor will respond to slow or fast varying data samples. If we fix Q to a higher value, the effect of Q is not so much prominent. But we should not choose Q much higher than 10. In that case, the predictor will dedicatedly follow the input data set and it will be very hard to detect the abrupt changes. The effect of Q on predictor performance is shown in Figure 4. For very small value of Q, the predictor will be too sensitive even for the noise. We have chosen Q=0.1 for rest of the experiments.

The measurement noise covariance R is most sensitive parameter and in general is taken as variance of measurement noise which is known. In our case we are not filtering out any measurement noise, but edges or intensity changes are treated similar to noise present in a data pattern and that is what

exactly we want to find out.

Fig. 3. Kalman predictor convergence with different

initial values of error covariance

Fig. 4. Kalman predictor convergence with different

values of process noise covariance

Fig. 5. Kalman predictor convergence with different values of measurement noise covariance

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The effect of R on predictor performance is opposite to that of Q. Higher values of R make the predictor very sensitive to changes. We have chosen R =0.001 for rest of the results. The effect of R on predictor performance is shown in Figure 5.

III. EDGE DETECTION ALGORITHM We have used the Kalman Predictor for detecting changes

in Intensity patterns in an image in both row-wise ( horizontal ) as well as column-wise(vertical) data. The basic step of change detection is described in previous section. We have used a few other steps to optimize the edge detectors. The algorithm is summarized below:

3.1. Procedure Edge Detection

// Input : Image I with dimension R x C

//Output: Edge Map E with dimension R x C

// Initialization

1. Initialize horizontal state variable (x0h),horizontal error covariance (P0h), vertical state variable (x0v),vertical error covariance (P0v),

2. Set Process and Measurement Noise Covariance Q and R.

3. Define lower and an upper threshold( tl and th) for edge detection

// Forward Estimation and Difference Calculation

4. For i=1;i<=R;i++

5. For j=1;j<=C,j++

6. If j==1

7. Initialize Kalman Filter.

8. else

9. Find predicted pixel value in horizontal direction using equation (3)-(7);

10. end If;

11. Calculate Horizontal Forward Difference( HFD) by absolute subtraction of predicted values and actual values.

12 end For.

13.end For;

14. Repeat steps 4-13 for vertical direction by interchanging rows and column access.

15. Calculate Vertical Forward Difference(VFD) by absolute subtraction of predicted values and actual values.

// Backward Estimation and Difference Calculation

16. For i=1;i<=R;i++

17. For j=1;j<=C,j++

18. Inverted Row Image Iir( i,C + 1- j)= I (i, j);

19. Inverted Column Image Iic(R +1- i, j)= I (i, j);

20. Repeat steps 4-15 with Iir and Iic to find Horizontal and Vertical Backward Differences : HBD and VBD.

// Edge Localization and Spurious point rejection using dual thresholding

21. For i=1;i<=R;i++

22. For j=1;j<=C,j++

23. Horizontal Difference, HD(i,j)= HFD(i,j) + HBD(i,j);

24. Vertical Difference, VD(i,j)= VFD(i,j) + VBD(i,j);

25. If HD(i,j)>tl and HD(i,j)<th

26. Horizontal Edge Eh(i,j)=1;

27. end If;

28. If VD(i,j)>tl and VD(i,j)<th

29. Vertical Edge Ev(i,j)=1;

30. end If;

31. Edge, E( i,j)= Eh(i,j) + Ev(i,j);

32. end For;

33. end For;

-end Procedure

The edge localization and spurious edge detection is done using dual direction (forward and backward) prediction and dual thresholding. For intensity step, the peak difference between actual and predicted values will be located in both sides of actual transition point in case of forward and backward predictions. In the transition location, the sum of difference will attain maximum value. Thus, by thresholding above a certain smaller difference value, these localized edge points can be detected. However, for spurious point both the forward and backward differences will be positioned in same location thus a bigger peak will be obtained for sum of difference. Thus by introducing a higher threshold, these spurious edges can be cut off. This fact is illustrated in the simulation of Figure 6, where only for step intensity profiles the edges are detected and the spurious points are rejected.

IV. RESULTS AND DISCUSSIONS

A. Kalman Filter Based Edge Detector and Comparison with other Standard Detectors We have first simulated our method with different standard

images and compared our results with the edges obtained from Sobel Operator, a standard edge detector. We have selected three standard gray scale images named Cameraman, Peppers and Spine to extract edges using our proposed method and Sobel Method. High end detectors like Canny, primarily rely on gradient based edge detectors and then other processing techniques are applied to extract true edges. We also did not apply any post processing for edge localization and edge linking. Thus we have compared our results to only basic gradient based edge detection operators like Sobel. The threshold selection of Sobel Detector is first determined using Matlab’s inbuilt function and then adjusted slightly by trial and

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error to get optimum result. The qualitative result of edge detection is shown in Figure 7. From the figure it is clear that proposed basic detector gives a better performance than Sobel Gradient based edge detectors with optimal threshold selection as discussed earlier. In our method, the lower and upper thresholds are kept at 10 and 50 respectively.

B. Edge Detection in Presence of Random Noise The main advantage of our method is its robustness against

the noise compared to gradient based methods. This is due to two reasons. First of all, the adaptability of Kalman Filter to a sequence of data pattern in presence of noise is very good. Secondly, the use of dual direction prediction and dual threshold helps to improve signal to noise ratio in detected edge map. We have first simulated our method with a standard image with different noise added. Coins image is selected and we have compared our result with other two standard gradient based edge detectors namely Sobel and Prewitt. The qualitative evaluation result is summarized in Figure 8. From Figure 8, we can see our algorithm performs better in presence of noise, and comparison gets more tilted towards our method as we increase the noise level.

Fig. 6. Spurious noise rejection using dual

prediction and dual thresholding

The performance evaluation is done quantitatively using Pratt Figure of Merit (PFOM) [13]. This FOM is defined as,

PFOM= (1/Im ) ∑ =

Ia

i 1( 1/(1+ a. di

2)) ;

where, Im= Max ( Ia, It) Ia= Actual edge points, It=True edge points, a= Scaling factor, di= Separation distance of actual edge point from the edge normal of true edge points. The FOM metrics are summarized in Table I. In all the cases, Canny edge detection of noiseless image is taken as reference.

Fig. 7. Algorithmic performance evaluation (a)

original image (b) proposed method output (c) Sobel output

(a) (b) (c)

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Fig. 8. Algorithmic performance evaluation for noisy images (a) original Image (b) proposed method

output (c) Sobel output

TABLE I. PRATT FIGURE OF MERIT FOR DIFFERENT NOISY IMAGES

Coin Image with type of noise

Proposed Method PFOM

Sobel PFOM

Prewitt PFOM

No noise 0.9175 0.8382 0.8263 Salt & Pepper with density

0.05

0.9108 0.6996 0.6817

Salt & Pepper with density 0.2

0.8274 0.3495 0.3118

Gaussian with 0 mean and 0.003

variance

0.9216 0.8114 0.8043

Gaussian with 0 mean and 0.01

variance

0.8908 0.7428 0.7315

Poisson 0.9131 0.8418 0.8103

C. Comparison with Method Proposed by Murphy. As discussed earlier the method proposed by Murphy is

also based on Kalman Filter , but it produces many false edges in case of intensity ramp due to its re-initialization. Moreover the noise robustness of their method is also poor in comparison to our method. This fact is illustrated in figure 9. A synthetic Ramp Intensity image is generated and Canny, our proposed algorithm and Murphy’s method is applied to detect edges. It is clear that Murphy’s method produces many false edges. The false edge detection probability of our method and Murphy’s Method for Coins image is 0.07 and 0.17 respectively.

Fig. 9. Ramp Intensity Image and Edge Detection by various methods

V. SUMMARY AND CONCLUSION In this paper we have introduced a Kalman Predictor based

edge detection scheme, which is an alternative for conventional gradient based edge detectors. As seen from the results, performance of this detector in noisy environment is far superior than any of the conventional gradient based edge detectors. Moreover, there is a greater scope for tuning this detector to adapt for different edge detection scenario. Using this scheme as basic edge detector and adding preprocessing, edge localization and edge linking process; further improved performance can be achieved.

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