A derivative based algorithm for image thresholding

Transcription

1 A derivative based algorithm for image thresholding André Ricardo Backes Bruno Augusto Nassif Travençolo Mauricio Cunha Escarpinati Faculdade de Computação - Universidade Federal de Uberlândia Av. João Naves de Ávila, 2121, CEP , Uberlândia, MG, Brasil Abstract This work proposes a different approach for threshold selection that uses the second derivative of the gray levels histogram of an image. The derivative is an useful tool to understand the behavior of a signal, here represented by the image histogram. By using it, it is possible to detect the peaks of a histogram, i.e., regions of the histogram that represent clusters of different gray levels. In a bimodal histogram, these peaks represent object and background. Both peaks can be detect by mapping the gray levels at the negative regions located between two consecutive zero crossings of the second derivative of the histogram, and the region between these peaks represents the diffuse separation between object and background on the image. We compared the proposed method with other threshold selection methods, such as Otsu. The obtained results show that our appraoch is a good alternative for threshold selection. the second derivative of a histogram it is possible to split the histogram into different regions and identify which of these regions are related to the object and background of the image. This also enables us to detect the region between peaks, which represents the diffuse separation between object and background on the image This paper is structured as follow: in Section 2, we present a detailed description of the derivative property of the Fourier Transform and the proposed approach. In order to validated the proposed method, we compare it with four different thresholding techniques described in the literature: Valley Emphasis [5], Modified Valley Emphasis [4], Balanced Histogram [3] and Otsu [6]. We applied these techniques in a set of different images. Section 3 presents these experiments and the obtained results. Section 4 presents the concluding remarks and discuss future work. 2. Second derivative threshold 2.1. Derivative Property of the Fourier Transform 1. Introduction Image segmentation is one of the most difficult task in image processing [7]. Among many existent techniques, thresholding is a simple and commonly used approach [8]. Its basic idea is to group the image pixels according to their gray-levels using a specific threshold. There exist many method to automatically select the optimal gray-level to use as a threshold value [5, 4, 3, 6], being the Otsu method one of the most used due its simplicity and excelent result in real world images [8]. In this work we propose a novel threshold selection method based on the second derivative of the histogram of the image. The method is based on a simple idea that combine two basics concepts widely applied in image processing: histogram analysis and the use of derivatives for detecting the peaks representing object and background. By using The Fourier transform is one of the most important tools in signal processing and image applications. It enables us to analyze the behavior of a signal from its frequency spectrum, so that, it is possible to analyze signal s characteristics from distinct regions of the spectrum, independently [2, 1]. Among its many properties, the Fourier transform has the derivative property. This property enable us to calculate the derivative of a signal u(t) from its frequency spectrum and it can be described by the following equation d a u(t) dt a = F 1 {F {u(t)} (j2πf) a } (1) where F and F 1 are, respectively, the Fourier transform and its inverse, f is the frequency, j is the imaginary number and a is the order of the derivative. Although simple, we must be consider some important aspects in order to apply the derivative property over a sig-

2 between sequential bins is always one. Then, it should be considered the presence of the Gibbs phenomenon in the derived signal. This is due to the discontinuity at the extremities of the signal (non-periodic signal) and the fact that the Fourier transform does not converge uniformly in the discontinuities [1]. In order to avoid this problem, we use a replication scheme and signal reflection, so that the signal becomes periodic and the Gibbs phenomenon is diminished. This replication process is illustrated in Figure 1b, where the original signal is shown in dark color in the interval [ ]. Finally, we must also consider the tendency that the differentiation methods have to emphasize high frequencies of a signal, including undesired noises [2, 1]. This setback can be easily avoided with the application of a low-pass filter, such as the Gaussian filter, during the derivative calculus (using the convolution property of the Fourier transform) as shown by the following equation: d a u(t) dt a = F 1 {F {u(t)} G(f, σ) (j2πf) a } (2) where G(t, σ) is the Fourier transform of a Gaussian with standard deviation σ Proposed approach Figure 1. (a) Example of gray level histogram of an image; (b) Histogram after replication and signal reflection. The dark part of the graph represents the original signal at [ ]; (c) Second derivative of the histogram. The light line indicates the zero crossings. nal. First of all, it should be ensured a good and uniform space sampling of the signal. In the case of gray-level histograms, this is not a problem as each bin of the histogram represents a different integer gray level and the difference In a bimodal histogram, we have the most of grayscale information concentrated in two intervals: one representing the object and, the other, the background. The derivative is an useful tool to understand the behavior of a signal, here included the histogram of a grayscale image. The second derivative of a histogram enable us to detect the peaks of a bimodal histogram. They are located between two zerocrossing of the derivative when its signal is negative (Figure 1c). By considering these two ranges of gray levels as the two peaks in a bimodal image, the intermediate values (i.e., the range of values where the derivative is positive between the two peaks) represents the set of values that do not belong to the object, neither the background. These values represent the fuzzy separation between object and background. In this context, we propose a threshold value located at the center of the valley between these two peaks, i.e., between the second and first zero crossings of the first and second peaks, respectively. 3. Experiments and results 3.1. The influence of the smoothing in the histogram In our method, we have used the Gaussian filter with standard deviation σ to smooth the histogram before com-

3 Figure 2. (a) Example of gray level histogram of an image; (b) Histograma of the image; (c)-(f) Second derivative and its zero crossings for different values of σ = {5, 10, 15, 20}. puting its second derivative. This is necessary as differentiation methods have a tendency to emphasize high frequencies of a signal, including undesired noises [2, 1]. However, the degree of smoothing that we apply over the signal also affects the number and position of the zero crossings in the second derivative and, as a consequence, the location of the selected threshold. As an example, Figure 2a shows an image, Figure 2b shows its histogram, while Figure 2c-f shows the second derivative and the zero crossings for different σ values. It is clear that the value of the parameter σ affects the second derivative as well as the number of zero crossings and the point of the selected threshold. Depending on the histogram aspect, a small σ will maintain small variations in the histogram, which will be detected as small peaks in the second derivative. In the case where more than two peaks are present, we use the two most central peaks to compute the threshold. In order to find out which σ will produce an optimal threshold, we have tested different values for this parameter in a small set of images. Figure 3 shows some segmentation results for various images and σ values while Table 1 presents the optimal thresholds computed for each image and σ value. We notice that, as the σ value increases, the value of the selected threshold tend to decrease and the segmentation becomes inefficient. However, the most noisy the histogram, larger the σ value will be necessary to achieve this result, as we can see in the camera picture. As a conclusion, based on our experiments and the image segmentation quality, we suggest to use σ = 4 as the best smoothing parameter for the proposed method. σ = 4 σ = 6 σ = 9 σ = 14 σ = 17 camera peppers boat house blobs Table 1. The optimal thresholds for various values of σ Comparison with other thresholding methods In this experiment, we aim to compare our method with four thresholding methods found in literature: Valley Emphasis [5], Modified Valley Emphasis [4], Balanced Histogram [3] and Otsu [6]. To accomplish this task, we use three different standard images: boat, peppers and lena. For this experiment we considered σ = 4 as the best smoothing parameter for our method. Figure 4 shows the histogram and the threshold values achieved for each method and image considered, while Figure 5 shows the segmentation results. We notice that our

4 Figure 3. Segmentation results for various images and σ values. From top to bottom: original image, and segmentation achieved for σ = {4, 6, 9, 14, 17}. method tends to compute a threshold value similar to Valley Emphasis and Otsu mehods, so that, these methods produce similar segmentation results. The exception is the lena image. This is due to a more complex variation among gray levels, which is characterized by a large set of peaks and valleys in the histogram. In this case, the use of a higher σ to produce a smoother curve would decrease the threshold value computed. However, this have the disadvantage that the segmentation tends to become inefficient as we see in Figure 5. Smaller threshold values, as those computed using the other aproaches, are not able to yield a good set of details at lena s face. This testifies that our method could get better results in comparison to other approaches in these cases.

5 Figure 5. Segmentation results for different images. From top to bottom: original image, Valley Emphasis, Modified Valley Emphasis, Balanced Histogram, Otsu, Proposed. Figure 4. Threshold computed for different images. (a) boat; (b) peppers; (c) lena. 4. Conclusion and Future Work In this paper, we proposed a threshold selection method based on the second derivative of the histogram. This novel method uses the zero crossings of the second derivative to detect the two peaks that represent the object and background on a grayscale image. Then, it selects a threshold value located at the center of the valley between these two peaks. The method also uses a Gaussian filter to smooth the histogram before computing its second derivative. This process enables us to reduce noisy information in the histogram and helps to locate the best threshold value. Comparison with other threshold techniques have demonstrated the effectiveness of the proposed method on different images. Thus, the presented method opens a new source of research in image segmentation to be explored. As future work, we aim to study a mechanism for automatic selection of the σ parameter, so that, the threshold could be selected based exclusively on the image content and without human interference.

6 Acknowledgements Prof. André R. Backes gratefully acknowledges the financial support of CNPq (National Council for Scientific and Technological Development, Brazil) (Grant #302416/2015-3) and FAPEMIG (Foundation to the Support of Research in Minas Gerais) (Grant #APQ ). References [1] E. O. Brigham. The Fast Fourier Transform and its applications. Prentice Hall, Englewood Cliffs, NJ, USA, [2] L. da F. Costa and R. M. Cesar Jr. Shape Analysis and Classification: Theory and Practice. CRC Press, [3] A. dos Anjos and H. Shahbazkia. Bi-level image thresholding - A fast method. In BIOSIGNALS (2), pages IN- STICC - Institute for Systems and Technologies of Information, Control and Communication, [4] J.-L. Fan and B. Lei. A modified valley-emphasis method for automatic thresholding. Pattern Recognition Letters, 33(6): , [5] H. F. Ng. Automatic thresholding for defect detection. Pattern Recognition Letters, 27(14): , [6] N. Otsu. A threshold selection method from gray level histograms. IEEE Trans. Systems, Man and Cybernetics, 9:62 66, [7] N. R. Pal and S. K. Pal. A review on image segmentation techniques. Pattern Recognition, 26(9): , [8] P. K. Sahoo, S. Soltani, and A. K. C. Wong. A survey of thresholding techniques. Computer Vision, Graphics and Image Processing, 41: , 1988.