A Class of Sharpness Measures

Junfeng Wang*, Xingzhao Liu

*Shanghai Jiaotong University, China. Email: junfengwang@sjtu.edu.cn

Abstract

In signal processing, it is often necessary to measure the sharpness of a distribution. A class of sharpness measures is studied in this paper. First the class is defined. Then the relation between sharpness and the class is clarified, and this justifies the class as sharpness measures. In addition, in general, we investigate the effect of the kernel on the sharpness measure and present a guide to select the kernel.

Keywords: Distribution, Sharpness, Measure.

1. Introduction

In signal processing, it is often needed to measure the sharpness of a distribution. In seismic reconstruction, one needs to measure the sharpness of reflectivity sequences and thus determine the deconvolution quality. In radar imaging, one needs to measure the sharpness of scattering images and thus determine the focus quality. Examples are also seen in other applications.

Different functions can be used to measure the sharpness of a distribution. A typical measure is the sum of squares. Muller used it to measure the sharpness of telescope images [1], Wiggins used it to measure the sharpness of seismic reflectivity sequences [2], and Herland used it to measure the sharpness of radar scattering images [3]. The negative entropy is also a typical measure. Originally, entropy was used by Shannon to measure the average information quantity of a random variable [4]. Actually, it can also be used to measurethe smoothness of a distribution, and thus its negative can be used to measure the sharpness of a distribution. The negative entropy was used by De Vries to measure the sharpness of seismic reflectivity sequences [5] and by Bocker to measure the sharpness of radar scattering images [6].

Efforts are made to unify different measures of sharpness. De Vries defined a class of sharpness measures [6]. However, his definition is incorrect. Fienup defined a class of sharpness measures correctly [7]. Besides, he studied the effect of the kernel on the sharpness measure in the context of radar imaging. However, his conclusion is flawed. Schulz derived the optimal kernel in the context of radar imaging [8].

In this paper, the class of sharpness measures, defined by Fienup, is studied. First the class is defined. Then the relation between sharpness and the class is clarified, and this justifies the class as sharpness measures. Moreover, in general, we investigate the effect of the kernel on the sharpness measure and present a guide to select the kernel.

2. Definition

Let an�0, n=1, 2, ..., N, be a distribution. The sharpness

of an can be measured by

��

���

���

N

n

n

Aa

s1

, (1)

where

��

�N

nnaA

1

, (2)

and �(x), called the kernel, is convex over [0, 1], i.e., ��(x)>0 for x [0, 1]. Note that s does not depend on the order of an’s. Therefore, it seems more appropriate to say that s is a measure of nonuniformity.

Different measures are obtained from different �(x)’s. If�(x)=x�, where �>1, then

��

�

���

��

N

n

n

Aa

s1

. (3)

Especially, when �=2, s is the sum of squares. If �(x)=�x�,where �<1, then

��

�

���

���

N

n

n

Aa

s1

. (4)

If �(x)=xln(x), s is the negative entropy

��

��

��

��

��

�N

n

nn

Aa

Aa

s1

ln . (5)

The definition in (1) is given by Fienup. Also, De Vries defines a class of sharpness measures, i.e.,

��

���

���

N

n

nn

Aa

Aa

h1

, (6)

where �(x) is a monotonically increasing function over [0, 1]. However, this definition is incorrect because there are counterexamples. Letting �(x)=�1/x, one obtains h=�N.Though �(x) is a monotonically increasing function over [0, 1], h cannot be used as a sharpness measure.

3. Relation of s to Sharpness

The tangent of �(x) at x=1/N has the equation

���

����

��

� ����

�����

NNx

Nx 111)(1

(7)

(figure 1). Over [0, 1], �(x) is convex, and thus �(x)��1(x), i.e.,

���

����

��

� ����

�����

NNx

Nx 111)( . (8)

Letting x=an/A, one obtains

���

����

��

� ����

�����

��

��

NNAa

NAa nn 111 . (9)

Accumulating (9) for n=1, 2, …, N, one obtains

���

���

NNs 1 . (10)

(10) gives the minimum of s. Evidently, s attains the

_____________________________ 978-1-4244-2732-1/09/$25.00 ©2009 IEEE

minimum when all an’s are equal, i.e., the distribution is smoothest.

Figure 1. Relation of s to Sharpness.

The line passing points (0, �(0)) and (1, �(1)) has the equation

)0()]0()1([)(2 ������� xx (11) (figure 1). Over [0, 1], �(x) is convex, and thus �(x)��2(x), i.e.,

)0()]0()1([)( ������� xx . (12) Letting x=an/A, one obtains

)0()]0()1([ ���������

��

Aa

Aa nn . (13)

Accumulating (13) for n=1, 2, …, N, one obtains )0()1()1( ����� Ns . (14)

(14) gives the maximum of s. It is easy to show that s attainsthe maximum when only one an is nonzero, i.e., the distribution is sharpest.

Let us consider the variation of s with sharpness (figure 1). When the distribution is smoothest, all (an/A)’s are 1/N. s,the sum of �(an/A)’s, attains the minimum N�(1/N), the sum of �1(an/A)’s. When the distribution becomes sharper, (an/A)’s spread in the x axis. s, the sum of �(an/A)’s, leaves the minimum N�(1/N), the sum of �1(an/A)’s, and tends to the maximum �(1)+(N�1)�(0), the sum of �2(an/A)’s. When the distribution is sharpest, one an/A is 1 and all other (an/A)’s are 0. s, the sum of �(an/A)’s, attains the maximum �(1)+(N�1)�(0), the sum of �2(an/A)’s. As we see, s can be used to measure the sharpness of a distribution.

In analyzing the relation of s to sharpness, Fienup gives and proves (10) only. His proof is based on Jensen’sinequality.

4. Selection of ��(x)

Different sharpness measures have similar but different performances [7]-[10]. In a particular application, one may be preferred to another. Therefore, it is significant to investigate the effect of the kernel on the sharpness measure and find a guide to select the kernel.

Let i and j be two values of n. Then, s is written as

��

���

�����

�

���

���

��

���

jin

nji

Aa

Aa

Aas

,

. (15)

If the sum of ai and aj is a constant, the derivative of s with respect to ai is

� � ������

�

���

����

���

����

��

����

�� Aa

Aa

ji

i

i

j

dxxAA

aAa

Aas /

/)(11 . (16)

(16) shows that the sensitivity of s to the “mass” transfer from aj to ai is determined by ��(x) over [aj/A, ai/A]. When ��(x) is

larger over [aj/A, ai/A], |�s/�ai| is larger, and therefore s is more sensitive to the “mass” transfer from aj to ai. This gives us a guide to select �(x).

Figure 2. Selection of �(x).

Assume that there exist two regions, T1 and T2, in the distribution (figure 2). They correspond to different ranges of an/A, denoted as R1 and R2, respectively. If we hope that T1 is estimated more accurately than T2, s should be more sensitive to the “mass” transfer in R1 than that in R2. This means that we should select �(x) such that ��(x) is larger over R1 than over R2. Here are some examples. If we hope that dark regions are estimated more accurately than bright regions, sshould be more sensitive to the “mass” transfer in dark regions than that in bright regions, and therefore we should select �(x) such that ��(x) is decreasing over [0, 1], like �(x)=x� with 1<�<2, �(x)=�x� with �<1 or �(x)=xln(x). If we hope that bright regions are estimated more accurately than dark regions, s should be more sensitive to the “mass”transfer in bright regions than that in dark regions, and therefore we should select �(x) such that ��(x) is increasing over [0, 1], like �(x)=x� with �>2. If we hope that various regions are estimated with the same accuracy, s should be equally sensitive to the “mass” transfer in various regions, and therefore we should select �(x) such that ��(x) is a constant over [0, 1], like �(x)=x2.

Fienup is the first to note the importance of ��(x). However, he think that if ��(x) is decreasing over [0, 1], the sharpness measure emphasizes making dark regions darker, and if ��(x) is increasing over [0, 1], the sharpness measure emphasizes making bright regions brighter. This statement is inappropriate. In fact, if ��(x) is decreasing over [0, 1], the sharpness measure makes dark regions estimated more accurately, and if ��(x) is increasing over [0, 1], the sharpness measure makes bright regions estimated more accurately.

Also, (16) further justifies s as a sharpness measure. Assume that ai>aj. Then, with the “mass” transfer from aj to ai, the distribution becomes sharper. Meanwhile, since ��(x)>0 and ai>aj, one has �s/�ai>0. This means that with the “mass” transfer from aj to ai, s becomes larger. Now assume that ai<aj. Then, with the “mass” transfer from aj to ai, the

xo

n

an/A

��(x)

o

T1

T2

R1 R2

R1 R2

�2(x)�1(x)

�(x)

1/N an/A 1 x

y

o

distribution becomes smoother. Meanwhile, since ��(x)>0 and ai<aj, one has �s/�ai<0. This means that with the “mass”transfer from aj to ai, s becomes smaller. s does be a rational measure of sharpness.

5. Examples

10 20 300

5

10

15

Index

Value

Figure 3. A Distribution.

10 20 300

5

10

15

Index

Value

Figure 4. Distribution Smoothed Slightly.

10 20 300

5

10

15

Index

Value

Figure 5. Distribution Smoothed Heavily.

OriginalDistribution

DistributionSmoothedSlightly

DistributionSmoothedHeavily

�(x)=x1.1 0.7074286 0.7073571 0.7072998�(x)=x2 0.03150826 0.03145087 0.03140496

�(x)=x3 1.000775�10�3

9.953943�10�4

9.910899�10�4

�(x)=�x0.5 �5.650995 �5.652298 �5.653340�(x)=xlnx �3.461598 �3.462518 �3.463254

Table 1. Sharpness Measures for Distributions in Figures 3-5.

Figure 3 shows a distribution. Figure 4 shows the distribution smoothed by a mean filter of length 3. Figure 5

shows the distribution smoothed by a mean filter of length 5. Table 1 shows the values of some sharpness measures for the three distributions. It can be seen that when the distribution becomes smoother and smoother, the sharpness measures become smaller and smaller. This indicates that the sharpness measures are rational.

10 20 300

5

10

15

Index

Value

Figure 6. Another Distribution.

10 20 300

5

10

15

Index

Value

Figure 7. Distribution with Dark Region Smoothed.

10 20 300

5

10

15

Index

Value

Figure 8. Distribution with Bright Region Smoothed.

OriginalDistribution

Distributionwith Dark

RegionSmoothed

Distributionwith Bright

RegionSmoothed

�(x)=x1.1 0.7472461 0.7468316 0.7472261�(x)=x2 0.05733471 0.05730601 0.05730601

�(x)=x3 3.422004�10�3

3.421760�10�3

3.416868�10�3

�(x)=�x0.5 �4.509948 �4.569323 �4.510194�(x)=xlnx �2.924907 �2.932140 �2.925147

Table 2. Sharpness Measures for Distributions in Figures 6-8.

Figure 6 shows another distribution. This distribution contains a dark region and a bright region. Figure 7 shows the

distribution with the dark region smoothed. Here, a mean filter of length 3 is used. Figure 8 shows the distribution with the bright region smoothed. Also, a mean filter of length 3 is used. Table 2 shows the values of some sharpness measures for the three distributions. Note that the sharpness measures with �(x)=x1.1, �(x)=�x0.5 and �(x)=xln(x) change more when the dark region is smoothed than when the bright region is smoothed. This indicates that if ��(x) is decreasing over [0, 1], the sharpness measure is more sensitive to dark regions than to bright regions. Note that the sharpness measure with �(x)=x3 changes more when the bright region is smoothed than when the dark region is smoothed. This demonstrates that if ��(x) is increasing over [0, 1], the sharpness measure is more sensitive to bright regions than to dark regions. Note that the sharpness measure with �(x)=x2 changes equally when different regions are smoothed. This indicates that if ��(x) is a constant over [0, 1], the sharpness measure is equally sensitive to different regions.

6. Conclusion

s is a rational measure of sharpness. The sensitivity of sto the “mass” transfer from aj to ai is determined by ��(x)over [aj/A, ai/A]. Assume that there exist two regions, T1 and T2, in the distribution. They correspond to different ranges of an/A, denoted as R1 and R2, respectively. If we hope that T1 is estimated more accurately than T2, s should be more sensitive to the “mass” transfer in R1 than that in R2, and therefore we should select �(x) such that ��(x) is larger over R1 than over R2.

References

[1] R. A. Muller and A. Buffington, “Real-Time Correction of Atmospherically Degraded Telescope Images through Image Sharpening,” Journal of Optical Society of America, Volume 64, Number 9, September 1974, Pages 1200-1210.

[2] R. A. Wiggins, “Minimum Entropy Deconvolution,” Geoexploration, Volume 16, Numbers 1-2, April 1978, Pages 21-35.

[3] E. A. Herland, “Seasat SAR Processing at the Norwegian Defense Research Establishment,” 1981 EARSeL-ESA Symposium, Pages 247-253.

[4] C. E. Shannon, “A Mathematical Theory of Communication,” Bell System Technical Journal, Volume 27, Number 3, July 1948, Pages 379-423.

[5] D. De Vries and A. J. Berkhout, “Velocity Analysis Based on Minimum Entropy,” Geophysics, Volume 49, Number 12, December 1984, Pages 2132-2142.

[6] R. P. Bocker, T. B. Henderson, S. A. Jones and B. R. Frieden, “A New Inverse Synthetic Aperture Radar Algorithm for Translational Motion Compensation,” Proceedings of SPIE, Volume 1569, 1991, Pages 298-310.

[7] J. R. Fienup and J. J. Miller, “Aberration Correction by Maximizing Generalized Sharpness Metrics,” Journal of Optical Society of America�A, Volume 20, Number 4, April 2003, Pages 609-620.

[8] T. J. Schulz, “Optimal Sharpness Function for SAR Autofocus,” IEEE Signal Processing Letters, Volume 14, Number 1, January 2007, Pages 27-30.

[9] M. D. Sacchi, D. R. Velis and A. H. Cominguez, “Minimum Entropy Deconvolution with Frequency-Domain Constraints,” Geophysics, Volume 59, Number 6, June 1994, Pages 938-945.

[10] J. Wang, X. Liu and Z. Zhou, “Minimum-Entropy Phase Adjustment for ISAR,” IEE Proceedings of Radar, Sonar and Navigation, Volume 151, Number 4, August 2004, Pages 203-209.