DIGITAL ELECTRONIC ARITHMETICS WITH APPLICATIONS

Eric Whartona, Dr. Karen Panettac, Dr. Sos AgaianbaTufts University, Department of Electrical and Computer Engineering, Medford, MA 02155

bThe University of Texas at San Antonio, 6900 North Loop 1604 West, San Antonio, TX 78249

ABSTRACT ~~~~~~~~~~~~~AdditionSystemABSTRACTUnifonnSource

In this paper we introduce a new parametric imageprocessing framework to more accurately process images. Absorption 2nd Absorption Screen showingThis Parameterized Digital Electronic Arithmetic (PDEA) Filter, gi(ij) Filter, g2(ij) results of addition,model replaces linear arithmetic with non-linear operations (a) f(j)which more accurately model the non-linear properties ofimage processing. In this paper, the parameterized model 1gg P(g)

* 7 * 7 7 - * 7 7 7 r 7 * > f~~~~~~~~~~~~~~~~~c(g) =-A(M) n'6 1-is presented, including experimental methods for choosing \ A)parameters. Grey-Tone Image p transform Transmitted Light

Function g(i,j) Intensity, I(i,j)Index Terms- Image Enhancement, Logarithmic Image (b)Processing, Measure of Enhancement, P1(g)=,Z(A1-t°tP2,,,) r lel

1. INTRODUCTIONTransmitted Light q-1 transform Grey-Tone ImageIntensity, I(i,j) Function g(i,j)

In many digital electronic arithmetic problems, there is a (c)need for a non-linear arithmetic to process signals. For Figure 1: System diagram of the Parameterized Digital Electronicexample, when processing heartbeat signals, there is a need Arithmetic, (a)Addition of two images, (b)p transform for multiplication,for a method of combining signals in a manner which is (c)Inverse 9 transformrepresentative of both. Images are another specific case of Concatenated Adder Systemsuch a problem.

Image processing is the system of mathematically Input d Atransforming an image from one form or another, generally Iaf(xg Ad Imageto change some characteristics of the image [1]. Thisincludes many applications, such as image enhancement,edge detection, object recognition, and noise reduction. Figure 2: An example system of arithmetic systems which requires theCurrent practice in image processing makes little use of any PDEA model for the output to be representative of the operationsstandardized, mathematically rigorous arithmetical structurespecifically designed for image manipulation [2]. andsitu be practical frifl.Te Logarithmic Imag

Traditionally, image processing makes use of linear oe LIP) model was developed to address these [4].operations to manipulate images, but since this is inherently The LIP model treats images as light absorption filters,a non-linear process, accuracy issues can arise. First, when as shown in figure 1 [4]. These filters can then be added byresulting pixel intensities lie outside the range [,M), they lacin multile filters in series, subtracted by removingare clipped, causing a loss of information. Second, linear filters, and so forth Figure 1 also shows the most basicoperations typically do not yield results consistent with the system in the LIP multiplication; the y transform.physical phenomena. By solving these common problems Despite the inherent advantages of the LIP model over

in an image processing context, the methods can be utilized linear image processing, there are systems for which neitherto improve many areas of signal processing. LIP nor linear arithmetic systems will work. Figure 2 gives

> . . ~~~~~~~~~~an example of one such system. Figure 2 shows aThe four fundamental requirements of an image F 2 oprocessing framework are given by Pinoli in [3]. These concatenated adder system, which takes an input image and

state~~~~ ~ ~~thaafrmwr mustbe ase onapysiall adds it to itself several times. It is easy to see that, usingrelevant image formation model, the mathematical linear arithmetic for the adder blocks, the result will be outoperations must be consistent with the physical nature of of the range for many Images, especially bright images. Ifimages, the operations must be computationally effective, thsiXen efre ih n8btacietr,ti

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qe) (f. )

Figure 3: Comparison of linear, LIP, and PLIP arithmetic to demonstrate the fifth requirement; (a)Linear addition of Lena and Clock, (b)Linearaddition of Lena and Plane, (c)LIP addition of Lena and Clock, (d)LIP addition of Lena and Plane, (e)PLIP addition of Lena and clock, (fPLIP additionof Lena and Plane; This shows that only the PLIP is able to perform both additions to satisfy the fifth requirement

information in the overflow is simply lost. If the LIP model 2. PARAMETERIZED DIGITAL ELECTRONICis used, it will not overflow, however the output image will ARITHMETICquickly be driven to saturation.

In this paper, we address these issues by proposing In this section we introduce proposed changes to arrive atmodifications to the LIP system which will make it the Parameterized Digital Electronic Arithmetic (PDEA).applicable to any signal processing problems. The LIP Originally, no method was used to select the optimal valuesystem has parameters, such as M, which were not of M, instead using the linear value of M = 256. Ourexperimentally derived. Instead, the values from linear research has shown that by varying M, better results can beimage processing are used. This paper will show that by obtained. In our previous research, we considered onlyparameterizing M, better results can be achieved. However, addition when choosing the optimal M value. In this paper,this parameterization leads to an under constrained problem. we consider addition, subtraction, and multiplicationThis can be addressed by introducing a fifth requirement to separately. This allows the LIP model to be fine tuned forthose of Pinoli. We propose that an image processing each case, yielding the most effective image processingframework must not visually damage either signal. When a framework possible.visually "good" image is added to another visually "good" We further propose the addition of an exponentialimage, the output image must also be visually "good." With coefficient, J, to the (p functions used for multiplication. Inthis requirement, the output image in figure 2 will always be this manner, we can fine tune the multiplication and adjusta "good" image, no matter how many systems it is the sensitivity to either the high end or the low end of theprocessed with inside the larger image processing system. pixel intensity range of [O,M). This better models theAlso, the system in figure 2 can also be replaced with any varying response of the human eye at high versus low lightother system of the arithmetic operation systems, and the intensity [7]. This allows for a more effective multiplicationresult will always be "good." This fifth requirement is operation which gives results that are more consistent withdemonstrated in figure 3. the physical model. To choose the ideal value, we use the

This paper is organized as follows: section 2 presents measure of image enhancement.the Parameterized Digital Electronic Arithmetic (PDEA), In this manner, we introduce the Parameterized LIPsection 3 presents experiments used to select parameters, model. It makes use of the same gray tone functionsection 4 presents applications, section 5 presents presented in equation (1). The arithmetical operations are:experimental results, and section 6 is a discussion of results .g1g2and some concluding comments are made. ®l( g2 =-g + g2- y((1)

kM g-2k(M)~~~~~~~2

6P(g) = -(M) -In'I - (4)A(M)) Linear Times

(p'(g) = (M.1exp1 2J (5) EMEc.86

The new parameters y(M), k(M), and A(M) are linearfunctions of the type y(M) = AM + B, with A and B areconstant parameters. Also,,is a parameter, and Eis a smallconstant introduced for the case where g2 = k(M) to avoiddividing by 0. LIP Times, A 1 PlUP Times, A 4

It is also important to point out many useful properties EMEE= 1.88 x 1010 EMEE =8.59 x 1012 (t)ofPDEA operations. It can be shown that PDEA operations (d) (e)have associativity, commutativity, a unit element, and Figure 4: Results of modifying the X(M) value for PLIP multiplication,

distributivity; much like their linear counterparts: (a)Moon image, (b)Clock image, (c)Linear multiplication, (d)LIPAssociativity: f (g ® h) ( g) h vs. X(M) for the LIP multiplication using a standard double and 15-bit

Af (g 8 h)= (f 8 g) 8 h floating point with a peak at X(M) = 1026

Commutativity: fig g.i.fCommutativty: |f * g =g f enhancement measures. This is done by performing a

Unit Element Oefof simulation using several algorithms which utilize the, (0)f =f logarithmic arithmetic. We tested for many values of, as

Distributivity: a * (f ( g) =(a * f) (a * g) well as all possible combinations of enhancementparameters, and assessed the output images using the

3. METHODS TO SELECT PARAMETERS measures of image enhancement. We expect a consistentpeak for all enhancement parameters at a single value of,.

In this section, we present methods used to select the As the graph in figure 5 shows, there are three consistentoptimal values for the PDEA parameters. In order todetermine the optimal value for y(M), k(M), and A(M); weattempt to maximize the information in the result of theoperations. To measure this, the operations are perform2ed010using a standard 64-bit double precision floating point and a 80015-bit floating point number, with the difference measured 00using the mean squared error (MSE). By selecting values 200

corresponding to the maximum MSE we maximize the 0300information in the output image. This study was performed 125 00for many different combinations of images using the three Threhold 3 °arithmetical operations. Figure 5: Logarithmic AME vs. threshold vs. for Lena processed using

The result of this experiment for multiplication, shown Bi-Histogram Equalization with LIP gray multiplicationin the graph in figure 4.f, shows several peaks. The largestpe~~~~akocur at X(M) 1026.. Tersutfradioshw peaks for alpha values, the largest and most consistent ofpeak occurs at k(M) = 1026. The result for addition shows whc ocusfr 2

a similar graph. These results, including the same peakswith similar relative sizes, were found in all of our 4. APPLICATIONSsimulations. This indicates that the maximum amount ofinformation is retained in the images when this value is In this section, we show parametric applications of theused. After investigations using the values from the peaks digital electronic arithmetic. We will present the Parametricand other values from the graph in figure 4, it was modified Lee's Algorithm and the Parametric modifieddetermined that this is the optimal value for algorithms and Bi-Histogram Equalization. We will show that, by only

mesues modifying these parameters, we have better imageThe ,B parameter gives greater control over the enhancement. We will also present the measure of image

logarithmic function used for the multiplication, which can enhancement, used to quantify results.assign increasing weight to the higher or lower end of thepixel intensity range. We determine these values using the

Original Classical LIP Parameterized LIP Original Classical LIP Parameterized LIPEMEE = 0.1518 EMEE = 1.7879 EMEE = 5.5749 EMEE = 0.2158 EMEE = 0.3576 EMEE = 1.2238

(a) (b) (c) (a) (b) (c)Figure 6: Comparison of classical LIP to PLIP using Bi-Histogram Figure 7: Comparison of classical LIP to PLIP using Lee's Algorithm:

Equalization (BHE): (a)Original parking lot image, (b)Enhanced image (a)Original Pentagon image, (b)Enhanced image using Lee's Algorithmusing BHE with classical LIP, (c)Enhanced image using BHE with PLIP with classical LIP, (c)Enhanced image using Lee's Algorithm with PLIP

4.1. Parametric modified Lee's Algorithm 1 kk2 (vw jwEME,k Im(a)=-ma, i|In

I max4 (9)

Lee's image enhancement algorithm functions by first ik2l/lklJink+C) ji,i+cdecomposing an image into two parts, a smoothed imageand the difference between this and the original image. Where 'maxk/ and win;k/ are the local maximum andThese images are then weighted and recombined [5]: mum ,rin t

minimum, respectively, within the block, a iS a constant,and c is a small constant to avoid dividing by 0.

5. EXPERIMENTAL RESULTSWheref'( i, j) is the output image,f( i, j) is the input image,a( i, j) is the average of the pixels surrounding the pixel at In this section, we present experimental results of the(i, j), and y, n are user defined operating parameters. parameterized algorithms. We use visual assessment as well

as the EMEE measure to decide which performs best.4.2. Parametric modified bi-histogram equalization Figure 6 shows the results of bi-histogram equalization

for the parking lot image, shown in figure 6.a. As can beBi-histogram equalization is based upon standard histogram seen, the image in figure 6.b, processed using the classicalequalization but introduces a thresholding parameter to give LIP, shows improved contrast and is more visuallybetter control over the enhancement and to obtain better appealing than the original. It also has an increased EMEEenhanced images [5]. Bi-histogram equalization first value. The image processed using PDEA, however,decomposes the image into two sub-images. These outperforms this both based upon the measure and a visualsub-images are then equalized separately and recombined, assessment. The image has far better contrast and all theaccording to the following formula: details can be seen.

Figure 7 shows the results of Lee's Algorithm for thef(x) {X i 33P(x)*(Threshold Y, ii) x<=Threshold (8) Pentagon image. Again, this shows that PLIP outperformsJ Threshold P(x) * (Yx - Threshold) x > Threshold classical LIP. The image in figure 7.b, processed with

classical LIP, shows improvement for both contrast andWhere f(x) is the output image, P(x) is the probability sharpness. The image in figure 7.c, processed with PLIP,distribution function such that P(M) = 1, Ymax and Ymin are shows more improvement and has a higher EMEE value.the desired maximum and minimum of the output range, and The image processed using PDEA clearly has betterthreshold is a user-defined operating parameter. sharpness and contrast, and is more visually appealing.

Also, this image has a much higher EMEE value. This4.3. Measure of Image Enhancement shows that the PDEA works well with standard test images

as well as more complicated images.In order to designate a result as optimal, it is necessary toestablish some objective criteria for quantifying results. 6. CONCLUSIONThe EMEE measure of image enhancement by entropy hasbeen proposed and used to achieve good results [6]. The In this paper, we presented the parameterized ParameterizedEMEE is calculated by dividing an image into k1 x k2 sized Digital Electronic Arithmetic (PDEA). This model has theblocks, processing each block with equation (9), and advantage of being fine tuned for signal processing usingaveraging the results. The EMEE is summarized as follows: mean squared error measurement and objective measures of

enhancement to achieve the more effective signalprocessing model. It was shown that, by simply modifyingthe parameters, one may have better image enhancement.

Also, with the new parameterization, the model is moreeffective for signal processing problems in general. Usingthe methods described herein, the PDEA can be fine-tunedfor any number of signal processing problems.

7. REFERENCES

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[3] J. C. Pinoli. "A General Comparative Study of theMultiplicative Homomorphic, Log-Ratio, and Logarithmic ImageProcessing Approaches," Signal Processing, vol. 58, number 1,1997, pp. 11 45.

[4] M. Jourlin, J. C. Pinoli. "Logarithmic Image Processing; TheMathematical and Physical Framework for the Representation andProcessing of Transmitted Images," Advances in Imaging andElectron Physics, vol. 115, 2001, pp. 129 196.

[5] E. Wharton, S. Agaian, and K. Panetta. "Comparative Study ofLogarithmic Enhancement Algorithms with PerformanceMeasure," in Proc. 2006 SPIE Electronic Imaging, San Jose, CA,paper #606412.

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