Bezier Curve Data Reconstruction overCompressive Sampling in Image Transmission

Syamsul Rizal, and Dong-Seong KimSchool of Electronic Engineering

Kumoh National Institute of Technology, South Koreae-mail: syamsulrizal123@gmail.com, dskim@kumoh.ac.kr

Abstract—This paper proposed a method for image recon-struction using Bezier Curve function. This method comparedwith the previous works, compressive sampling and discreteFourier transform (DFT). Compressive sampling is a methodwhich has small number of data sample and DFT used to performFourier analysis in many practical applications in digital signalprocessing. One of the causes of errors in image reconstructionis lack of sampling. In order to solve that problem, this paperproposed a method by using Bezier Curve to interpolate databetween sampling points to reduce the error. The result showsthat image reconstruction using Bezier curve has close to theoriginal image compared to compressive sampling and DFT.

Index Terms—Bezier curve, compressive sampling, DFT, imagereconstruction.

I. INTRODUCTION

Image or video processing is any form of signal processingwhere the input is an image, such as photographs or videoframe, while the output of image processing can be either animage or a set of characteristics or parameters related to image.Many techniques can be used in image processing such ascompressive sampling, discrete Fourier transform (DFT), etc.

Compressive sampling or compressive sensing (CS) is asignal processing method for efficiently and reconstructiona signal but contradicted with the common method in dataacquisition. This technique can reduces power consumptionby requiring acquisition and transmission of fewer samples[1]. The signal can be reconstructed by using fewer samplingcompared to traditional method such as Nyquist theorem,where the sampling data must be at least twice the maximumfrequency used in the signal [2], [3].

Besides compressive sampling, Discrete Fourier Transformalso a technique used for image processing. DFT is used totransform original image from spatial to frequency domain[4], [5]. In image processing, the samples can be the valuesof pixels along a row or column of an image.

This paper proposed a technique to reconstructed the imageusing Bezier curve function to interpolate the data betweeneach sampling point. Bezier curve is a set of independentvariables that represent the coordinates of several points incurved line between two points. Hence, the data betweensampling points can be predicted and reconstructed using thismethod. Using image processing technique, the image sizewill become smaller than the original image with good quality.These techniques can increases energy efficient and send the

Fig. 1. The data frame structure: a) Compressive data transmission, b)Functional data transmission

Fig. 2. Cubic Bezier curve.

image faster than the original due to the small size of theimage [2].

The remain of this paper is organized as follows: Section IIdescribes the compressive sampling and DFT, section III ex-plains the proposed method. Results and analysis are explainedin section IV. Finally, section V concludes the paper.

II. IMAGE PROCESSSING TECHNIQUES

Image processing is a technique to compressed and recon-structed the image to become the image size was smaller thanthe original image but with the image that resembles the actualimage. Many techniques can be used to reconstructed imagein image processing, such as compressive sampling, DFT, etc.

A. Compressive Sampling Overview

Compressive sampling is a compression method with arandom collection an amount of data which is determinedby the value of the rate measurement. This method used 2transformations; sparsity and incoherent transform [2], [3]. Asensing mechanism in which information about signal f(t) isobtained by linear functional recording this value.

yk = 〈f, ψk〉 k = 1, ...,m. (1)

That correlates the object that we wish to acquire with thewaveforms ψk(t). y is a vector of sampled values of f in

the time or space domain, Dirac delta functions (spikes) aresensing waveform. If the sensing waveforms are indicatorfunctions of pixels, then y is the image data typically collectedby sensors in a digital camera. If the sensing waveforms aresinusoids, then y is a vector of Fourier coefficients.

1) Sparsity: Sparsity expresses the idea that the informa-tion rate of a continuous signal may be much smaller thansuggested by its bandwidth. Mathematically, a vector f ∈ Rn

(such as the n-data) which expanded in orthonormal basis(such as a wavelet basis) Ψ = [ψ1, ψ2, ..., ψn] as follows:

f(t) =

n∑i=1

xiψi(t), (2)

where x is the coefficient sequence of f, xi = (f, ψi). It willbe convenient to express f as ψx, where Ψ is the nxn matrixwith ψ1, ..., ψn as columns. The implication of sparsity is nowclear: when a signal has a sparse expansion, one can discardthe small coefficients without much perceptual loss.

2) Incoherence: Incoherence extend the duality betweentime and frequency and expressed the idea that objects havinga sparse representation in Ψ must be spread out in the domainin which they are required. Suppose there is a pair (Φ,Ψ) oforthobases of Rn. The first basis Φ is used for sensing theobject f as in (1) and the second is used to represent f . Thecoherence between the sensing basis Φ and the representationbasis Ψ is

µ(Φ,Ψ) =√n. max

1≤k,j≤n|〈ϕk, ψj〉|. (3)

The coherence measures the target correlation between anytwo elements of Φ and Ψ. If Φ and Ψ contain correlatedelements, the coherence is large. Otherwise, it is small. Asfor how large or small, it follows from linear algebra thatµ(Φ,Ψ) ∈ [1,

√n].

B. Fourier Transform Overview

In mathematics, the discrete Fourier transform (DFT) is oneof the spesific form of Fourier analysis. It transforms onefunction to another, which is called the frequency domainrepresentation of the time domain. Yet the DFT requires aninput function that is discrete and whose non-zero values havea limited (finite) duration.

Since the input function is a finite sequence of real orcomplex numbers, the DFT is ideal for processing informa-tion stored in computers. In particular, the DFT is widelyemployed in signal processing and related fields to analyzethe frequencies contained in a sampled signal, to solve partialdifferential equations, and to perform other operations such asconvolutions. The DFT can be computed efficiently in practiceusing a fast Fourier transform (FFT) algorithm.

FFT algorithms are so commonly employed to computethe DFT, the two terms are often used interchangeably incolloquial settings, although there is a cleal distinction: ”DFT”refers to a mathematical transformation, regardless of how itis computed, while ”FFT” refers to any one of several efficientalgorithms for the DFT.

Fig. 3. Bezier curve transmission process

The sequence of N complex numbers x0,...,xN−1 is trans-formed into the sequence on N complex numbers X0,...,XN−1

by the DFT according to the formula: where is a primitive N th

root of unity.

III. SYSTEM MODEL

Data frame structure is a basic method to describe thisproposed method. Figure 1 shows compressive sampling andproposed method data frame structure. The data frame isconsisted of data and protector part. Protector part usuallyrepresented as the error correction such as CRC, ECC, etc.This paper proposed a progressive data transmission usingBezier curve formula, which is described in Figure 1.

A. Bezier Curve

Bezier curve is a set of independent variables that representthe coordinates of several points in curved line between twoor more points. A Bezier curve is defined by a set of controlpoints P0 through Pn, where n is Bezier’s order (n=1 forlinear, 2 for quadratic, etc). The first and last control points arealways the end points of the curve. However, the intermediatecontrol points generally do not lie on the curve.

Consider the following cubic-Bezier formulation. Define Ais the set of number where a0 = |A|min and an−1 = |A|max.Therefore, there exist such a P that represents A where |P | ≤|A| and A ∈ C.

Consider the cubic-Bezier where the curve is representedby 4 points. Figure 2 shows the possibilities to cover a data

——

——

——

a ———————— b ———————— c ———————— d

Fig. 4. image reconstruction : a. Original, b. DFT, c. Compressive Sampling, d. Bezier Curve

which consists of 5 numbers data that lies on the curve withina set that can be represented 4 numbers of cubic-Bezier. Itmeans that by sending 4 data from Bezier function, we canconstruct 5 data on the receiver.

The extended version of the cubic-Bezier, double cubicBezier, where 6 numbers of Bezier function, can be used torepresent 9 numbers of data. This process can be describedas the subdividing process of the curve. In other words, everycurve can be scooped within the Bezier function depends onthe depth of the curve. Therefore, the points of one Beziercurve can be extended as follows,

Pji = (1− τ)Pj−1

i + τPj−1i+1 . (4)

Eq.(4) shows that every data can be approached by anyBezier curved. The detail level is formulated correspondsdirectly with the level of Bezier curve division.

The transmission process of Bezier curve can be seen atFigure 3. If the errors occurs during transmission, the Beziercurve construction is performed by the following procedure.

Algorithm 1 Constructing Bezier Curvet is ratio between 0-1for an order n curve do

Take all lines between the curve defining pointsPlace markrers along each of these line, at distance tif t=0.2 then

Place mark at 20% from the startend ifForm lines between those points, this gives n-l linesRepeat this until have only one line left

end for

Bezier Curve can be constructed using Algorithm 1. Treatt as a ratio, t=0 is 0% along a line t=100% along a line. Takeall lines between the curve’s defining points. For an order nline, that is n lines. Place markers along each of these line, atdistance t. So if t=0.2, place a mark at 20% from the start, 80%from the end. Now form lines between those points. This gives

n-l lines. Place markers along each of these line at distance t.Form lines between those points, this will be n-2 lines. Repeatthis until have only one line left. The point of t on that linecoincides with the original curve point at t.

B. Proposed Method

Starting from the S received data, the receiver createssampling data by using compressive sampling in P numberof sampling which is much less than S. For every 4 P pointsof sampling data (P0,P1,P2,P3), Bezier curve is applied tointerpolate the data. Let P0 and P3 are the start and the endpoint of Bezier curve, P1 and P2 are intermediate pointswhich are used as control points to create the Bezier curve, Fig.2. This algorithm is repeated until the Bezier curve covers allP sampling points. After Bezier curve is obtained, the receiverextracts its function and reconstructs the data based on Beziercurve.

IV. RESULTS AND ANALYSIS

In this simulation, we consider image 100x100 pixels tobe reconstructed. We compare the reconstructed image byusing compressive sampling, DFT, and Bezier curve. We usestandard DFT and 10 percent of data sampling in compressivesampling. Figure 4 shows 4 simulation results of imagereconstruction. Figures 4a are the original image which wouldbe reconstructed. In both of the Figures 4b and 4c show theimage cannot be reconstructed well. The quality of the imagesare poor. In Figures 4d show the quality of image are betterthan the other method (CS and DFT). With the proposedmethod the image reconstructed clear and well.

Beside the image quality, compressed technique is usefulfor reduce the size of image. Table 1 shows the size differentfrom each method. From that Table we can see the image sizeusing proposed method is smaller than compressive samplingand DFT methods. Using this method we can reduce the imagesize become smaller but with good image quality.

The size of an image will affect the transmission timein the network. Figure 5 shows the transmission time forevery method. In assumption, the bandwidth in the networkis 10KBps. The graph shows that the image which has beenreconstructed using Bezier curve is faster than the othermethods. It means we can save time to transfer image fromthe source to the destination with good image quality.

V. CONCLUSION

In this paper, an approach to reconstruct data by takingadvantage of Bezier curve in compressive sampling and DFTwas presented. Compressive sampling is a sampling methodthat against sampling paradigm that the number of sampling ismuch less than the original data. This method caused data errorin reconstruction. DFT is a method which is used commonlyin image processing. This paper presented a method to helpimage reconstruction by adding a function called Bezier curveto interpolate data between sampling points the reconstructdata. The cubic Bezier curve is used in this paper, thismethod takes 4 sampling points to interpolate and construct

No. Image Size (KByte)1 Original a1 24

Original a2 14Original a3 19Original a4 23

2 DFT b1 24DFT b2 18DFT b3 20DFT b4 18

3 CS c1 10CS c2 32CS c3 32CS c4 31

4 Bezier Curve d1 3Bezier Curve d2 2Bezier Curve d3 3Bezier Curve d4 3

TABLE IIMAGE SIZE

Fig. 5. Transmission time of images

the image (100x100 pixels). By using this method, imagereconstruction is closer to the original data compared to onlyuse the compressive sampling or DFT methods. For futurestudy, we will develop a method to suppress the constructeddata to be same as original data. Another way is by addingerror correction in data it self to help data reconstruction.

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