Soo-Chang Pei

Effective color interpolation in CCD color filter arrays using signal correlation

We propose an effective color filter array (CFA) interpolation method for digital still cameras (DSCs) using a simple image model that correlates the R,G,B channels. In this model, we define the constants K/sub R/ as green minus red and K/sub B/ as green minus blue. For real-world images, the contrasts of K/sub R/ and K/sub B/ are quite flat over a small region and this property is suitable for interpolation. The main contribution of this paper is that we propose a low-complexity interpolation method to improve the image quality. We show that the frequency response of the proposed method is better than the conventional methods. Simulation results also verify that the proposed method obtain superior image quality on typical images. The luminance channel of the proposed method outperforms by 6.34-dB peak SNR the bilinear method, and the chrominance channels have a 7.69-dB peak signal-to-noise ratio improvement on average. Furthermore, the complexity of the proposed method is comparable to conventional bilinear interpolation. It requires only add and shift operations to implement.

Discrete fractional Fourier transform based on orthogonal projections

The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform.

Closed-form discrete fractional and affine Fourier transforms

The discrete fractional Fourier transform (DFRFT) is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. We introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by the FFT. Since the continuous FRFT can be generalized into the continuous affine Fourier transform (AFT) (the so-called canonical transform), we also extend the DFRFT into the discrete affine Fourier transform (DAFT). We derive two types of the DFRFT and DAFT. Type 1 is similar to the continuous FRFT and AFT and can be used for computing the continuous FRFT and AFT. Type 2 is the improved form of type 1 and can be used for other applications of digital signal processing. Meanwhile, many important properties continuous FRFT and AFT are kept in the closed-form DFRFT and DAFT, and some applications, such as filter design and pattern recognition, are also discussed. The closed-form DFRFT we introduce has the lowest complexity among all current DFRFTs that is still similar to the continuous FRFT.

Relations between fractional operations and time-frequency distributions, and their applications

The fractional Fourier transform (FRFT) is a useful tool for signal processing. It is the generalization of the Fourier transform. Many fractional operations, such as fractional convolution, fractional correlation, and the fractional Hilbert transform, are defined from it. In fact, the FRFT can be further generalized into the linear canonical transform (LCT), and we can also use the LCT to define several canonical operations. In this paper, we discuss the relations between the operations described above and some important time-frequency distributions (TFDs), such as the Wigner distribution function (WDF), the ambiguity function (AF), the signal correlation function, and the spectrum correlation function. First, we systematically review the previous works in brief. Then, some new relations are derived and listed in tables. Then, we use these relations to analyze the applications of the FRPT/LCT to fractional/canonical filter design, fractional/canonical Hilbert transform, beam shaping, and then we analyze the phase-amplitude problems of the FRFT/LCT. For phase-amplitude problems, we find, as with the original Fourier transform, that in most cases, the phase is more important than the amplitude for the FRFT/LCT. We also use the WDF to explain why fractional/canonical convolution can be used for space-variant pattern recognition.

Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT

The concepts of quaternion Fourier transform (QFT), quaternion convolution (QCV), and quaternion correlation, which are based on quaternion algebra, have been found to be useful for color image processing. However, the necessary computational algorithms and their complexity still need some attention. We develop efficient algorithms for QFT, QCV, and quaternion correlation. The conventional complex two-dimensional (2-D) Fourier transform (FT) is used to implement these quaternion operations very efficiently. With these algorithms, we only need two complex 2-D FTs to implement a QFT, six complex 2-D FTs to implement a one-side QCV or a quaternion correlation and 12 complex 2-D FTs to implement a two-side QCV, and the efficiency of these quaternion operations is much improved. Meanwhile, we also discuss two additional topics. The first one is about how to use QFT and QCV for quaternion linear time-invariant (QLTI) system analysis. This topic is important for quaternion filter design and color image processing. Besides, we also develop the spectrum-product QCV. It is an improvement of the conventional form of QCV. For any arbitrary input functions, it always corresponds to the product operation in the frequency domain. It is very useful for quaternion filter design.

A weighted least-squares method for the design of stable 1-D and 2-D IIR digital filters

We present a new approach to the least-squares design of stable infinite impulse response (IIR) digital filters. The design is accomplished by using an iterative scheme in which the denominator polynomial obtained from the preceding iteration is treated as a part of the weighting function, and each iteration is carried out by solving a standard quadratic programming problem that yields a stable rational function. When the iteration converges, a stable and truly least-squares solution is obtained. The method is then extended to address the least-squares design of stable IIR two-dimensional (2-D) filters. Examples are included to illustrate the proposed design techniques.

Improved discrete fractional Fourier transform

The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFTs) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.

The discrete fractional cosine and sine transforms

This paper is concerned with the definitions of the discrete fractional cosine transform (DFRCT) and the discrete fractional sine transform (DFRST). The definitions of DFRCT and DFRST are based on the eigen decomposition of DCT and DST kernels. This is the same idea as that of the discrete fractional Fourier transform (DFRFT); the eigenvalue and eigenvector relationships between the DFRCT, DFRST, and DFRFT can be established. The computations of DFRFT for even or odd signals can be planted into the half-size DFRCT and DFRST calculations. This will reduce the computational load of the DFRFT by about one half.

Color image processing by using binary quaternion-moment-preserving thresholding technique

This paper presents a new moment-preserving thresholding technique, called the binary quaternion-moment-preserving (BQMP) thresholding, for color image data. Based on representing color data by the quaternions, the statistical parameters of color data can be expressed through the definition of quaternion moments. Analytical formulas of the BQMP thresholding can thus be determined by using the algebra of the quaternions. The computation time for the BQMP thresholding is of order of the data size. By using the BQMP thresholding, quaternion-moment-based operators are designed for the application of color image processing, such as color image compression, multiclass clustering of color data, and subpixel color edge detection. The experimental results show that the proposed operator for color image compression can have output picture quality acceptable to human eyes. In addition, the proposed edge operator can detect the color edge at the subpixel level. Therefore, the proposed BQMP thresholding can be used as a tool for color image processing.

Eigenfunctions of linear canonical transform

The linear canonical transform (the LCT) is a useful tool for optical system analysis and signal processing. It is parameterized by a 2/spl times/2 matrix {a, b, c, d}. Many operations, such as the Fourier transform (FT), fractional Fourier transform (FRFT), Fresnel transform, and scaling operations are all the special cases of the LCT. We discuss the eigenfunctions of the LCT. The eigenfunctions of the FT, FRFT, Fresnel transform, and scaling operations have been known, and we derive the eigenfunctions of the LCT based on the eigenfunctions of these operations. We find, for different cases, that the eigenfunctions of the LCT also have different forms. When |a+d| 2, the eigenfunctions become the chirp multiplication and chirp convolution of self-similar functions (fractals). Besides, since many optical systems can be represented by the LCT, we can thus use the eigenfunctions of the LCT derived in this paper to discuss the self-imaging phenomena in optics. We show that there are usually varieties of input functions that can cause the self-imaging phenomena for an optical system.