Human Face Recognition

1D Phase only correlation




                  This section defines the 1D POC function and a set of techniques for high-accuracy image matching. Let I and J be rectified stereo images as illustrated in Fig. 1. Given a reference point p in the image I, the problem is to find the corresponding point q in the image J. In the image I, we first extract the 1D image signal f(n) centered at the reference point p along the epipolar line. Similarly, in the image J, we extract the 1D image signal g(n) centered at q_ — the initial estimate for the true corresponding point q. The points q and q_ should be on the common epipolar line corresponding to p. Let n ( {−M , −(M−1), · · · , 0, · · · , (M−1),M}) be the discrete spatial index for the 1D image signals f(n) and g(n), whereM is a positive integer. The signal length N is given by N = 2M + 1. Note that we assume here the sign symmetric index range {−M, · · · ,M} for mathematical  simplicity. The discussion could be easily generalized to non-negative index ranges with power-of-two signal length.

The 1D Discrete Fourier Transforms (1D DFTs) of f(n) and g(n) are given by

where k = −M · · ·M and WN = e−j 2πN . AF and AG are amplitude components, and ejθF (k)   and ejθG(k) are phase components.The cross-phase spectrum R(k) is defined as


where G(k) denotes the complex conjugate of G(k) and θ(k) =θF (k)−θG(k). The 1D POC function r(n) between f(n) and g(n) is the 1D Inverse DFT (1D IDFT) of R(k) and is given by


 In the following, we derive the analytical peak model for the 1D POC function between the same signals that are minutely displaced  with each other. Now consider fc(x) as a 1D image signal defined in continuous space with real-number index x. Let δ represents minute (sub-pixel) displacement of fc(x). So, the displaced 1D image signal can be represented as fc(x−δ). Assume that f(n) and g(n) are spatially sampled signals of fc(x) and fc(x − δ), respectively, and are defined as


where T is the spatial sampling interval, and index range is given by n= −M,· · · ,M. For simplicity, we assume T = 1.



The POC function r(n) between f(n) and g(n) is given by



where α = 1. The above Eq. (7) represents the shape of the peak for the 1D POC function between the same 1D image signals that are minutely displaced with each other. This equation gives a distinct sharp peak. (When δ = 0, the 1D POC function r(n) becomes the Kronecker delta function.) We can show that the peak value α decreases (without changing the function shape itself), when small noise components are added to the original images. Hence, we assume α ≤ 1 in practice. The peak position n = −δ of the 1D POC function reflects the displacement between the two 1D image signals.



Thus, we can compute the displacement δ between extracted signals f(n) and g(n) by estimating the true peak position of the 1D POC function r(n). Then, the corresponding point q for the reference point p is determined from q’ and δ as

                                                                   q = q’ + (δ, 0),                       (8)

 where q and q_ in this equation are regarded as the coordinate vectors of the true corresponding point q and its initial estimate q_, respectively. Listed below are important techniques for improving the accuracy of 1D image matching for sub-pixel  correspondence search.


(i) Function fitting for high-accuracy estimation of peak position

                  We use Eq. (7) — the closed-form peak model of the POC function — directly for estimating the peak position by function fitting. By calculating the POC function, we can obtain a data of r(n) for each discrete index n. It is possible to find the location of the peak that may exist between image pixels by fitting the function Eq. (7) to the calculated data array around the correlation peak, where α and δ are fitting parameters.


(ii)Windowing to reduce boundary effects

                  Due to the DFT’s periodicity, a signal can be considered to “wrap around” at an edge, and therefore discontinuities, which are not supposed to exist in real world, occur at every border in 1D DFT computation. We reduce the effect of discontinuity at signal border by

applying 1D window function to 1D image signals. For this purpose, we employ 1D Hanning window. 

(iii) Spectral weighting for reducing aliasing and noise effects

                  For natural images, typically the high frequency components may have less reliability (low S/N) compared with the low frequency components. We could improve the estimation accuracy by applying a low-pass-type weighting function to 1D POC function in frequency domain and eliminating the high frequency components with low reliability.

For this purpose, we use the Gaussian-type spectral weighting function. The peak model Eq. (7) for function fitting should be modified correspondingly.


(iv) Averaging 1D POC functions to improve peak-to-noise ratio

                  When image quality is poor, a single 1D POC function is not sufficient for estimating accurate correspondence q due to degraded Peak-to-Noise Ratio (PNR). We can improve PNR by averaging a set of 1D POC functions evaluated at distinct positions around p and

q_. Figure 2 illustrates a typical situation. We extract B distinct 1D image signals fi(n) (i = 1, 2, · · · ,B) around the reference point p in the image I. Similarly, we extract 1D image signals gi(n) (i = 1, 2, · · · ,B) around the initial estimate q_ in the image J. Then, we compute the B distinct 1D POC functions ri(n) between fi(n) and gi(n). By taking the average of ri(n) for i = 1, 2, · · · ,B, we have the overall correlation surface r(n) with significantly improved .


PNR. Figure 2 illustrates a typical case of B = 5, which can be easily generalized to arbitrary arrangement of 1D image signals. Figure 3 shows an example of PNR improvement through averaging.



This paper presents a technique for high-accuracy correspondence search between two rectified images using 1D POC. The use of 1D POC in 1D correspondence search makes possible to significantly reduce computational cost without sacrificing reconstruction accuracy compared with the conventional 2D POC-based approach. Through some experimental evaluations, we demonstrate that the stereo vision system employing the proposed technique achieves sub-mm ( 0.48 mm) accuracy in 3D measurement.


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