A vehicle detection scheme based on two-dimensional HOG features in the DFT and DCT domains

Abstract

Histogram of oriented gradients (HOG) are often used as features for object detection in images, since they are robust to changes in illumination and environmental conditions. However, these features are not invariant to changes in the resolution of input images. A 2D representation of these features, referred to as 2DHOG features, has been used since it preserves the relations among the neighboring pixels or cells. In this paper, a new vehicle detection scheme using transform-domain 2DHOG features is proposed. The method is based on extracting the 2DHOG features from the input image and applying to it 2D discrete Fourier or cosine transform. This is followed by a truncation process through which only the low frequency coefficients, referred to as the transform-domain 2DHOG (TD2DHOG) features, are retained. It is shown that the TD2DHOG features obtained from an image at the original resolution and a downsampled version from the same image are approximately the same within a multiplicative factor. This property is then utilized in our scheme for the detection of vehicles of various resolutions using a single classifier rather than multiple resolution-specific classifiers. Experimental results show that the use of the single classifier in the proposed detection scheme reduces drastically the training and storage cost over the use of a classifier pyramid, yet providing a detection accuracy similar to that obtained using TD2DHOG features with a classifier pyramid. Furthermore, the proposed method provides a detection accuracy that is similar or even better than that provided by the state-of-the-art techniques.

Appendix: Derivation of Equation (21)

The 2DDCT for a grayscale image in the spatial domain, $x\in \mathbb {R}^2$, is given by

$$\begin{aligned} X_{N,M}[u,v]= & {} \hat{\varGamma }_{N}[u]\hat{\varGamma }_{M}[v]\sum _{m=0}^{M-1}\sum _{n=0}^{N-1}x[n,m] \cos \left( \frac{\pi (2n+1)u}{2N}\right) \nonumber \\&\times \cos \left( \frac{\pi (2m+1)v}{2 M}\right) \end{aligned}$$

(1)

where $0\le u \le N-1$, $0\le v \le M-1$, $\hat{\varGamma }_{N}[k]=\sqrt{1/N}$ for $k=0$ and $\hat{\varGamma }_{N}[k]=\sqrt{2/N}$ for $0< k \le N-1$. Let N and M be even multiples of $K_1$, and $K_2$, respectively, where $K_1$ and $K_2$ are the downsampling factors in the y and the x directions, respectively. Let x[n, m] be a bandlimited sequence, and the sequence $y\in \mathbb {R}^2$, of size $(2N \times 2M)$, be defined as

$$\begin{aligned} \small {\begin{aligned} y[n,m]&={\left\{ \begin{array}{ll}x[n,m], &{} 0\le n \le N-1, 0\le m \le M-1\\ 0, &{} \text {otherwise} \\ \end{array}\right. } \end{aligned}} \end{aligned}$$

(2)

The N $\times$ M-point 2DDCT can be computed by 2N $\times$ 2M-point 2DDFT for a sequence, y[n, m], as follows. First, the 2DDFT is employed on y[n, m] in order to obtain $Y_{2N,2M}$. Similar to the 1DDCT case, the relation between the signal in the 2DDCT domain $X_{N,M}[u,v]$, and $Y_{2N,2M}[u,v]$ can be expressed as

$$\begin{aligned} X_{N,M}[u,v]=\hat{\varGamma }_{N}[u]\hat{\varGamma }_{M}[v] \text {Re} \left( Y_{2N,2M}[u,v] e^{-j\left( \frac{\pi u}{2N}+\frac{\pi v}{2M}\right) }\right) \end{aligned}$$

(3)

where $0\le u \le N-1$, $0\le v \le M-1$. Let $c_1$, $c_2$ denote the maximum frequencies retained by the truncation operator, where $c_1 < \hat{N}$, $c_2 <\hat{M}$, $\hat{N}=N/K_1$, and $\hat{M}= M/K_2$. Assume $Y_{2N,2M}$ is bandlimited to the maximum frequencies $(\hat{N}, \hat{M})$. Then, the downsampled signal in the 2DDCT domain, $\hat{X}_{\hat{N},\hat{M}}$, can be obtained as

$$\begin{aligned} \hat{X}_{\hat{N},\hat{M}}[u,v]&=\frac{1}{K_1 K_2} \hat{\varGamma }_{\hat{N}}[u]\hat{\varGamma }_{\hat{M}}[v] \text {Re}\left( Y_{2N,2M}[u,v] e^{-j\left( \frac{\pi u}{2N}+\frac{\pi v}{2M}\right) }\right) \end{aligned}$$

(4)

$$\begin{aligned}&= \frac{\hat{\varGamma }_{\hat{N}}[u]\hat{\varGamma }_{\hat{M}}[v]}{K_1 K_2\hat{\varGamma }_{{N}}[u]\hat{\varGamma }_{{M}}[v]} \hat{\varGamma }_{{N}}[u]\hat{\varGamma }_{{M}}[v] \text {Re}(Y_{2N,2M}[u,v]\nonumber \\&\quad \times e^{-j\left( \frac{\pi u}{2N}+\frac{\pi v}{2M}\right) }) \end{aligned}$$

(5)

$$\begin{aligned}&=\frac{\sqrt{1/\hat{N}}{\sqrt{1/\hat{M}}}}{K_1 K_2 \sqrt{1/N} \sqrt{1/M}} X_{N,M}[u,v] \end{aligned}$$

(6)

$$\begin{aligned}&= \frac{1}{\sqrt{K_1K_2}} X_{N,M}[u,v] \end{aligned}$$

(7)

where $0\le u \le c_1-1$ and $0\le v \le c_2-1$. Thus, the relation between the 2DDCT coefficients of the original image and that of the downsampled version is given by

$$\begin{aligned} X_{N,M}[u,v]= \sqrt{K_1K_2} \hat{X}_{\hat{N},\hat{M}}[u,v] \end{aligned}$$

(8)