# Fast digital zooming system using directionally adaptive image interpolation and restoration

- Wonseok Kang
^{1}, - Jaehwan Jeon
^{1}, - Soohwan Yu
^{1}and - Joonki Paik
^{1}Email author

**Received: **13 August 2014

**Accepted: **17 November 2014

**Published: **6 December 2014

## Abstract

This paper presents a fast digital zooming system for mobile consumer cameras using directionally adaptive image interpolation and restoration methods. The proposed interpolation algorithm performs edge refinement along the initially estimated edge orientation using directionally steerable filters. Either the directionally weighted linear or adaptive cubic-spline interpolation filter is then selectively used according to the refined edge orientation for removing jagged artifacts in the slanted edge region. A novel image restoration algorithm is also presented for removing blurring artifacts caused by the linear or cubic-spline interpolation using the directionally adaptive truncated constrained least squares (TCLS) filter. Both proposed steerable filter-based interpolation and the TCLS-based restoration filters have a finite impulse response (FIR) structure for real time processing in an image signal processing (ISP) chain. Experimental results show that the proposed digital zooming system provides high-quality magnified images with FIR filter-based fast computational structure.

### Keywords

Image interpolation Image restoration Edge orientation Digital zooming## Introduction

A digital zooming system can increase spatial resolution without a high density image sensor or a high cost optical zoom lens. Recently, most digital cameras adopt a digital zooming system, which is commonly implemented by convolving an up-sampled version of the low-resolution (LR) image with a small kernel using proper weighting coefficients. Popular convolution-based up-sampling methods include linear and cubic-spline interpolation (Wick et al. 2004). Linear interpolation simply averages four neighboring pixels with weights that are inversely proportional to the distance from the target pixel. On the other hand, cubic-spline interpolation determines the pixel intensity value using the weighted average of sixteen neighboring pixels with weights determined by the two dimensional (2D) cubic function. However, neither linear nor cubic-spline interpolation can avoid jagged artifacts in the slanted edge region and blurring artifacts due to the nature of the rectangular-shaped interpolation kernel (Papker et al. 1983; Unser et al. 1991).

To solve this problem, advanced interpolation algorithms have been proposed. Li et al. estimated local covariance coefficients from an LR image and used the estimated coefficients to adapt the interpolation effect based on the geometric duality between the pair of LR and High-resolution (HR) image covariance (Li et al. 2001). Zhang et al. proposed the edge-guided nonlinear interpolation using directional filtering and data fusion (Zhang et al. 2006). Giachetti et al. proposed an up-scaling algorithm based on two-step grid filling and iterative correction of the interpolated pixels by minimizing an objective function depending on the second-order directional derivatives of the image intensity (Giachetti et al. 2011). Zhou et al. proposed the improved cubic-spline interpolation algorithm based on the estimation of the strong edge for a missing pixel location (Zhou et al. 2012). These advanced interpolation algorithms are, however, unsuitable for a fast digital zooming system which has limiter computational power and memory space.

The proposed digital zooming system consists of directionally adaptive image interpolation and restoration. After estimating the edge orientation using steerable filters with edge refinement (Kang et al. 2013a, [b]), the input LR image is adaptively interpolated along the estimated edge orientation using the directionally weighted linear or adaptive cubic-spline interpolation function. The blurring artifacts caused by the interpolation process are then restored using the proposed directionally adaptive truncated constrained least-squares (TCLS) filter (Kim et al. 2009). Both proposed interpolation and restoration filters have a finite impulse response (FIR) structure that is suitable for real-time digital zooming in an image signal processing (ISP) chain.

## The directionally steerable and truncated constrained least-squares (TCLS) filters

The main advantage of the proposed digital zooming system is implemented simply based on the FIR structure for real-time processing in many digital imaging systems. In this section, our proposed four-direction steerable filters and directionally adaptive TCLS restoration filters are described.

### Four-direction steerable filters

where scaling and normalization constants have been set to unity for notational simplicity.

*G*

^{ θ }be the 1D derivative of

*G*(

*x*,

*y*) in the direction with angle

*θ*. For example, the first-order derivative with

*θ*= 0 is expressed as

*θ*can be synthesized by taking linear combination of ${G}^{{0}^{\xb0}}$ and ${G}^{{90}^{\xb0}}$ as

*θ*and sin

*θ*terms are used to express an arbitrary direction. For reducing the computational load of edge orientation, four 5 × 5 steerable filters are used with standard deviation

*σ*= 1.0. The proposed steerable filter coefficients are generated as shown in Table 1.

**Directionally steerable filter coefficients**

Angle θ | 5 × 5 steerable filter coefficients (σ = 1.0) | |||||
---|---|---|---|---|---|---|

${G}^{{0}^{\xb0}}$ | Horizontal | |||||

Vertical | -0.0084 | -0.0377 | -0.0621 | -0.0377 | -0.0084 | |

-0.0188 | -0.0844 | -0.1392 | -0.0844 | -0.0188 | ||

0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||

0.0188 | 0.0844 | 0.1392 | 0.0844 | 0.0188 | ||

0.0084 | 0.0377 | 0.0621 | 0.0377 | 0.0084 | ||

${G}^{{45}^{\xb0}}$ | Horizontal | |||||

Vertical | -0.0114 | -0.0384 | -0.0420 | -0.0127 | 0.0000 | |

-0.0384 | -0.1142 | -0.0942 | 0.0000 | 0.0127 | ||

-0.0420 | -0.0942 | 0.0000 | 0.0942 | 0.0420 | ||

-0.0127 | 0.0000 | 0.0942 | 0.1142 | 0.0382 | ||

0.0000 | 0.0127 | 0.0420 | 0.0382 | 0.0144 | ||

${G}^{{90}^{\xb0}}$ | Horizontal | |||||

Vertical | 0.0084 | 0.0188 | 0.0000 | -0.0188 | -0.0084 | |

0.0377 | 0.0844 | 0.0000 | -0.0844 | -0.0377 | ||

0.0621 | 0.1392 | 0.0000 | -0.1392 | -0.0621 | ||

0.0377 | 0.0844 | 0.0000 | -0.0844 | -0.0377 | ||

0.0084 | 0.0188 | 0.0000 | -0.0188 | -0.0084 | ||

${G}^{{135}^{\xb0}}$ | Horizontal | |||||

Vertical | 0.0000 | -0.0127 | -0.0420 | -0.0382 | -0.0144 | |

0.0127 | 0.0000 | -0.0942 | -0.1142 | -0.0382 | ||

0.0420 | 0.0942 | 0.0000 | -0.0942 | -0.0420 | ||

0.0382 | 0.1142 | 0.0942 | 0.0000 | -0.0127 | ||

0.0114 | 0.0382 | 0.0420 | 0.0127 | 0.0000 |

### Directionally adaptive TCLS restoration filter

In order to remove the blurring artifacts caused by the interpolation process, an image restoration filter is needed. Kim et al. proposed the original version of the TCLS restoration filter for removing spatially adaptive image degradation followed by a spatially adaptive noise smoothing filter (Kim et al. 2009). This sub-section presents a directionally adaptive version of the TCLS restoration filter that removes the blurring artifacts caused by the interpolation process.

where *g*(*x*, *y*) and *η*(*x*, *y*) respectively represent 2D arrays of LR image and additive white Gaussian noise. *f*(*p*, *q*) represents the HR image, and *h*(*p*, *q*) the space-variant point spread function (PSF), which plays a role in anti-aliasing filter for the subsequent subsampling operation denoted as *T*[·].

*H*(

*u*,

*v*) be the frequency response of the PSF

*h*(

*p*,

*q*), then the frequency response of the constrained least-squares (CLS) restoration filter is given as (Katsaggelos, 1989)

where *C*(*u*, *v*) represents a frequency response of the high-pass filter, and *λ* the regularization parameter the controls the relative amount of data fidelity and the smoothing constraint.

*C*

^{ θ }(

*u*,

*v*), for

*θ*∈ {0°, 45°, 90°, 135°, Flat}, are generated using the four directional high-pass filter in the spatial domain as (Kang et al. 2013a, [b])

*C*

^{ θ }(

*u*,

*v*), for

*θ*∈ {0°, 45°, 90°, 135°, Flat}, plays a role of directionally adaptive smoothness constraints, and

*λ*= 0.2 is experimentally used. The spatial-domain counterpart of ${R}_{CLS}^{\theta}\left(u,v\right)$ is its inverse DFT expressed as

*F*

^{- 1}[·] represents the inverse DFT operation. For reducing the computational load of restoration processing, ${r}_{\mathit{TCLS}}^{\theta}\left(x,y\right)$ is truncated an

*m*×

*m*FIR filter. Table 2 shows five truncated constrained least-squares (TCLS) filter when

*m*= 5.

**Directionally adaptive TCLS filter coefficients**

Angle θ | 5 × 5 TCLS filter coefficients (λ = 0.2) | |||||
---|---|---|---|---|---|---|

${r}_{\mathit{TCLS}}^{{0}^{\xb0}}$ | Horizontal | |||||

Vertical | 0.0001 | -0.0013 | -0.0084 | -0.0013 | 0.0001 | |

-0.0015 | -0.0130 | 0.0636 | -0.0130 | -0.0015 | ||

-0.0073 | 0.0826 | 0.8018 | 0.0826 | -0.0073 | ||

-0.0015 | -0.0130 | 0.0636 | -0.0130 | -0.0015 | ||

0.0001 | -0.0013 | -0.0084 | -0.0013 | 0.0001 | ||

${r}_{\mathit{TCLS}}^{{45}^{\xb0}}$ | Horizontal | |||||

Vertical | 0.0002 | -0.0014 | -0.0095 | -0.0010 | 0.0002 | |

-0.0014 | -0.0057 | 0.0686 | -0.0130 | -0.0010 | ||

-0.0095 | 0.0686 | 0.8099 | 0.0686 | -0.0095 | ||

-0.0010 | -0.0130 | 0.0686 | -0.0057 | -0.0014 | ||

0.0002 | -0.0010 | -0.0095 | -0.0014 | 0.0002 | ||

${r}_{\mathit{TCLS}}^{{90}^{\xb0}}$ | Horizontal | |||||

Vertical | 0.0001 | -0.0015 | -0.0073 | -0.0015 | 0.0001 | |

-0.0013 | -0.0130 | 0.0826 | -0.0130 | -0.0013 | ||

-0.0084 | 0.0636 | 0.8018 | 0.0636 | -0.0084 | ||

-0.0013 | -0.0130 | 0.0826 | -0.0130 | -0.0013 | ||

0.0001 | -0.0015 | -0.0073 | -0.0015 | 0.0001 | ||

${r}_{\mathit{TCLS}}^{{135}^{\xb0}}$ | Horizontal | |||||

Vertical | 0.0002 | -0.0014 | -0.0095 | -0.0010 | 0.0002 | |

-0.0014 | -0.0057 | 0.0686 | -0.0130 | -0.0010 | ||

-0.0095 | 0.0686 | 0.8099 | 0.0686 | -0.0095 | ||

-0.0010 | -0.0130 | 0.0686 | -0.0057 | -0.0014 | ||

0.0002 | -0.0010 | -0.0095 | -0.0014 | 0.0002 | ||

${r}_{\mathit{TCLS}}^{\mathrm{Flat}}$ | Horizontal | |||||

Vertical | 0.0002 | -0.0012 | -0.0088 | -0.0012 | 0.0002 | |

-0.0012 | -0.0109 | 0.0701 | -0.0109 | -0.0012 | ||

-0.0088 | 0.0701 | 0.8077 | 0.0701 | -0.0088 | ||

-0.0012 | -0.0109 | 0.0701 | -0.0109 | -0.0012 | ||

0.0002 | -0.0012 | -0.0088 | -0.0012 | 0.0002 |

## Combined directionally adaptive image interpolation and restoration

A typical digital imaging system consists of four functional modules: (i) a set of optical lenses, (ii) the analog front-end (AFE) module including a color filter array (CFA), a complementary metal-oxide-semiconductor image sensor (CIS), and an analog-to-digital converter (ADC), (iii) the digital back-end (DBE) module including various image signal processing subsystems, and (iv) the display devices as shown in Figure 2. The proposed digital zooming system belongs to the DBE module in the ISP chain.

### Edge orientation estimation and refinement

where *f*_{
L
}(*m*, *n*) represents a 5 × 5 local block of the input image centered at (*x*, *y*) and *G*^{
θ
}(*x*, *y*) the 5 × 5 FIR steerable filters rotated by angle *θ* ∈ {0°, 45°, 90°, 135°}.

*d*

^{ θ }(

*x*,

*y*) as (Kang et al. 2013a, [b])

*D*

^{ θ }(

*x*,

*y*) is less than a pre-specified threshold, the corresponding pixel is considered to be in the non-slanted edge region. In this work, the threshold value of 0.075 was used for the empirically optimum sensitivity of steerable filters. Given an initial edge orientation

*θ*

_{ I }(

*x*,

*y*), the refined edge orientation is selected among eighteen directions. The proposed edge refinement algorithm is summarized as.

*θ*

^{∗}is finally quantized with the interval of 10°. Figure 4 shows the results of edge orientation estimation using four directionally steerable filters followed by edge refinement. As shown in Figure 4c, the proposed method provides more accurate and continuous edge orientation, which make the result of the proposed directionally adaptive interpolation looks more natural.

### Directionally adaptive image interpolation

*θ*

^{∗}as shown in Figure 5.

*v*

_{2}and four pixels on the same horizontal line as

*f*(·) represents a cubic-spline weight function defined as

where *a* represents a cubic-spline weight function parameter.

*a*= - 1 was used. For the spatially adaptive interpolation without blurring artifacts, the cubic-spline weight function parameter is changed according to the strength of the edge using the activity-map (Efstratiadis et al. 1990) as

where the tuning parameter *σ* is chosen so that *α*_{
MAP
}(*x*, *y*) distributes as uniformly as possible in [0,1], and Var(*x*, *y*) is the local variance of a pixel located at (*x*, *y*). In this work, the tuning parameter of σ = 250 was used*.*

where *α*_{
MAP
}(*x*, *y*) represents the activity value at *v*_{2}.

where *w* 1 represents the distance between P1 and P3, and *w* 2 the distance between P2 and P3.

For reducing the computational load of the interpolation process, a simple cubic-spline interpolation is used with *a* = - 0.5, if a pixel is not on the salted edge region. By using the directionally optimized interpolation, the proposed method can significantly reduce jagging artifacts in the slanted edge region.

### Directionally adaptive image restoration

*C*

^{ θ }(

*u*,

*v*) according to the estimated edge orientation. To remove blurring artifacts caused by the interpolation process, the proposed restoration method performs 2D convolution using five 5 × 5 directionally adaptive TCLS filters according to the edge orientation

*θ*as

where ∗ represents the 2D convolution operator, *ĝ*(*x*, *y*) is the interpolated image, ${r}_{\mathit{TCLS}}^{\theta}\left(x,y\right)$ is the impulse response of the directionally TCLS filter of orientation *θ*, and $\widehat{f}\left(x,y\right)$ is the restored HR image.

## Experimental results

For evaluating the performance of the proposed digital zooming method, we used a set of standard images of size 512 × 512, and outdoor test images of size 1280 × 720 acquired by a mobile phone camera. The performance of the proposed method is evaluated with PSNR, SSIM and processing time in seconds on a personal computer with 3.4 GHz CPU and 8GB memory.

*a*= - 0.5. The magnification results show that the proposed method can better remove jagging and blurring artifacts in the slanted edge region than the cubic-spline interpolation method as shown in Figure 6.

For additional experiments, to evaluate the performance and speed of the six interpolation algorithms such as cubic-spline interpolation, Li’s method (Li et al. 2001), Zhang’s method (Zhang et al. 2006), Giachetti’s method (Giachetti et al. 2011), Zhou’s method (Zhou et al. 2012), and the proposed method, standard images are generated by down-sampling 512 × 512 images by a factor of two in both horizontal and vertical directions.

**PSNR, SSIM, and CPU times of two interpolation methods using standard test images**

Image Type | Interpolation Type | PSNR | SSIM | Time (Sec) |
---|---|---|---|---|

Lena | Cubic-spline | 33.7420 | 0.9693 | 0.281 |

Li’s method | 33.9460 | 0.9729 | 16.286 | |

Zhang’s method | 33.8423 | 0.9723 | 12.979 | |

Giachetti’s method | 34.0461 | 0.9701 | 97.249 | |

Zhou’s method | 34.3944 | 0.9851 | 3.462 | |

Proposed method | 34.1497 | 0.9728 | 0.608 | |

Barbara | Cubic-spline | 23.9285 | 0.8873 | 0.281 |

Li’s method | 22.1196 | 0.8665 | 17.006 | |

Zhang’s method | 25.1406 | 0.8989 | 13.058 | |

Giachetti’s method | 22.9126 | 0.8691 | 101.076 | |

Zhou’s method | 23.3526 | 0.9851 | 3.452 | |

Proposed method | 24.3669 | 0.8919 | 0.405 | |

Boat | Cubic-spline | 28.9056 | 0.9320 | 0.296 |

Li’s method | 29.3233 | 0.9393 | 17.160 | |

Zhang’s method | 29.3349 | 0.9401 | 13.151 | |

Giachetti’s method | 28.8491 | 0.9329 | 101.463 | |

Zhou’s method | 29.5240 | 0.9835 | 3.439 | |

Proposed method | 29.2777 | 0.9388 | 0.390 | |

Crowd | Cubic-spline | 21.6893 | 0.9370 | 0.281 |

Li’s method | 21.8357 | 0.9356 | 17.709 | |

Zhang’s method | 22.2017 | 0.9409 | 12.963 | |

Giachetti’s method | 21.7158 | 0.9367 | 99.841 | |

Zhou’s method | 22.2346 | 0.7517 | 3.347 | |

Proposed method | 22.0536 | 0.9414 | 0.460 |

For additional experiments, 1280 × 720 high-definition (HD) images are generated from 320 × 180 LR mobile camera images by four times magnification using the six advanced interpolation algorithms.

## Conclusion

This paper presents novel directionally adaptive image interpolation and restoration algorithms for a fast digital zooming system in digital cameras. The proposed interpolation algorithm analyzes the edge orientation using computationally efficient steerable filters followed by the edge refinement process. The selective use of directionally weighted 1D interpolation and 2D adaptive cubic-spline interpolation can enhance the image quality without jagging artifacts in the slanted edge region. Blurring artifacts are also removed using directionally adaptive TCLS filters. Both proposed steerable filter-based interpolation and the TCLS-based restoration filters have an FIR structure for real-time processing in an ISP chain. Experimental results show that the proposed method can provide high-quality magnified images without jagging and blurring artifacts. Furthermore, the proposed method gives higher PSNR and SSIM values than existing state-of-the-art interpolation methods with reduced computation load.

## Declarations

### Acknowledgements

This work was supported in part by the Technology Innovation Program (Development of Super Resolution Image Scaler for 4K UHD) under Grant K10041900, by the ICT R&D program of MSIP/IITP [14-824-09-002, Development of global multi-target tracking and event prediction techniques based on real-time large-scale video analysis], by the MSIP(Ministry of Science, ICT&Future Planning), Korea, under the ITRC(Information Technology Research Center) support program (NIPA-2014-H0301-14-1044) supervised by the NIPA(National ICT Industry Promotion Agency), and by Ministry of Culture, Sports and Tourism(MCST) and Korea Creative Content Agency(KOCCA) in the Culture Technology(CT) Research & Development Program.

## Authors’ Affiliations

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