# Reading: OISR — ODE-Inspired Schemes to Super-Resolution Network Designs (Super Resolution)

In this story, **ODE-inspired Network Design for Single Image Super-Resolution (OISR)**, by Chinese Academy of Sciences, University of Chinese Academy of Sciences, CAS, Alibaba Group, is presented. In this paper:

**An ordinary differential equation (ODE)-inspired design scheme**is adopted for single image super-resolution, which have brought**a new understanding of ResNet**in classification problems.**Two types of network structures are derived: LF-block and RK-block**, which correspond to the Leapfrog method and Runge-Kutta method in numerical ordinary differential equations.

This is a paper in **2019 CVPR **with over **20 citations**. (Sik-Ho Tsang @ Medium)

# Outline

**ODE-Inspired Network Design****Derivation of OISR-Blocks****Overall Network Architecture and Block Design****Experimental Results**

**1. ODE-Inspired Network Design**

- From a dynamical system perspective, it defines a map that takes input status forward
*x*units of time in the phase space. - In CNN semantics, time horizon
*x*corresponds to layers that can be adaptively chosen, while the final status is restricted by labels. - Considering the dynamical systems which can be described as an ODE:

- This system gives a map:

- with initial status
*y*0 ∈*Rd*. Suppose*p*(*y*0) is the distribution of input feature*y*0 on a domain, if we regard CNN-based SISR as such a dynamical system, then we are supposed to minimize:

- where
*Φ*is a map should be learned in SISR. - As the system is non-linear, there is no simple formula describing the map, numerical methods are used — forward Euler method:

- which provides the approximation.
**It can be seen as a numerical ODE using the approximation to the integral of***y*′ over an interval of width. **And residual block takes a similar form:**

- The above suggests the relationship and establish the bridge by defining:

- Thus,
**mapping forward Euler to a residual block.**

# 2. **Derivation of O**ISR-**Blocks**

**To learn such a map***Φ***, it may take many steps to reach the final status, each step corresponds to a CNN block.**- (The CNN blocks as shown above will be derived below. Also the
*G*block is composed of convolutions and activation functions which will be defined in the next section after deriving the above CNN block variants.) - Either increasing the number of steps or refining motion of each step helps to achieve the goal, corresponding to increasing block numbers and designing finer blocks.
**Higher-order methods are supposed to bring about some merits.**

## 2.1. LF-Block

- LeapFrog method is a
**second-order linear 2-step method**, as a refinement of forward Euler scheme. - By doubling the time interval
*h*, the approximation of*y*′ can be rewritten in the form of*y*′ ≈ (*yn*+1−*yn*−1)/2*h*. Thus,*yn*+1 is:

**In order to retain flexibility and obtain a block architecture, every three formulas above are grouped into a block as**:

- As mentioned:

- where
*G*is some kinds of convolutional blocks.

## 2.2. RK2-block

- Consider the Runge-Kutta family, which is widely used in numerical analysis. Making use of
**trapezoidal formula**:

- We obtain a block structure:

- In mathematics, these formulas are referred as Heun’s method, which is also a two-stage second-order Runge- Kutta method.

## 2.3. RK3-Block

**Higher-order methods**should obtain a smaller local truncation error.- Explicit iterative Runge-Kutta methods can be
**extended to arbitrary**:*n*stages

- In particular,
**3-stage Lunge-Kutta with third order**is:

- Generally,
**higher-order methods tend to generate more complicated blocks.** - (Please note that I am not expert in ODE. Please feel free to read the paper if interested.)

# 3. **Overall Network Architecture and Block Design**

## 3.1. Network Architecture

- The dimension of input and output feature maps of
*G*is kept unchanged. - The OISR Blocks are at the middle of the network.
- Residual learning is used.
- Pixel-Shuffle in ESPCN is used at the end to get the SR.

## 3.2. Block Design (*G*)

- There are different types of
*G*tried. There is a large searching space to search*G*. - Here, only choose three different forms are chosen to illustrate the general effectiveness of ODE-inspired schemes.
- Each of these designs keeps at least one activation function and one convolutional layer, thus promising the nonlinearity.

**4. Experimental Results**

## 4.1. Ablation Study

- First 800 images in DIV2K are used for training. L1 loss is used.
- Ablation study is performed on DIV2K validation set.

- ”PReLU+Conv”, namely G-v2, is suitable for LF-blocks.
- RK2-blocks should be equipped with G-v3.

## 4.2. SOTA Comparison

- The small-scale network designs are suffixed by ”-s”.
- The small-scale models,
**OISR-RK2-s and OISR-LF-s outperform other methods such as****FSRCNN****,****DRRN****,****MemNet****,****SelNet****and****CARN**, on different upscaling factors and datasets, except a slightly behind on Urban100 with upscaling factor ×2. - In addition, the middle-scale models,
**OISR-RK2 and OISR-LF, surpass****MSRN****with only two exceptions**on B100 and Urban100 SSIM when the upscaling factor is 2.

- For current state-of-the-art deep residual methods,
**OISR-RK3 achieves the best performances in most cases, outperforms****LapSRN****,****VDSR****,****DRCN****,****MDSR****,****RDN****, and****EDSR****.**

- OISRs can reconstruct more detailed images with less blurring.

This is the 20th story in this month.

## Reference

[2019 CVPR] [OISR]

ODE-inspired Network Design for Single Image Super-Resolution

## Super Resolution

[SRCNN] [FSRCNN] [VDSR] [ESPCN] [RED-Net] [DnCNN] [DRCN] [DRRN] [LapSRN & MS-LapSRN] [MemNet] [IRCNN] [WDRN / WavResNet] [MWCNN] [SRDenseNet] [SRGAN & SRResNet] [SelNet] [CNF] [BT-SRN][EDSR & MDSR] [MDesNet] [RDN] [SRMD & SRMDNF] [DBPN & D-DBPN] [RCAN] [ESRGAN] [CARN] [IDN] [ZSSR] [MSRN] [SR+STN] [SRFBN] [OISR]