A wavelet based method for image reconstruction from gradient data with applications

Abstract

In this paper, an algorithm for image reconstruction from gradient data based on the Haar wavelet decomposition is proposed. The proposed algorithm has two main stages. First, the Haar decomposition of the image to be reconstructed is obtained from the given gradient data set. Then, the Haar wavelet synthesis is employed to produce the image. The proposed algorithm is based on the relationship between the Haar analysis and synthesis filters and the model for the discretized gradient. The approach presented here is based on the one by Hampton et al. (IEEE J Sel Top Signal Process 2(5):781–792, 2008) for wavefront reconstruction in adaptive optics. The main strength of the proposed algorithm lies in its multiresolution nature, which allows efficient processing in the wavelet domain with complexity ${\fancyscript{O}}(N)$. In addition, obtaining the wavelet decomposition of the image to be reconstructed provides the possibility for further enhancements of the image, such as denoising or smoothing via iterative Poisson solvers at each resolution during Haar synthesis. To evaluate the performance of the proposed algorithm, it is applied to reconstruct ten standard test images. Experiments demonstrate that the algorithm yields results comparable in terms of solution accuracy to those produced by well-known benchmark algorithms. Further, experiments show that the proposed algorithm is suitable to be employed as a final step to reconstruct an image from a gradient data set, in applications such as image stitching or image morphing.

Appendices

Appendix 1: Algorithm proof

Given the Hudgin derivatives components $\Phi ^H_x$ and $\Phi ^H_y$, the wavelet decomposition of $\Phi$ can be obtained using Eq. (19)–(26).

Proof

From wavelet theory, it is known that the HH subband of the Haar wavelet decomposition of a 2D surface is given by:

$$\begin{aligned} {}_{HH}^{M-1}\Phi =\downarrow _2\left[ \Phi H_H\left( z_h \right) H_H\left( z_v \right) \right] \end{aligned}$$

(33)

where $H_H\left( z\right)$ is the highpass analysis Haar wavelet filter described by (2) and the variables $z_h$ and $z_v$ indicate the horizontal and vertical direction of filtering, respectively.

With the connection between the Haar wavelet filters and discretization model in mind, we can write Eq. (33) as follows:

$$\begin{aligned} {}_{HH}^{M-1}\Phi =\downarrow _2 \left[ \frac{\sqrt{2}}{2} \cdot \sqrt{2} \cdot \Phi H_H\left( z_h \right) H_H\left( z_v\right) \right] \end{aligned}$$

Next, according to the observation made in Sect. 2.2, we can replace $\sqrt{2} \cdot \Phi H_H \left( z_h\right)$ with the given $\Phi _x^H$. Doing so yields:

$$\begin{aligned} {}_{HH}^{M-1}\Phi&= \downarrow _2\left[ \frac{\sqrt{2}}{2} \Phi _x^HH_H\left( z_v\right) \right] \end{aligned}$$

(34)

$$\begin{aligned}&= \downarrow _2\left[ \frac{\sqrt{2}}{4}\cdot 2\cdot \Phi _x^HH_H \left( z_v\right) \right] \end{aligned}$$

(35)

Further, we note that:

$$\begin{aligned} \Phi _x^H H_H\left( z_v\right)&= \sqrt{2}\Phi H_H\left( z_h\right) H_H\left( z_v\right) \\&= \sqrt{2}\Phi H_H\left( z_v\right) H_H\left( z_h\right) =\Phi _y^H H_H\left( z_h\right) \end{aligned}$$

This observation allows us to replace $2\Phi _x^H H_H \left( z_v\right)$ with $\Phi _x^HH_H\left( z_v\right) + \Phi _y^HH_H\left( z_h\right)$, and so Eq. (34) becomes:

$$\begin{aligned} {}_{HH}^{M-1}\Phi =\, \downarrow _2 \left\{ \frac{\sqrt{2}}{4} \left[ \Phi _y^H H_H(z_h)+ \Phi _y^HH_H(z_h)\right] \right\} \end{aligned}$$

(36)

This concludes the proof of Eq. (21). Please note that the downsampler in front of the square brackets of the right hand-side term is two directional (i.e., the downsampling is done horizontally and vertically). The proof of Eq. (19)–(20), and (22)–(22) can be found in Hampton et al. (2008).

Appendix 2: Performance comparison

See Tables 2 and 3

Table 2 Relative reconstruction error in the presence of white Gaussian noise

Table 3 Relative reconstruction error in the presence of outliers