Quantum image edge detection using improved Sobel mask based on NEQR

Abstract

Recently, image edge detection using quantum image processing has focused due to having less circuit complexity and storage capacity. Edge extraction using the Sobel operator has the restriction in vertical and horizontal directions, which provides less edge information. In this paper, we introduce a quantum improved Sobel edge detection algorithm with non-maximum suppression and double threshold techniques for novel enhanced quantum representation method. We have analyzed the quantum circuit of realizing the edge detection algorithm, the number of edge pixels, simulation results, and circuit complexity. Thereafter, we have compared with the classical and some existing quantum edge detection algorithms. Our proposed algorithm can achieve a significant improvement in the case of edge information and circuit complexity.

Appendices

Appendix 1: Classical Sobel edge extraction algorithm

Sobel operator is an individually separate and distinct differential operator. Sobel operator has two sets of 3 × 3 masks and mainly used for edge detection of an image (Fig. 12).

Fig. 12

a 3 × 3 pixel neighborhood templates, b and c two operators of Sobel algorithm

If G^H and G^V represent the image gradient values of the original image into the horizontal and vertical directions, then the calculation of G^H & G^V is defined as

$$ \begin{aligned} G^{\text{H}} & = - 1 \times S\left( {Y - 1,X - 1} \right) + 0 \times S\left( {Y - 1,X} \right) + 1 \times S\left( {Y - 1,X + 1} \right) + \left( { - 2} \right) \\ & \quad \times S\left( {Y,X - 1} \right) + 0 \times S\left( {Y,X} \right) + 2 \times S\left( {Y,X + 1} \right) + \left( { - 1} \right) \times S\left( {Y + 1,X - 1} \right) + 0 \times S\left( {Y + 1,X} \right) + 1 \\ & \quad \times S(Y + 1,X + 1) \\ & = S\left( {Y - 1,X + 1} \right) + 2S\left( {Y,X + 1} \right) + S\left( {Y + 1,X + 1} \right) - S\left( {Y - 1,X - 1} \right) \\ & \quad - 2S\left( {Y,X - 1} \right) - S(Y + 1,X - 1) \\ \end{aligned} $$

(13)

$$ \begin{aligned} G^{\text{V}} & = - 1 \times S\left( {Y - 1,X - 1} \right) + \left( { - 2} \right) \times S\left( {Y - 1,X} \right) + \left( { - 1} \right) \times S\left( {Y - 1,X + 1} \right) + 0 \\ & \quad \times S\left( {Y,X - 1} \right) + 0 \times S\left( {Y,X} \right) + 0 \times S\left( {Y,X + 1} \right) + 1 \times S\left( {Y + 1,X - 1} \right) + 2 \\ & \quad \times S\left( {Y + 1,X} \right) + 1 \times S\left( {Y + 1,X + 1} \right) \\ & = S\left( {Y + 1,X - 1} \right) + 2S\left( {Y + 1,X} \right) + S\left( {Y + 1,X + 1} \right){-}S\left( {Y - 1,X - 1} \right) \\ & \quad - 2S(Y - 1,X) - S\left( {Y - 1,X + 1} \right) \\ \end{aligned} $$

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The total gradient for each pixel is as follows

$$ G \cong |G^{\text{V}} | + |G^{\text{H}} | $$

(15)

The pixel will be the part of edge if G ≥ T_H (Threshold).

Appendix 2: Quantum image cyclic shift transformations

The cyclic shift (X shift and Y shift) transformation is used to shift the position of whole image so that every pixel of the image will get the information of its neighborhood simultaneously [18, 19, 39]. As an example, if we shift one unit upward, the pixel of the image will be transformed S(x, y) into S(x, y+ 1). The cyclic shift transformation (CT) of quantum image $|I\rangle$ can be expressed as

$$ {\text{CT}}(X \pm )|I\rangle = \frac{1}{{2^{n} }}\mathop \sum \limits_{Y = 0}^{{2^{n} - 1}} \mathop \sum \limits_{X = 0}^{{2^{n} - 1}} |CT_{{YX^{\prime}}} \rangle |Y\rangle |X \pm 1 mod2^{n} \rangle $$

(16)

$$ CT(Y \pm )|I\rangle = \frac{1}{{2^{n} }}\mathop \sum \limits_{Y = 0}^{{2^{n} - 1}} \mathop \sum \limits_{X = 0}^{{2^{n} - 1}} |CT_{{Y^{\prime}X}} \rangle |Y \pm 1 mod2^{n} \rangle |X\rangle $$

(17)

where X′ = ($X \mp 1$) mod2^n,, Y′ = $\left( {Y \mp 1} \right) mod2^{n}$, CT_(X+) & CT_(Y+) = $\left[ {\begin{array}{*{20}c} 0 & 1 \\ {I_{2}^{n} - 1} & 0 \\ \end{array} } \right]$ and CT_(X−) & CT_(Y−) = $\left[ {\begin{array}{*{20}c} 0 & {I_{2}^{n} - 1} \\ 1 & 0 \\ \end{array} } \right]$

See Fig. 13 and Table 2.

Fig. 13

a An example of original image of size $4 \times 4$, b the image transformed after CT(X+) operation, c the image transformed after CT (X −) operation. d Quantum circuit realization of CT_(X+) and CT_(Y−) for n-qubit sequence

Table 2 Computation prepared algorithm for shifting the image

Appendix 3: Quantum arithmetic operations

3.1 3.1 Quantum absolute value (QAV) module

Quantum absolute value module is used to estimate the absolute value of two integer number in quantum circuit. Basically, it consists of reversible parallel subtractor (RPS) module and complement operation (CO) module [32, 41, 42]. Quantum absolute value module for qubits |Y〉 = $|Y_{n - 1} Y_{n - 2} \ldots Y_{1} Y_{0} \rangle$ and |X〉 = $|X_{n - 1} X_{n - 2} \ldots X_{1} X_{0} \rangle$ is shown in Fig. 14. For more information referred to [4, 5].

Fig. 14

Quantum circuit of QAV module

3.2 3.2 Parallel-controlled NOT (CNOT) operation

CNOT operations are used to copy a quantum states. Parallel CNOT can be used to copy n-qubit quantum states into $|0\rangle$^⊗n ancillary qubits. The quantum circuit for parallel q-CNOT is shown in Fig. 15.

Fig. 15

Quantum circuit for parallel CNOT operations

3.3 3.3 Quantum operation for multiplied by 2ⁿ

Quantum operation for the qubit of an integer binary bits (i.e., |A〉 = |a_n−1 a_n−2… a₁a₀〉) multiplied by 2ⁿ can be realized as follows.

2ⁿ|A〉 = |a_n−1 a_n−2…a₁a₀ $\underbrace {0 \ldots 0\rangle }_{n}$, where $|0\rangle$^⊗n ancillary qubits are added after the lowest qubit.

3.4 3.4. Quantum ripple-carry adder (QRCA)

QRCA module [40] can be used to estimate the sum of two n-bit numbers A and B, where A = a_n−1a_n−2…a₀, B = b_n−1b_n−2…b₀. It consists of two basic modules of MAJ (Majority) gate and UMA (Un Majority and Add) gate [40], which start from low-order bits of the input added with carry to next order bits. In QRCA module, some ancillary and garbage qubits are omitted for simplification. The simplified QRCA module is shown in Fig. 16.

Fig. 16

Quantum circuit realization for a n-bit simple ripple-carry adder, b MAJ module, c UMA module and d simplified QRCA module

3.5 3.5 Quantum comparator (QC)

Quantum comparator [43] is used to compare the relations between two numbers. The output of QC module for two qubit sequence $|A\rangle = |a_{n - 1} a_{n - 2} \ldots a_{1} a_{0} \rangle$ and $|B\rangle = |b_{n - 1} b_{n - 2} \ldots b_{1} b_{0} \rangle$ can be represented as follows

• If C₁C₀ = 10, then |A〉 > |B〉

• If C₁C₀ = 01, then |A〉 < |B〉

• If C₁C₀ = 00, then |A〉 = |B〉

See Fig. 17.

Fig. 17

a Quantum circuit for quantum comparator. b Quantum module for quantum comparator

Appendix 4: Peak-signal-to-noise ratio (PSNR) and mean square error (MSE)

To compare the fidelity of a retrieval image with its original version, the peak-signal-to-noise ratio (PSNR) [44,45,46,47,48] is often used. The PSNR representing an evaluation metric for estimating fidelity of quantum is written as

$$ {\text{PSNR}} = 20\log_{10} \left( {\frac{{Q{\text{MAX}}_{I} }}{{\sqrt {\text{MSE}} }}} \right) $$

(18)

where $Q{\text{MAX}}_{I}$ the maximum pixel value of quantum is mechanically represented image and MSE [44,45,46,47,48] is the mean square error of quantum mechanically represented image. For two m $\times$ n images, MSE can be defined as follows

$$ {\text{MSE}} = \frac{1}{mn}\mathop \sum \limits_{X = 0}^{m - 1} \mathop \sum \limits_{Y = 0}^{n - 1} \left[ {\left( {Q\left( {X,Y} \right) - QR\left( {X,Y} \right)^{2} } \right)} \right] $$

(19)

We have considered m = n for simulation. Where Q(X, Y) = Quantum mechanically represented original image with X–Y position coordinates.

Figure 18a shows more PSNR obtained from all four images by using our algorithm than that obtained by the other two existing techniques. More improvement of PSNR value is noticed in walk bridge images than that in other images by using our algorithm, due to having more edge boundary and local changes of intensity in walk bridge image. So our algorithm is comparatively more effective in edge extraction of an image having more boundary and local changes than the other two algorithms. Figure 18b compares the MSE of extracted images between our algorithm and existing algorithms. The MSE of all extracted images obtained from our algorithm is less than that obtained using existing techniques due to less false edge information (Table 3).

Fig. 18

a PSNR graph for classical Sobel, quantum Sobel and our edge detection algorithm. b MSE graph for classical Sobel, quantum Sobel and our edge detection algorithm

Table 3 PSNR and MSE values for Sobel-based edge extraction algorithms