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3548 KiB  
Article
Metric for Estimating Congruity between Quantum Images
by Abdullah M. Iliyasu, Fei Yan and Kaoru Hirota
Entropy 2016, 18(10), 360; https://doi.org/10.3390/e18100360 - 9 Oct 2016
Cited by 23 | Viewed by 5994
Abstract
An enhanced quantum-based image fidelity metric, the QIFM metric, is proposed as a tool to assess the “congruity” between two or more quantum images. The often confounding contrariety that distinguishes between classical and quantum information processing makes the widely accepted peak-signal-to-noise-ratio (PSNR) ill-suited [...] Read more.
An enhanced quantum-based image fidelity metric, the QIFM metric, is proposed as a tool to assess the “congruity” between two or more quantum images. The often confounding contrariety that distinguishes between classical and quantum information processing makes the widely accepted peak-signal-to-noise-ratio (PSNR) ill-suited for use in the quantum computing framework, whereas the prohibitive cost of the probability-based similarity score makes it imprudent for use as an effective image quality metric. Unlike the aforementioned image quality measures, the proposed QIFM metric is calibrated as a pixel difference-based image quality measure that is sensitive to the intricacies inherent to quantum image processing (QIP). As proposed, the QIFM is configured with in-built non-destructive measurement units that preserve the coherence necessary for quantum computation. This design moderates the cost of executing the QIFM in order to estimate congruity between two or more quantum images. A statistical analysis also shows that our proposed QIFM metric has a better correlation with digital expectation of likeness between images than other available quantum image quality measures. Therefore, the QIFM offers a competent substitute for the PSNR as an image quality measure in the quantum computing framework thereby providing a tool to effectively assess fidelity between images in quantum watermarking, quantum movie aggregation and other applications in QIP. Full article
(This article belongs to the Collection Quantum Information)
►▼ Show Figures

Figure 1

Figure 1
<p>Circuit structure for comparing similarity between two FRQI quantum images (figure adapted from [<a href="#B15-entropy-18-00360" class="html-bibr">15</a>,<a href="#B23-entropy-18-00360" class="html-bibr">23</a>]).</p> Full article ">Figure 2
<p>Generalised circuit structure for parallel comparison of similarity between FRQI quantum images (figure adapted from [<a href="#B15-entropy-18-00360" class="html-bibr">15</a>,<a href="#B23-entropy-18-00360" class="html-bibr">23</a>]).</p> Full article ">Figure 3
<p>(<b>A</b>) Notation for a single qubit projective measurement operation and (<b>B</b>) description of the ancilla-driven measurement operation (figures and explanations in the text are adapted from [<a href="#B2-entropy-18-00360" class="html-bibr">2</a>,<a href="#B14-entropy-18-00360" class="html-bibr">14</a>,<a href="#B24-entropy-18-00360" class="html-bibr">24</a>]).</p> Full article ">Figure 4
<p>Layout of proposed QIFM framework to assess fidelity between two (or more) quantum images.</p> Full article ">Figure 5
<p>Flowchart for executing the proposed QIFM framework to compare two (or more) quantum images.</p> Full article ">Figure 6
<p>QIFM sub-circuit to execute the Binary check operation (<span class="html-italic">BCO</span>) of the QIFM image metric.</p> Full article ">Figure 7
<p>QIFM sub-circuit to execute the Bit error rate operation (<span class="html-italic">BO</span>) of the QIFM image metric.</p> Full article ">Figure 8
<p>Dataset of images paired for the FPS analysis. (<b>A</b>) Lena; (<b>B</b>) Inverted Lena; (<b>C</b>) Blonde Lady; (<b>D</b>) Peppers; (<b>E</b>) Scarfed Lady; (<b>F</b>) Baboon; (<b>G</b>) Brunette Lady; (<b>H</b>) Cameraman; (<b>I</b>) Man; (<b>J</b>) Couple; (<b>K</b>) Aeroplane; (<b>L</b>) House; (<b>M</b>) Pentagon; (<b>N</b>) Fingerprint; (<b>O</b>) Bridge; (<b>P</b>) Trees.</p> Full article ">Figure 9
<p>Comparison between PSNR and QIFM for watermarked images. (<b>A</b>) PSAU watermark logo; (<b>B</b>) original Lena image; (<b>C</b>) original Blonde Lady image; (<b>D</b>) watermarked version of Lena image; (<b>E</b>) watermarked version of Blonde Lady image.</p> Full article ">
157 KiB  
Editorial
Special Issue on Entropy-Based Applied Cryptography and Enhanced Security for Ubiquitous Computing
by James (Jong Hyuk) Park and Wanlei Zhou
Entropy 2016, 18(9), 334; https://doi.org/10.3390/e18090334 - 13 Sep 2016
Viewed by 4604
Abstract
Entropy is a basic and important concept in information theory. It is also often used as a measure of the unpredictability of a cryptographic key in cryptography research areas. Ubiquitous computing (Ubi-comp) has emerged rapidly as an exciting new paradigm. In this special [...] Read more.
Entropy is a basic and important concept in information theory. It is also often used as a measure of the unpredictability of a cryptographic key in cryptography research areas. Ubiquitous computing (Ubi-comp) has emerged rapidly as an exciting new paradigm. In this special issue, we mainly selected and discussed papers related with ore theories based on the graph theory to solve computational problems on cryptography and security, practical technologies; applications and services for Ubi-comp including secure encryption techniques, identity and authentication; credential cloning attacks and countermeasures; switching generator with resistance against the algebraic and side channel attacks; entropy-based network anomaly detection; applied cryptography using chaos function, information hiding and watermark, secret sharing, message authentication, detection and modeling of cyber attacks with Petri Nets, and quantum flows for secret key distribution, etc. Full article
(This article belongs to the Special Issue Entropy-Based Applied Cryptography and Enhanced Security for Ubiquitous Computing)
3717 KiB  
Review
Quantum Computation-Based Image Representation, Processing Operations and Their Applications
by Fei Yan, Abdullah M. Iliyasu and Zhengang Jiang
Entropy 2014, 16(10), 5290-5338; https://doi.org/10.3390/e16105290 - 10 Oct 2014
Cited by 51 | Viewed by 11941
Abstract
A flexible representation of quantum images (FRQI) was proposed to facilitate the extension of classical (non-quantum)-like image processing applications to the quantum computing domain. The representation encodes a quantum image in the form of a normalized state, which captures information about colors and [...] Read more.
A flexible representation of quantum images (FRQI) was proposed to facilitate the extension of classical (non-quantum)-like image processing applications to the quantum computing domain. The representation encodes a quantum image in the form of a normalized state, which captures information about colors and their corresponding positions in the images. Since its conception, a handful of processing transformations have been formulated, among which are the geometric transformations on quantum images (GTQI) and the CTQI that are focused on the color information of the images. In addition, extensions and applications of FRQI representation, such as multi-channel representation for quantum images (MCQI), quantum image data searching, watermarking strategies for quantum images, a framework to produce movies on quantum computers and a blueprint for quantum video encryption and decryption have also been suggested. These proposals extend classical-like image and video processing applications to the quantum computing domain and offer a significant speed-up with low computational resources in comparison to performing the same tasks on traditional computing devices. Each of the algorithms and the mathematical foundations for their execution were simulated using classical computing resources, and their results were analyzed alongside other classical computing equivalents. The work presented in this review is intended to serve as the epitome of advances made in FRQI quantum image processing over the past five years and to simulate further interest geared towards the realization of some secure and efficient image and video processing applications on quantum computers. Full article
(This article belongs to the Section Entropy Reviews)
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<p>Commonly used quantum gates (NOT, Z, Hadamard and CNOT).</p> Full article ">
<p>A single qubit measurement gate.</p> Full article ">
<p>A 2 × 2 FRQI quantum image, its circuit structure and quantum state.</p> Full article ">
<p>Preparation of FRQI state through the unitary transform <span class="html-italic">℘</span>.</p> Full article ">
<p>Generalized circuit design for geometric transformations on quantum images.</p> Full article ">
<p>Single qubit gates applied on the color wire.</p> Full article ">
<p>General circuit of MCQI quantum images.</p> Full article ">
<p>A 2 × 2 MCQI quantum image, its circuit structure and MCQI state.</p> Full article ">
<p>The general quantum circuit of <span class="html-italic">U<sub>X</sub></span> operations, including: (<b>a</b>) <span class="html-italic">U<sub>R</sub></span>; (<b>b</b>) <span class="html-italic">U<sub>G</sub></span>; (<b>c</b>) <span class="html-italic">U<sub>B</sub></span>; and (<b>d</b>) <span class="html-italic">U<sub>α</sub></span>.</p> Full article ">
3725 KiB  
Review
Towards Realising Secure and Efficient Image and Video Processing Applications on Quantum Computers
by Abdullah M. Iliyasu
Entropy 2013, 15(8), 2874-2974; https://doi.org/10.3390/e15082874 - 26 Jul 2013
Cited by 59 | Viewed by 9754
Abstract
Exploiting the promise of security and efficiency that quantum computing offers, the basic foundations leading to commercial applications for quantum image processing are proposed. Two mathematical frameworks and algorithms to accomplish the watermarking of quantum images, authentication of ownership of already watermarked images [...] Read more.
Exploiting the promise of security and efficiency that quantum computing offers, the basic foundations leading to commercial applications for quantum image processing are proposed. Two mathematical frameworks and algorithms to accomplish the watermarking of quantum images, authentication of ownership of already watermarked images and recovery of their unmarked versions on quantum computers are proposed. Encoding the images as 2n-sized normalised Flexible Representation of Quantum Images (FRQI) states, with n-qubits and 1-qubit dedicated to capturing the respective information about the colour and position of every pixel in the image respectively, the proposed algorithms utilise the flexibility inherent to the FRQI representation, in order to confine the transformations on an image to any predetermined chromatic or spatial (or a combination of both) content of the image as dictated by the watermark embedding, authentication or recovery circuits. Furthermore, by adopting an apt generalisation of the criteria required to realise physical quantum computing hardware, three standalone components that make up the framework to prepare, manipulate and recover the various contents required to represent and produce movies on quantum computers are also proposed. Each of the algorithms and the mathematical foundations for their execution were simulated using classical (i.e., conventional or non-quantum) computing resources, and their results were analysed alongside other longstanding classical computing equivalents. The work presented here, combined together with the extensions suggested, provide the basic foundations towards effectuating secure and efficient classical-like image and video processing applications on the quantum-computing framework. Full article
(This article belongs to the Special Issue Quantum Information 2012)
►▼ Show Figures

Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>The three stages of the circuit model of quantum. The figure was adapted from [<a href="#B35-entropy-15-02874" class="html-bibr">35</a>] from where additional explanation can be obtained.</p> Full article ">Figure 2
<p>A simple FRQI image and its quantum state.</p> Full article ">Figure 3
<p>Generalised circuit showing how information in an FRQI quantum image state is encoded.</p> Full article ">Figure 4
<p>Illustration of the two steps of the PPT theorem to prepare an FRQI image.</p> Full article ">Figure 5
<p>Colour and position transformations on FRQI quantum images. The * in (c) indicates the 0 or 1 control-conditions required to confine <span class="html-italic">U<sub>3</sub></span> to a predetermined sub-block of the image.</p> Full article ">Figure 6
<p>Left: The circuit design for the horizontal flip operation, <span class="html-italic">F<sup>X</sup></span>, and on the right that for the coordinate swap operation, <span class="html-italic">S<sub>I</sub></span>.</p> Full article ">Figure 7
<p>(<b>a</b>) Original 8×8 image, and its resulting output images after applying in (<b>b</b>) the vertical flip <span class="html-italic">F<sup>Y</sup></span>, (<b>c</b>) the horizontal flip <span class="html-italic">F<sup>X</sup></span>, and in (<b>d</b>) the coordinate swap <span class="html-italic">S<sub>I</sub></span> operations, respectively.</p> Full article ">Figure 8
<p>Circuit to rotate the image in <a href="#entropy-15-02874-f007" class="html-fig">Figure 7</a>a through an angle of 90° and (on the left) the resulting image.</p> Full article ">Figure 9
<p>The 8 × 8 synthetic and Lena images before and after the application of the <math display="inline"> <semantics> <mrow> <mi>R</mi> <mo>(</mo> <mstyle scriptlevel="+1"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> <mo>)</mo> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>R</mi> <mo>(</mo> <mstyle scriptlevel="+1"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mstyle> <mo>)</mo> </mrow> </semantics> </math> on the upper half and lower half of their content.</p> Full article ">Figure 10
<p>Circuit to execute the <math display="inline"> <semantics> <mrow> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mstyle scriptlevel="+1"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mstyle scriptlevel="+1"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> colour operations on the upper half and lower half of the 8 × 8 synthetic and Lena images.</p> Full article ">Figure 11
<p>General circuit design for transforming the geometric (G) and colour (C) content of FRQI quantum images.</p> Full article ">Figure 12
<p>Demonstrating the use of additional control to target a smaller sub-area in an image.</p> Full article ">Figure 13
<p>The control on the <span class="html-italic">y<sub>n-1</sub></span> qubit in the circuit on the left divides an entire image into its upper and lower halves. Using this control, this circuit shows how the flip operation can be confined to the lower half of an image, while the figure to its right shows the effect of such a transformation on the 8×8 binary image in <a href="#entropy-15-02874-f007" class="html-fig">Figure 7</a>(a). (The image on the right corrects the image for the same example in [<a href="#B18-entropy-15-02874" class="html-bibr">18</a>]).</p> Full article ">Figure 14
<p>Circuit to realise high fidelity version of the image in <a href="#entropy-15-02874-f007" class="html-fig">Figure 7</a>(a). On the left is the circuit to confine the flip operation to the predetermined 2 × 2 sub- area, <span class="html-italic">i.e.</span> left lower-half, of the image in <a href="#entropy-15-02874-f007" class="html-fig">Figure 7</a>(a); and to its right, the resulting transformed image. (The image on the right corrects the image for the same example in [<a href="#B18-entropy-15-02874" class="html-bibr">18</a>]).</p> Full article ">Figure 15
<p>A 4×4 image showing sub-blocks labelled a–e within which the transformations <span class="html-italic">U<sub>a</sub></span>, <span class="html-italic">U<sub>b</sub></span>, <span class="html-italic">U<sub>c</sub></span>, <span class="html-italic">U<sub>d</sub></span> and <span class="html-italic">U<sub>e</sub></span> should be confined.</p> Full article ">Figure 16
<p>Circuit showing the layers to confine the operations <span class="html-italic">U<sub>a</sub></span>, <span class="html-italic">U<sub>b</sub></span>, <span class="html-italic">U<sub>c</sub></span>, <span class="html-italic">U<sub>d</sub></span> and <span class="html-italic">U<sub>e</sub></span> to the layers labelled “a” to “e” of the image in <a href="#entropy-15-02874-f015" class="html-fig">Figure 15</a>. MSQ and LSQ indicate the most and least significant qubits of the FRQI representation encoding the image.</p> Full article ">Figure 17
<p>Original Lena image with labelled sub-blocks.</p> Full article ">Figure 18
<p>The original Lena image and the two different output images using <math display="inline"> <semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mo>(</mo> <mfrac> <mi>π</mi> <mn>12.5</mn> </mfrac> <mo>)</mo> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mo>(</mo> <mfrac> <mi>π</mi> <mn>125</mn> </mfrac> <mo>)</mo> </mrow> </semantics> </math> as discussed in the text.</p> Full article ">Figure 19
<p>The quantum circuit to realise the output images in <a href="#entropy-15-02874-f018" class="html-fig">Figure 18</a>.</p> Full article ">Figure 20
<p>Watermark embedding procedure of the WaQI scheme.</p> Full article ">Figure 21
<p>Merger of 2×2 sub-block entries from the first to the 2nd iteration.</p> Full article ">Figure 22
<p>Merging the content of 2×2 sub-block entries to realise (<b>i</b>) e = 1and (<b>ii</b>) e = –1values as explained in step 3 of the watermark-embedding algorithm.</p> Full article ">Figure 23
<p>(<b>a</b>) the a–d Alphabet test image—HTLA text logo watermark pair, and (<b>b</b>) the watermarked version of the a–d Alphabet test image.</p> Full article ">Figure 24
<p>Watermark map for a–d alphabet–HTLA text watermark pair.</p> Full article ">Figure 25
<p>Watermark embedding circuit for the a–d alphabet/HTLA text logo pair in <a href="#entropy-15-02874-f023" class="html-fig">Figure 23</a>(a).</p> Full article ">Figure 26
<p>Decomposing layer 1 of the watermark-embedding circuit in <a href="#entropy-15-02874-f025" class="html-fig">Figure 25</a> into its two sub-layers.</p> Full article ">Figure 27
<p>Merging flip gates to realise revised <span class="html-italic">F<sup>X</sup></span>, and <span class="html-italic">F<sup>Y</sup></span> operations for (i) <span class="html-italic">R &gt; L</span> and (ii) <span class="html-italic">R &lt; L.</span></p> Full article ">Figure 28
<p>Merger of watermark map content to realise the revised GTQI operations for <span class="html-italic">R = L</span>. The operation <span class="html-italic">G<sub>I</sub></span> could be any of the operations from our <span class="html-italic">r</span>GTQI library comprising of the flip operations, <span class="html-italic">F<sup>X</sup></span> or <span class="html-italic">F<sup>Y</sup></span>; the coordinate swap operation, <span class="html-italic">S</span>; or the do nothing operation, <span class="html-italic">D</span>.</p> Full article ">Figure 29
<p>Merging of flip gates to realise the revised <span class="html-italic">F<sup>X</sup></span> and <span class="html-italic">F<sup>Y</sup></span> flip operations for <span class="html-italic">R = L.</span></p> Full article ">Figure 30
<p>Quantum watermarked image authentication procedure.</p> Full article ">Figure 31
<p>Dataset comprising of images and watermark signals used for simulation-based experiments on WaQI.</p> Full article ">Figure 32
<p>Top row shows the watermark maps for the image paired with different watermark signals HTLA text, Baboon, and Noise image. Below is the watermarked version for each pair and their corresponding PSNR values.</p> Full article ">Figure 33
<p>Variation of watermarked image quality (PSNR) with the size of the Lena–Noise image pair. The size of each point in the watermark maps in the top row varies with the size of the image–watermark pairs. It is 8×8 for the 256×256 and 512×512 pairs; and 16×16 for the 1024×1024 Lena–Noise pair.</p> Full article ">Figure 34
<p>Variation of watermarked image quality (PSNR) with size of image–watermark pair.</p> Full article ">Figure 35
<p>Relationship between the colour angle <span class="html-italic">θ<sub>i</sub></span> and greyscale value |<span class="html-italic">G<sub>i</sub></span>〉 in an FRQI image.</p> Full article ">Figure 36
<p>Greyscale spectrum showing the correlation between the greyscale values and changes in their values that can be perceived by the HVS.</p> Full article ">Figure 37
<p>General schematic for two-tier watermarking and authentication of greyscale quantum images.</p> Full article ">Figure 38
<p>Generalised circuit for the two-tier watermarking of greyscale FRQI images. The visible and invisible watermark embedding transformations <span class="html-italic">T<sub>α</sub></span> and <span class="html-italic">T<sub>β</sub></span> are confined to predetermined areas of the cover image using the control-conditions specified by <span class="html-italic">I<sub>Rl</sub></span> and <span class="html-italic">I<sub>S</sub></span> respectively.</p> Full article ">Figure 39
<p>(<b>a</b>)–(<b>d</b>) Cover images and (<b>e</b>) watermark logo used for experiments on the proposed scheme.</p> Full article ">Figure 40
<p>Watermark embedding circuit for the Lena-Titech logo pair.</p> Full article ">Figure 41
<p>(Top row) shows the four watermarked images while (Bottom row) shows the magnified visible watermarked windows and PSNR for each pair.</p> Full article ">Figure 42
<p>Watermark recovery circuit for the Lena- Titech logo pair.</p> Full article ">Figure 43
<p>Results for the Lena-Titech logo pair based on the revised watermark embedding circuit for the scheme-designated watermark window on the left and one whose watermark window has been assigned to the extreme lower-right corner by default.</p> Full article ">Figure 44
<p>Revised watermark-embedding circuit for the Lena-Titech logo pair using the scheme-designated watermark window.</p> Full article ">Figure 45
<p><span class="html-italic">m</span>-shots from a movie showing the key |<span class="html-italic">F<sub>m</sub></span>〉, makeup <math display="inline"> <semantics> <mrow> <mo>|</mo> <msubsup> <mi>K</mi> <mi>c</mi> <mi>m</mi> </msubsup> <mo>〉</mo> </mrow> </semantics> </math>, and viewing |<span class="html-italic">F<sub>mq</sub></span>〉 frames.</p> Full article ">Figure 46
<p>Circuit structure to encode the input of a movie strip.</p> Full article ">Figure 47
<p>Circuit structure to encode the input of a movie strip Circuits for SMO. Depending on the motion axis <span class="html-italic">Z<sub>n</sub></span> = <span class="html-italic">x</span> or <span class="html-italic">y</span>) the circuit on the left is used to accomplish the <math display="inline"> <semantics> <mrow> <msubsup> <mi>M</mi> <mi>F</mi> <mi>c</mi> </msubsup> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msubsup> <mi>M</mi> <mi>D</mi> <mi>c</mi> </msubsup> </mrow> </semantics> </math> operations when applied along the <span class="html-italic">x</span><sup>−</sup> and <span class="html-italic">y</span><sup>−</sup> axis, respectively. Similarly, the circuit on the right is used to accomplish the <math display="inline"> <semantics> <mrow> <msubsup> <mi>M</mi> <mi>B</mi> <mi>c</mi> </msubsup> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msubsup> <mi>M</mi> <mi>U</mi> <mi>c</mi> </msubsup> </mrow> </semantics> </math> operations when applied the <span class="html-italic">x</span><sup>−</sup> and <span class="html-italic">y</span><sup>−</sup> axis, respectively.</p> Full article ">Figure 48
<p>SMOs on the key frame in (<b>a</b>) to mimic the movement of the + shaped ROI on a constant white background and its viewing frames after applying (<b>b</b>) the forward motion operation <math display="inline"> <semantics> <mrow> <msubsup> <mi>M</mi> <mi>F</mi> <mi>c</mi> </msubsup> </mrow> </semantics> </math>, (<b>c</b>) the upward motion operation <math display="inline"> <semantics> <mrow> <msubsup> <mi>M</mi> <mi>U</mi> <mi>c</mi> </msubsup> </mrow> </semantics> </math>, and (<b>d</b>) a somewhat zigzag movement of the + ROI.</p> Full article ">Figure 49
<p>Movie scenes to demonstrate SMO operations. The panels in (a) and (b) show the transcribed scripts for scene 1 and 2, (c) shows the key frame for scene 1, and (d)-(l) show the resulting viewing frames.</p> Full article ">Figure 50
<p>Movie sub-circuit to realise the first three viewing frames of scene 1 (of the example in <a href="#entropy-15-02874-f049" class="html-fig">Figure 49</a>). The layers separated by short-dashed lines labelled “a” indicate SMO operations, while the layers grouped and labelled as “b” indicate CTQI transformations on the key frame. Layers labelled <math display="inline"> <semantics> <mrow> <msubsup> <mi>M</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <msubsup> <mi>M</mi> <mn>2</mn> <mn>0</mn> </msubsup> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msubsup> <mi>M</mi> <mn>3</mn> <mn>0</mn> </msubsup> </mrow> </semantics> </math> indicate sub-circuits of the movie reader to recover the classical readout of frames |<span class="html-italic">f</span><sub>0,1</sub>〉, |<span class="html-italic">f</span><sub>0,2</sub>〉 and |<span class="html-italic">f</span><sub>0,3</sub>〉.</p> Full article ">Figure 51
<p>Restricting the movie operation <math display="inline"> <semantics> <mrow> <msubsup> <mi>M</mi> <mi>F</mi> <mi>C</mi> </msubsup> </mrow> </semantics> </math> in order to move the ROI <span class="html-italic">R<sub>1</sub></span> from node 0 to node 1 as specified by the movie script.</p> Full article ">Figure 52
<p>Movie sub-circuit for scene 2 in <a href="#entropy-15-02874-f049" class="html-fig">Figure 49</a>(b). The labels 5 through 7 and 5′ through 7′ for <span class="html-italic">R<sub>1</sub></span> and <span class="html-italic">R<sub>2</sub></span> indicate the circuit layers to perform the operations that yield the viewing frames in <a href="#entropy-15-02874-f049" class="html-fig">Figure 49</a>(j) –(l).</p> Full article ">Figure 53
<p>Circuit on the <span class="html-italic">m</span>FRQI strip axis to perform the frame-to-frame transition operation.</p> Full article ">Figure 54
<p>The cyclic shift transformation for the case <span class="html-italic">c</span> = 1 and <span class="html-italic">n</span> = 5.</p> Full article ">Figure 55
<p>A single qubit measurement gate.</p> Full article ">Figure 56
<p>Exploiting the position information |<span class="html-italic">i</span>〉 of the FRQI representation to predetermined the 2D grid location of each pixel in a transformed image <span class="html-italic">G<sub>I</sub></span>(|<span class="html-italic">I(θ)</span>〉).</p> Full article ">Figure 57
<p>Control-conditions to recover the readout of the pixels of a 2<span class="html-italic"><sup>n</sup></span>×2<span class="html-italic"><sup>n</sup></span> FRQI quantum image.</p> Full article ">Figure 58
<p>Predetermined recovery of the position information of an FRQI quantum image. The * between the colour |<span class="html-italic">c</span>(<span class="html-italic">θ<sub>i</sub></span>)〉 and ancilla |<span class="html-italic">a</span>〉 qubits indicates the dependent ancilla-driven measurement as described in <a href="#entropy-15-02874-f059" class="html-fig">Figure 59</a> and Theorem 4.</p> Full article ">Figure 59
<p>Circuit to recover the content of the single-qubit colour information of an FRQI quantum image. This circuit represents each of the * between the colour and ancilla qubit in <a href="#entropy-15-02874-f058" class="html-fig">Figure 58</a>.</p> Full article ">Figure 60
<p>Reader to recover the content of a 2<span class="html-italic"><sup>n</sup></span>×2<span class="html-italic"><sup>n</sup></span> FRQI quantum image.</p> Full article ">Figure 61
<p>Movie reader sub-circuit to recover pixel <span class="html-italic">p</span><sub>0</sub> and <span class="html-italic">p</span><sub>1</sub> for frame |<span class="html-italic">f</span><sub>0,1</sub>〉 corresponding to <a href="#entropy-15-02874-f049" class="html-fig">Figure 49</a>e.</p> Full article ">Figure 62
<p>Movie reader to recover pixels <span class="html-italic">p</span><sub>4</sub>, <span class="html-italic">p</span><sub>6</sub>, <span class="html-italic">p</span><sub>8</sub>, <span class="html-italic">p</span><sub>9</sub>, <span class="html-italic">p</span><sub>10</sub> of viewing frame |<span class="html-italic">f</span><sub>0,1</sub>〉.</p> Full article ">Figure 63
<p>Readout of the new state of pixel <span class="html-italic">p</span><sub>4</sub> as transformed by sub-circuit 1 in <a href="#entropy-15-02874-f052" class="html-fig">Figure 52</a>.</p> Full article ">Figure 64
<p>Movie reader sub-circuit to recover the content of pixels <span class="html-italic">p</span><sub>2</sub>, <span class="html-italic">p</span><sub>3</sub>, <span class="html-italic">p</span><sub>7</sub>, <span class="html-italic">p</span><sub>11</sub>, <span class="html-italic">p</span><sub>12</sub>, <span class="html-italic">p</span><sub>13</sub> and <span class="html-italic">p</span><sub>15</sub>.</p> Full article ">Figure 65
<p>Key and makeup frames for the scene “The lonely duck goes swimming”. See text and [<a href="#B19-entropy-15-02874" class="html-bibr">19</a>] for additional explanation.</p> Full article ">Figure 66
<p>Key and makeup frames for the scene “The cat and mouse chase”. See text and [<a href="#B19-entropy-15-02874" class="html-bibr">19</a>] for additional explanation.</p> Full article ">Figure 67
<p>Framework for quantum movie representation and manipulation.</p> Full article ">
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