Row‐column factorial designs with multiple levels Rahim, Fahim; Cavenagh, Nicholas J.
Journal of combinatorial designs,
November 2021, 2021-11-00, 20211101, Letnik:
29, Številka:
11
Journal Article
Recenzirano
Odprti dostop
An
m
×
n row‐column factorial design is an arrangement of the elements of a factorial design into a rectangular array. Such an array is used in experimental design, where the rows and columns can act ...as blocking factors. Formally, for any integer
q, let
q
=
{
0
,
1
,
…
,
q
−
1
}. The
q
k (full) factorial design with replication
α is the multiset consisting of
α occurrences of each element of
q
k; we denote this by
α
×
q
k. A regular
m
×
n row‐column factorial design is an arrangement of the elements of
α
×
q
k into an
m
×
n array (which we say is of type
I
k
(
m
,
n
;
q
)) such that for each row (column) and fixed vector position
i
∈
k
, each element of
q
occurs
n
∕
q times (respectively,
m
∕
q times). Let
m
≤
n. We show that an array of type
I
k
(
m
,
n
;
q
) exists if and only if (a)
q
∣
m and
q
∣
n; (b)
q
k
∣
m
n; (c)
(
k
,
q
,
m
,
n
)
≠
(
2
,
6
,
6
,
6
), and (d) if
(
k
,
q
,
m
)
=
(
2
,
2
,
2
) then 4 divides
n. Godolphin showed the above is true for the case
q
=
2 when
m and
n are powers of 2. In the case
k
=
2, the above implies necessary and sufficient conditions for the existence of a pair of mutually orthogonal frequency rectangles (or
F‐rectangles) whenever each symbol occurs the same number of times in a given row or column.
A structure of a compact ultra-wideband monopole antenna has been presented. The antenna consists of a microstrip-fed rectangle radiator as well as a ground plane with a rectangle slit and an ...L-shaped stub. The critical factor in achieving a small size is a careful design procedure involving numerical optimisation of all geometry parameters of the antenna aiming at explicit size reduction while maintaining acceptable electrical performance. The final design exhibits dimensions of only 9.45 × 18.5 mm and a footprint of 175 mm2. Experimental validation and comparisons with competitive designs are also provided.
Geometric Matching for Cross-Modal Retrieval Wang, Zheng; Gao, Zhenwei; Yang, Yang ...
IEEE transaction on neural networks and learning systems,
04/2024, Letnik:
PP
Journal Article
Despite its significant progress, cross-modal retrieval still suffers from one-to-many matching cases, where the multiplicity of semantic instances in another modality could be acquired by a given ...query. However, existing approaches usually map heterogeneous data into the learned space as deterministic point vectors. In spite of their remarkable performance in matching the most similar instance, such deterministic point embedding suffers from the insufficient representation of rich semantics in one-to-many correspondence. To address the limitations, we intuitively extend a deterministic point into a closed geometry and develop geometric representation learning methods for cross-modal retrieval. Thus, a set of points inside such a geometry could be semantically related to many candidates, and we could effectively capture the semantic uncertainty. We then introduce two types of geometric matching for one-to-many correspondence, i.e., point-to-rectangle matching (dubbed P2RM) and rectangle-to-rectangle matching (termed R2RM). The former treats all retrieved candidates as rectangles with zero volume (equivalent to points) and the query as a box, while the latter encodes all heterogeneous data into rectangles. Therefore, we could evaluate semantic similarity among heterogeneous data by the Euclidean distance from a point to a rectangle or the volume of intersection between two rectangles. Additionally, both strategies could be easily employed for off-the-self approaches and further improve the retrieval performance of baselines. Under various evaluation metrics, extensive experiments and ablation studies on several commonly used datasets, two for image-text matching and two for video-text retrieval, demonstrate our effectiveness and superiority.
•A special RPAMP with central rectangles is described and named CR-RPAMP.•A novel heuristic algorithm called HACR is presented for solving CR-RPAMP.•For describing easily, some new definitions are ...introduced in HACR.•Detailed strategies of rectangles packing are proposed to meet the requirements.•HACR is applied to the layout research of drilling equipment of drilling platforms.
The rectangle packing area minimization problem (RPAMP) has a wide range of applications in the industrial production. A special RPAMP with central rectangles that must be located in the center of the final layout is proposed and named CR-RPAMP in which the length-width ratio of the final layout can be changed legitimately within a reasonable scope. In this paper, for the purpose of solving the CR-RPAMP, a novel heuristic algorithm called HACR is presented. In HACR, by constraining the aspect ratio of enveloping rectangle, the length-width ratio of the final rectangular frame can meet the requirements. Besides, by constraining the betweenness centrality of central rectangle, the central rectangle can be located in the center of the final layout. In order to minimize the area of the enveloping rectangle, the solution procedure of HACR has been projected based on defining the priority ofcandidate rectangle. Strategies of padding inner space are put forward to improve the filling rate of the final layout. Comprehensive experiments were conducted on 34 international instances reported in the literature. Simulation results show that the proposed novel heuristicalgorithm was effective and practicable. At last, the proposed HACR is applied to research the layout of drilling equipment in deep water semi-submersible platforms.
Oriented object detection has achieved a significant progress in image processing. Compared with horizontal detection methods, oriented detectors add the orientation parameter in regression to locate ...objects. However, the existing rotation and quadrilateral representations are not appropriate for oriented two-stage methods to generate efficient oriented proposals. In this letter, we propose a novel framework to detect oriented objects, termed short-side excursion detection (SSEDet). Inspired by the circle theorem, we propose a transformation method from horizontal rectangles to oriented ones to accurately describe oriented objects. To be specific, we exploit the offset of short sides relative to the top-right vertex to represent the orientation of rectangle. Compared with the horizontal rectangle, the representation parameters of oriented rectangle have only one more orientation parameter. Under the action of the orientation parameter, the one-to-one correspondence between representation parameters and oriented rectangle can be realized. Experimental results on commonly used datasets verify that the SSEDet can generate high-quality oriented proposals.
Mutually orthogonal frequency rectangles Rahim, Fahim; Cavenagh, Nicholas J.
Discrete mathematics,
December 2023, 2023-12-00, Letnik:
346, Številka:
12
Journal Article
Recenzirano
Odprti dostop
A frequency rectangle of type FR(m,n;q) is an m×n matrix such that each symbol from a set of size q appears n/q times in each row and m/q times in each column. Two frequency rectangles of the same ...type are said to be orthogonal if, upon superimposition, each possible ordered pair of symbols appear the same number of times. A set of k frequency rectangles in which every pair is orthogonal is called a set of mutually orthogonal frequency rectangles, denoted by k–MOFR(m,n;q). We show that a k–MOFR(2,2n;2) and an orthogonal array OA(2n,k,2,2) are equivalent. We also show that an OA(mn,k,2,2) implies the existence of a k–MOFR(2m,2n;2). We construct (4a−2)–MOFR(4,2a;2) assuming the existence of a Hadamard matrix of order 4a.
A k–MOFR(m,n;q) is said to be t–orthogonal, if each subset of size t, when superimposed, contains each of the qt possible ordered t-tuples of entries exactly mn/qt times. A set of vectors over a finite field Fq is said to be t-independent if each subset of size t is linearly independent. We describe a method to obtain a set of t–orthogonal k–MOFR(qM,qN,q) corresponding to a set of t–independent vectors in (Fq)M+N. We also discuss upper and lower bounds on the sizes of sets of t–independent vectors and give a table of values for binary vectors of length N⩽16.
A frequency rectangle of type FR(n,n;q) is called a frequency square and a set of k mutually orthogonal frequency squares is denoted by k–MOFS(n;q) or k–MOFS(n) when there is no ambiguity about the symbol set. For p an odd prime, we show that there exists a set of (p−1) binary MOFS(2p), hence improving the lower bounds in (Britz et al. 2020) for p⩾19.
A squared rectangle is a rectangle dissected into squares. Similarly a rectangled rectangle is a rectangle dissected into rectangles. The classic paper ‘The dissection of rectangles into squares’ of ...Brooks, Smith, Stone and Tutte described a beautiful connection between squared rectangles and harmonic functions. In this paper we count dissections of a rectangle into a set of integral squares or a set of integral rectangles. Here, some squares and rectangles may have the same size. We introduce a method involving a recurrence relation of large sized matrices to enumerate squared and rectangled rectangles of a given sized rectangle and propose the asymptotic behavior of their growth rates.
•A method to efficiently reconstruct 3D indoor scene from a single image is presented.•Our method not only can estimate room layout of scene, but also can reconstruct excellent details of scene. The ...method can cope with clutter without prior training, making it practical and efficient for a navigating robot.•The method can estimate locations and positions of spatial rectangles in 3D scenes, without the knowledge of camera’s intrinsic parameters, nor of the relation between the camera and world.
Understanding of indoor scenes has considerable value in mission planning and monitoring in robots. This has become one of the biggest challenges in computer vision because of the diversity and changeability of 3D indoor scenes. Indoor scenes can be considered compositions of many planes in which most common external surfaces are rectangles, such as doors, windows, walls, tables. These spatial rectangles are projected into 2D projections with special geometric configurations, which may enable us to estimate their original orientation and position in 3D scenes. In this paper, the study presents a method to efficiently reconstruct 3D indoor scene without any knowledge of camera’s internal calibration. The approach first found quadrangles composed of lines. Through the projection of spatial rectangles, our method not only can estimate room layout of scene, but also can reconstruct excellent details of scene. Due to simple geometric inferences, our method can cope with clutter without prior training, making it practical and efficient for a navigating robot. We compare the room layout estimated by our algorithm against room box ground truth, measuring the percentage of pixels that were classified correctly. Furthermore, we evaluate our ability to fit the indoor scene by comparing against the details that were reconstructed correctly in scene. The experiments showed that our method is capable of reconstructing various structures of indoor environments and that the accuracy and speed of this method meet the requirements a of indoor robot navigation.
Towards Better Analysis of Deep Convolutional Neural Networks Liu, Mengchen; Shi, Jiaxin; Li, Zhen ...
IEEE transactions on visualization and computer graphics,
2017-Jan., 2017-01-00, 2017-1-00, 20170101, Letnik:
23, Številka:
1
Journal Article
Recenzirano
Odprti dostop
Deep convolutional neural networks (CNNs) have achieved breakthrough performance in many pattern recognition tasks such as image classification. However, the development of high-quality deep models ...typically relies on a substantial amount of trial-and-error, as there is still no clear understanding of when and why a deep model works. In this paper, we present a visual analytics approach for better understanding, diagnosing, and refining deep CNNs. We formulate a deep CNN as a directed acyclic graph. Based on this formulation, a hybrid visualization is developed to disclose the multiple facets of each neuron and the interactions between them. In particular, we introduce a hierarchical rectangle packing algorithm and a matrix reordering algorithm to show the derived features of a neuron cluster. We also propose a biclustering-based edge bundling method to reduce visual clutter caused by a large number of connections between neurons. We evaluated our method on a set of CNNs and the results are generally favorable.
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