Dr.P.S.Jagadeesh Kumar Stanford University California, United States

1. ISSN: 0002-9327 American Journal of Mathematics 12 (1)
Copyright © 2020 University Press Journals 29
Continuous and Discrete Crooklet Transform
P.S.Jagadeesh Kumar, Thomas Binford, J.Nedumaan, J.Tisa
Stanford University, California, United States
J.Ruby, Susan Daenke
University of Oxford, Oxford, United Kingdom
J.Lepika
Harvard University, Cambridge, United States
ABSTRACT
This paper introduces the prototype of a novel and powerful transform, the 'Crooklet Transform'
in overcoming the drawbacks of wavelet and curvelet transform. In image processing, one has to deal with the fact
that interesting phenomena occur along curves or crooks of images. Wavelets are certainly suitable for dealing
with objects where the interesting portents, e.g., singularities, are associated with exceptional points but they are
ill-suited for detecting, organizing, or providing a compact representation of intermediate dimensional structures.
Given the significance of such intermediate dimensional phenomena, there has been a vigorous research effort to
provide better-adapted alternatives by combining ideas from geometry with ideas from the traditional multiscale
analysis. Curvelet Transform consisted of special filtering process and multi-scale Ridgelet Transform could fit
image properties well but they are computationally complex. Continuous and Discrete Crooklet Transform
proposed in this paper is less intricate and effectual compared to Curvelet Transform and Wavelet Transform,
henceforth laying a foundation for third-generation transforms.
KEYWORDS: Continuous Crooklet Transform, Discrete Crooklet Transform, Wavelet Transform, Curvelet
Transform, Third Generation Transforms, Curves and Crooks.
1. INTRODUCTION
Wavelets are a powerful statistical tool which can be used for a wide range of applications.
Wavelet transforms are now being adopted for a vast number of applications, often trading the conventional
Fourier Transform. Wavelet Transforms can also be used in the field of image compression, feature extraction,
image denosing and other medical image technology [7]. Many areas of physics have seen this paradigm shift,
including molecular dynamics, astrophysics, density matrix localization, seismic geophysics, optics, turbulence
and quantum mechanics. This change has also occurred in image processing, blood-pressure, heart-rate and ECG
analysis, DNA analysis, protein analysis, climatology, general signal processing, speech recognition, computer
graphics and multifractal analysis. The signal which is described by a continuous function (for example a
recording of a speech signal or music signal, which measures the current in the cable to the loudspeaker as a
function of time) is called continuous signals [6]. The signal which is described by a sequence of numbers or pairs
of numbers is called discrete signals. A digital black-white photo consists of a splitting of the picture into a large
number of small squares, called pixels; to each pixel, the camera associates a light intensity, measured on a scale

2. P.S.Jagadeesh Kumar et al. American Journal of Mathematics 12 (1)
ISSN: 0002-9327 30
from, say, 0 (completely white) to 256 (completely black). Put together, this data constitutes the picture. Thus,
mathematically a photo consists of a sequence of pairs of numbers, namely, a numbering of the pixels together
with the associated light intensity. Curvelet Transform consisted of special filtering process and multi-scale
Ridgelet Transform. It could fit image properties well. Nevertheless, Curvelet Transform had complicated digital
realization, includes sub-band division, smoothing block, normalization, Ridgelet analysis and so on. Curvelet‟s
pyramid decomposition brought immense data redundancy [10]. Forward Fast Curvelet Transform (FCT) was the
Second Generation Curvelet Transform which was simpler and easily understandable. On the other hand,
Wavelets are frequently used to remove noise from music recordings. The main idea is to think about a music
signal as consisting of the music itself to which some noise is added. The music signal itself describes how the
music changes in time; we can think about the signal as the current through the loudspeaker when we play a
recording. The noise contribution is usually small compared to the music, but irritating for the ears; it also
contributes to the coefficients, but usually less than the music itself. The idea is now to remove the coefficients
which are smaller than a certain threshold value (this procedure is not applied on the signal itself, but on its so
called wavelet transform).
A special member of the emerging family of multiscale geometric transforms is the curvelet
transform, which was developed in the last few years in an attempt to overcome inherent limitations of traditional
multiscale representations such as wavelets [11]. Conceptually, the curvelet transform is a multiscale pyramid
with many directions and positions at each length scale, and needle-shaped elements at fine scales. This pyramid
is nonstandard, however. Indeed, curvelets have useful geometric features that set them apart from wavelets and
the likes. For instance, curvelets obey a parabolic scaling relation which says that at scale 2−j
, each element has an
envelope which is aligned along a “ridge” of length 2−j/2
and width 2−j
. Fast Discrete Curvelet Transforms (FDCT)
describes two digital implementations of a new mathematical transform, namely, the second generation curvelet
transform in two and three dimensions. The first digital transformation is based on unequally-spaced fast Fourier
transforms (USFFT) while the second is based on the wrapping of specially selected Fourier samples. The two
implementations essentially differ by the choice of spatial grid used to translate curvelets at each scale and angle.
Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an
orientation parameter, and a spatial location parameter [9]. Both implementations are fast in the sense that they
run in O(n2
log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion
algorithms of about the same complexity. The digital transformations improve upon earlier implementations based
upon the first generation of curvelets, in the sense that they are conceptually simpler, faster and far less redundant.
The next section introduces the concept, design and development of Crooklet Transform.
2. CONTINUOUS CROOKLET TRANSFORM (CCrT)
In definition, the continuous crooklet transform is a convolution of the input data sequence with
a set of functions generated by the principal crooklet. The convolution can be computed by using a Fast Fourier
Transform (FFT) algorithm. Normally, the output is a real valued function except when the principal crooklet is
complex. A complex principal crooklet will convert the continuous crooklet transform to a complex valued
function. The power spectrum of the continuous crooklet transform can be represented by | ( )| . The

3. Continuous and Discrete Crooklet Transform
Copyright © 2020 University Press Journals 31
Continuous Crooklet Transform of a function ( ) at a scale ( ) and translational value is
expressed by the following integral;
( ) | | ∫ ( ) ̅ [ ]
where ( ) is a continuous function in both the time domain and the frequency domain and the overline
represents operation of complex conjugate. The main purpose of ( ) is to provide a source function to generate
its descendant crooklet, which are simply the translated and scaled versions of ( ), the principal crooklet.
Inverse Continuous Crooklet Transform
To recover the original signal ( ), the first inverse continuous crooklet transform can be exploited.
( ) ∫ ∫ ( ) | | ̌ [ ]
̌( ) is the dual function of ( ) and
∫ ( ̅( ) ̌( ) | |)
is admissible constant, where hat means Fourier transform operator. Sometimes, ̌( ) ( ), then the
admissible constant becomes;
∫
| ̌( )|
| |
Traditionally, this constant is called single admissible constant. A crooklet whose admissible constant satisfies;
is called an admissible crooklet. An admissible crooklet implies that ̌( ) , so that an admissible crooklet
must integrate to zero. To recover the original signal ( ) , the second inverse continuous wavelet transform can
be exploited.
( )
̌( )
∫ ∫ ( ) [ ]
This inverse transform suggests that a wavelet should be defined as,
̌( ) ( ) ( )
where ( ) is a window. Such defined crooklet can be called as an analyzing crooklet, because it admits to time-
frequency analysis. An analyzing crooklet is unnecessary to be admissible. The signal graph of Continuous
Crooklet Transform between the radial window and the angular window curves from 2-t
to 2-t/2
, a phenomenal
property of curvelet transforms achieved in crooklet domain.

4. P.S.Jagadeesh Kumar et al. American Journal of Mathematics 12 (1)
ISSN: 0002-9327 32
3. DISCRETE CROOKLET TRANSFORM (DCrT)
The DCrT of a signal is given by;
∑
DCrT is calculated by passing through a series of filters. First the samples are passed through a low pass filter
with impulse response resulting in a convolution of the two:
( )
∑ ( )
The signal is also decomposed simultaneously using a high-pass filter . The outputs give the
detail coefficients (from the high-pass filter) and approximation coefficients (from the low-pass). It is important
that the two filters are related to each other and they are known as square root mirror filter.
( )
∑ ( )
This decomposition has doubled the time resolution since only square root of each filter output
characterizes the signal series. However, each output has double the frequency band of the input, so the frequency
resolution has been squared similar to in curvelet transform.
Time complexity
The filter bank implementation of the Discrete Crooklet Transform (DCrT) takes only O(N), as
compared to O(N log N) for the fast Fourier transform. The filter bank does each of these two O(N) convolutions.
4. APPLICATIONS OF CROOKLET TRANSFORM
Crooklet transform is a powerful statistical tool which can be used for a wide range of
applications, namely: Signal processing, Data compression, Smoothing and image denoising, Fingerprint
verification, Biology for cell membrane recognition, To distinguish the normal from the pathological membranes,
DNA analysis, Protein analysis, Blood-pressure, Heart-rate and ECG analysis, Finance for detecting the properties
of quick variation of values, In Internet traffic description, For designing the services size, Industrial supervision
of gear-wheel, Speech recognition, Computer graphics and multifractal analysis, and In many areas of physics
(molecular dynamics, astrophysics, etc.). The applications of the Crooklet transform were extended to image
contrast enhancement and astronomical image representation to the fusion of satellite images. The applications of
Crooklets showed continuous improvement in many fields involving image/video presentation, denoising and
classification. The first model introduced in Euro Robotics 2019, Crooklet Transform lay the fundamental block
in building self-learning robots since their computational cost is significantly lower. Crooklet transform overlooks
the challenge of space-domain formulation of curvelet transform [10].

5. Continuous and Discrete Crooklet Transform
Copyright © 2020 University Press Journals 33
Digital Image Processing Applications:
a) Fingerprint Recognition:
Fingerprint verification is one of the most reliable personal identification methods and it plays
an essential role in forensic and civilian applications. For automatic identification, it is one of the primogenital
and most reliable methods because of invariance of the fingerprint features over the age of the subject. Facsimile
scans of the impressions are distributed among law enforcement agencies, but the digitization quality is often low.
Because a number of jurisdictions are experimenting with digital storage of the prints, incompatibilities between
data formats have recently become a problem [1]. This problem led to a demand in the criminal justice community
for a digitization and a compression standard, to overcome this problem. Crooklet Transform epitomized
predominant usage in all finger print related compression and image processing applications.
b) Image Compression:
Image compression is one of the central and successful applications of the Crooklet Transform.
The rapid increase in the range and use of electronic imaging justifies attention for systematic design of an image
compression system and for providing the image quality needed in different applications. Image compression
algorithms aim to remove redundancy in data in a way which makes image reconstruction possible. This basically
means that image compression algorithms attempt to exploit redundancies in the data; they calculate which data
needs to be kept in order to reconstruct the original image and therefore which data can be thrown away. By
removing the redundant data, the image can be represented in a smaller number of bits, and hence can be
compressed [3]. With the inherent features of Crooklet Transform, it provides multi-resolution functionality and
better compression performance at very low bit-rate compared to other image and video compression standards.
c) Image Denoising:
An image is often corrupted by noise in its acquisition and transmission. Image denoising is
used to remove the additive noise while retaining as much as possible the important signal features. Crooklet
Transform provides us with one of the methods for image denoising. Crooklet Transform, due to its excellent
localization property, has rapidly become an indispensable signal and image processing tool for a variety of
applications, including denoising and compression. Crooklet denoising attempts to remove the noise present in the
signal while preserving the signal characteristics, regardless of its frequency content. Crooklet thresholding is one
of the signal estimation techniques that exploit the capabilities of Crooklet Transform for signal denoising [4]. It
removes noise by killing coefficients that are trivial relative to some threshold. Researchers have developed
various techniques for choosing denoising parameters and so far there is no best universal threshold determination
technique. The Crooklet Transform domain noise filtration technique preserves edges and removes noise. Noise is
especially removed from the Crooklet Transform data at a given scale by comparing the data at that scale to the
correlation of the data at that scale with those at larger scales. Features are identified and retained because they are
strongly correlated across scale in the Crooklet Transform domain. Noise is identified and removed because it is
poorly correlated across scale in the Crooklet Transform domain. Features remain relatively undistorted because
they are very well localized in space in the Crooklet Transform domain; therefore, edges remain sharp after
filtration.

6. P.S.Jagadeesh Kumar et al. American Journal of Mathematics 12 (1)
ISSN: 0002-9327 34
d) Face Recognition:
Face recognition such as identification of person using Credit cards, Passport check, Criminal
investigations etc. The human face is an important object in image and video databases, because it is a unique
feature of human beings and is ubiquitous in photos, news videos, and video telephony. Face detection can be
regarded as a more general case of face localization. In face localization, the task is to find the locations and sizes
of a known number of faces (usually one). In face detection, one does not have this additional information. There
are several application areas where automated face recognition is a relatively new concept. Developed in the
1960s, the first semi-automated system for face recognition required the administrator to locate features (such as
eyes, ears, nose, and mouth) on the photographs before it calculated distances and ratios to a common reference
point, which were then compared to reference data [5]. Face recognition can be used for both verification and
identification (open-set and closed-set). Linear Discriminant Analysis (LDA) with Crooklet transformation can be
regarded as one of the principal techniques for face recognition systems. LDA is well-known arrangement for
feature extraction and dimension reduction. LDA using Crooklet Transform approach enhances performance as
regards accuracy and time complexity.
e) Image Fusion:
Extracting more information from multi-source images is an attractive thing in remotely sensed
image processing, which is recently called the image fusion [8]. Image Fusion using Crooklet Transform is used
to improve the geometric resolution of the images, in which two images to be processed are firstly decomposed
into sub-images with different frequency and then the information fusion is performed using these images under
certain conditions and finally these sub-images are reconstructed into the result image with plentiful information.
The successful fusion of images acquired from different modalities or Instruments is of great importance in many
applications such as medical imaging, microscopic imaging, remote sensing, computer vision, and robotics [2].
For the remotely-sensed images, some have good spectral information, and the others have high geometric
resolution, how to integrate the information of these two kinds of images into one kind of images is very attractive
thing in image processing, which is called image fusion. For the purpose of realization of this task, we often need
some algorithms to fuse the information of these two kinds of images. Crooklet Transform used in the fields of
graphics and image fusion is been proved to be an effective tool to process the signals in multiscale spaces.
Some Advantages of Crooklet Transform:
 One of the main advantages of Crooklet Transform is that they offer a simultaneous localization in time
and frequency domain.
 The second main advantage of Crooklet Transform is that it is computationally feasible.
 Crooklet Transform has the great advantage of being able to separate the fine details in a signal.
 It is possible to obtain a good approximation of the given function „f‟ by using only a few coefficients
which are the great achievement in comparison to other transform.
 Crooklet Transform is capable of revealing aspects of data that other signal analysis techniques miss the
aspects like trends, breakdown points, and discontinuities in higher derivatives and self-similarity.
 It can efficiently compress or denoise a signal without appreciable degradation.

7. Continuous and Discrete Crooklet Transform
Copyright © 2020 University Press Journals 35
5. EVALUATION OF CROOKLET TRANSFORM
Evaluating well established and many decades old mathematical transform model with a newly
proposed transform model is painstaking and not an easy task. The authors have made their best practices with
real-time and practical applications in comparing the proposed mathematical transform model with existing
models since the end user of any mathematical transform model is the real time applications. Some of the most
commonly used real time applications were taken into consideration in evaluating the Crooklet Transform as
discussed in the earlier section and the remaining is left to the readers for their concern.
Table 1: Performance Evaluation of Crooklet Transform
In Table 1, five major areas of digital image processing applications such as image compression,
image denoising, face recognition, fingerprint recognition and image fusion were analyzed for the performance of
Crooklet Transform alongside the other major transforms. The relevant metrics were carefully measured for each
application with respective mathematical transformation model. The illustrated results portray that the proposed
transform convincingly out performs the other major transforms in all the concerned applications in the area of
signal and image processing. Particularly when utilized for robotic applications, the Crooklet transform is less
intricate and effectual. Crooklet Transform lay the principal block in edifying self-learning robots since their
computational cost is considerably lower. Crooklet transform superintends the contest of space-domain design of
curvelet transform. Undoubtedly, the Crooklet Transform will be deliberated as a powerful tool that provides
significant improvements in all arenas of science, engineering, and medicine. At any stage of problem solving and
modeling, numerical analysis and computational tools are required as a part of the mathematical foundations
needed to work in computer science. The ultimate aim of developing Crooklet Transform is to build a platform for
'Third-generation Transforms' in overpowering the drawbacks of Second-generation Wavelet and Curvelet
Transforms.
Transform
Fingerprint
Recognition
Image
Compression
Image Denoising Face Recognition Image Fusion
True
Positive
(TP)
True
Negative
(TN)
Mean
Squared
Error
(MSE)
Peak
Signal
To
Noise
Ratio
(PSNR)
Structural
Similarity
(SSIM)
Feature
Similarity
Index
(FSIM)
Precision
Rate
(PR)
Recall
Rate
(RR)
Correlation
Coefficient
(CC)
Per-pixel
Deviation
(PD)
Wavelet
Transform
84.32% 83.37% 0.19 50.81 0.9132 0.8985 83.63% 89.32% 0.8010 0.0345
Curvelet
Transform
86.12% 86.35% 0.39 23.05 0.8932 0.8722 87.33% 87.25% 0.8696 0.0126
Crooklet
Transform
95.61% 94.65% 0.10 60.65 0.9499 0.9364 93.92% 94.56% 0.9143 0.0016

8. P.S.Jagadeesh Kumar et al. American Journal of Mathematics 12 (1)
ISSN: 0002-9327 36
CONCLUSION
Curvelet transform descends from the wavelet transform, and it overwhelms the deficits of
wavelet transform in the manifestation of image edges. The anisotropic features of curvelet transform provide an
idyllic depiction of the object edges. However, both wavelet and curvelet transforms were not fair enough to
provide efficient representation in all the domains of signal and image processing. The proposed prototype of
Continuous and Discrete Crooklet Transform is anticipated to be a powerful mathematical transformation tool in
all arenas of science, engineering, and medicine. The Crooklet Transform embraces the imperative phenomena‟s
from both the wavelet and the curvelet transforms. The incompetence and shortfalls of the major transforms were
endowed by the proposed Crooklet Transform. As the name indicates crooklets are efficient in dealing with the
contours, edges, curves and other aspects related to image and video processing as well. To instigate, the Crooklet
Transform will amateur a basis for third-generation transforms.
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