Computational complexity of O(n2) versus O(n log2 n).
\r\n\t
",isbn:"978-1-83962-718-7",printIsbn:"978-1-83962-717-0",pdfIsbn:"978-1-83962-754-5",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!0,hash:"4df95c7f295de7f6003e635d9a309fe9",bookSignature:"Dr. Yajuan Zhu, Dr. Qinghong Luo and Dr. Yuguo Liu",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/8969.jpg",keywords:"Water Cycle, Water Use Strategy, Vegetation Dynamics, Plant Community, Precipitation, Carbon Emission, Soil Respiration, Autotrophic Respiration, Algae Crust, Wind, Temperature, Vegetation Stability",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"January 26th 2021",dateEndSecondStepPublish:"February 23rd 2021",dateEndThirdStepPublish:"April 24th 2021",dateEndFourthStepPublish:"July 13th 2021",dateEndFifthStepPublish:"September 11th 2021",remainingDaysToSecondStep:"2 days",secondStepPassed:!0,currentStepOfPublishingProcess:3,editedByType:null,kuFlag:!1,biosketch:"Dr. Zhu holds a Ph.D. in Ecology and is currently an Associate Research Professor at the Chinese Academy of Forestry at the Institute of Desertification Studies, she has led a number of national projects while working there.",coeditorOneBiosketch:"Dr. Luo holds a Ph.D. in Physical Geography and is currently a Research Professor at the Institute of Afforestation and Sand Control, Xinjiang Academy of Forestry. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"46546",title:"Traumatic Cloaca",doi:"10.5772/57377",slug:"traumatic-cloaca",body:'Traumatic cloacal deformities are a result of major obstetric injury; these usually occur after a fourth-degree perineal laceration. They occur in one in 5000 vaginal deliveries. [1] This deformity is characterized by three anatomic lesions: complete disruption of the perineal body, anterior defect of the internal and external anal sphincter, and loss of the distal rectovaginal and/or anovaginal septum. [2-4] Thus, a common genitourinary and gastrointestinal outlet ensues, as in congenital cloaca. This is the cause of symptoms and disorders related to this condition.
Resulting disorders are mostly functional. These patients suffer from severe incontinence to flatus, liquid and solid stool. Recurrent urinary tract and vaginal infection may also result. Sexual dysfunction is significant; patients often complain of dyspareunia [4] or refrain from sexual activity altogether. Partners are also affected. All these disorders may lead to psychological disorders and social distress associated with incapacitating incontinence as well as a sense of shame from the “deformity”. [3-6] A nihilistic approach from both the patient and physicians may lead to delay in treatment, thus burdening the patient with years of suffering and diminished quality of life.
There are several reports and series in the literature describing techniques for repairing traumatic cloacal deformities. There is considerable variety regarding the use or not of flaps, the use or not of fecal diversion. In any case, repair of these deformities can be challenging and complex, and patients may present years after the initial injury. Some may have had a previous repair, thus adding difficulty to definitive correction.
Traumatic cloaca is usually the result of obstetric injury. The most common injuries involved are third and fourth-degree perineal lacerations. Third-degree lacerations are characterized by disruption of the internal and external anal sphincter. In fourth-degree lacerations, disruption of the anal and/or rectal epithelium is included in addition to sphincter injury. Surgical correction of these types of injuries involves precise, layered approximation of damaged structures and tissues. Failure of repair may be a result of hematoma, infection or technical mishaps. Partial failure may result in a rectovaginal fistula; complete failure leads to a cloacal deformity. [8] Traumatic cloaca may occur if these obstetric injuries of this nature are not repaired, or after failure of surgical repair. This may be secondary to poor surgical technique or a complication such as infection and disruption of the layered repair.
The resulting deformity is three-dimensional absence of the perineal body, disruption of the sphincter complex and loss of the anovaginal/anorectal septum of varying length.
Most, if not all, women with traumatic cloacae have severe incontinence to flatus, liquid and solid stool. These patients also suffer from sequelae of the communication of the gastrointestinal and genitourinary tract; namely, severe perineal skin irritation, as well as recurrent urinary tract and vaginal infections. Sexual function is seriously affected. Many patients complain of dyspareunia, and abstinence from sexual activity is not uncommon. Thus, spouses and sexual partners are also affected by this disorder.
Traumatic cloaca may lead to psychological disorders and social distress associated with incapacitating incontinence as well as a sense of shame from the “deformity”. Many patients with this disorder isolate themselves socially and suffer from anxiety and depression. [3-6] A nihilistic approach from both the patient and physicians may lead to delay in treatment, thus burdening the patient with years of suffering and diminished quality of life.
A careful obstetric history and detailed physical exam of the perineal area is enough to make the diagnosis of traumatic cloaca. Inspection reveals the diminution or absence of the perineal body and the common genitourinary-gastrointestinal outlet. (Fig1, 2) A digital exam will reveal the length of the remaining anovaginal/rectovaginal septum as well as the relationship of the cervix with the rectum. This will aid in planning the appropriate surgical repair. The examining physician will also note a significant decrease in resting anal sphincter tone as well as a marked loss of voluntary squeeze pressure.
Traumatic cloaca with small bridge of skin between anus and vagina.
Traumatic cloaca. Note the incomplete anovaginal septum.
Additional diagnostic tests can be performed but are not mandatory. Endoanal ultrasound can be utilized to define the degree of the external and internal anal sphincter defect. In most cases, a significant defect of the anterior portion of both sphincters is present. Anal physiologic testing such as manometry documents the decrease in resting tone and squeeze pressure. Improvement in these parameters can be assessed after surgical repair of traumatic cloaca. Finally, transanal pudendal nerve stimulation provides evaluation of the neuromuscular integrity of the pelvic floor. [11] Pudendal nerve terminal motor latency can be used to predict the success of sphincteroplasty in incontinent patients with anterior sphincter defects. [12] Assessment of fecal incontinence can be performed with any number of validated tests such as the Cleveland Clinic Florida-Wexner Score or the St. Marks’ incontinence score. [1, 13]
Women with a traumatic cloaca defect are incapacitated in many aspects. Severe incontinence may lead to social isolation. Sexual dysfunction may result in depression and other psychological disorders. Both the patient and the obstetrician may be unaware that such defects can be repaired; therefore, traumatic cloaca may go untreated for many years.
The mainstay of conservative treatment is manipulation of stool consistency and frequency. This is done with fiber bulking and constipating medication such as loperamide. While this type of therapy is valid, results are not satisfactory. The anatomic defects that are associated with traumatic cloaca are such that severe incontinence persists despite conservative therapy. Surgical correction is usually necessary.
A variety of surgical techniques have been utilized to repair traumatic cloaca.
Surgical correction of these defects invariably involves the following steps:
Separation of the anorectum from the vagina by dissecting the rectovaginal septum.
Performance of an overlapping sphincteroplasty using both the external and internal anal sphincter, thus repairing the anterior defect.
Reconstruction of the perineal body by plicating the puborectalis muscle to the midline and skin closure.
This surgical technique was originally described by Corman [6] and was used both for traumatic cloaca repair as well as rectovaginal fistula repair. It is the technique most commonly used at our institution.
Preoperatively, all patients are given a full bowel preparation. Intravenous broad-spectrum antibiotics are administered preoperatively and continued for 24 hours post-surgery. The procedure is performed under general anaesthesia and the patient is placed in the prone-jacknife position with the buttocks taped apart and a urinary catheter in place.
The planned incisions are marked on the skin, where an “X” is outlined across the perineum, intersecting at the point of the anovaginal septum. (Fig 3)
The skin incision is performed, and flaps are developed to the ischiorectal fossae bi-laterally. Dissection is then carried out in the ischiorectal fossa cephalad up to the level of the levator muscles. (Fig 4)
Outline of “X” incision on perineum.
Flaps have been developed, and dissection up to the level of the puborectalis (arrow) is performed.
The next step is division of the ano-vagial/rectovaginal septum. Initially, traction sutures are placed in the anal and vaginal mucosa. Then, the septum is divided with electrocautery, and dissection is carried out cephalad up to the level of the puborectalis muscle. (Fig 5) This division allows for mobilization of both the rectal and vaginal “tubes”. During the following step, both edges of the sphincter complex are identified. Mobilization of the sphincter complex is performed lateral to medial. An overlapping sphincteroplasty was then done with interrupted sutures. At all times, a small retractor is placed in the anal canal, in order to prevent stenosis after repair. Midline approximation of the levators is performed with interrupted 2-0 polypropylene sutures, thus filling the space between the anal and vaginal tubes and re-creating a perineal “body”. Overlapping of the “X” flaps is then undertaken. The anterior wall of the anus and the posterior wall of the vagina are sutured to the skin of the newly constructed perineum. Layered closure with absorbable sutures is performed. (Fig 6) No drains are placed, and we do not routinely create a diverting stoma.
Postoperatively, betadine ointment is applied to the wounds daily. Metronidazole is given orally for 7 days. Patients are allowed moderate ambulation immediately; however, sitting is discouraged. Pharmacological bowel confinement is achieved by administering loperamide thrice daily for 5 days. A soft diet is started on the second postoperative day. Fiber supplements are given after discontinuation of loperamide.
Divided anovaginal septum, with posterior vaginal wall superiorly.
Completed repair, with separation of anus and vagina and restoration of perineal body.
Repair at one month after surgical correction.
An alternative technique, especially in patients with a low-profile perineum, (Fig 8) utilizes a transverse incision, made between the anus and vagina. This incision may also be curvilinear. Dissection of the ischiorectal fossae, division of the anovaginal septum, levatoroplasty and overlapping sphincteroplasty are performed in a fashion similar to the patients with the “X” flaps. The transverse incision is closed in a vertical fashion, thus creating a high-profile perineum. (Fig 9)
Patient with a low-profile perineum after obstetric injury. This patient underwent surgical correction utilizing a transverse-to vertical perineal incision.
Postoperative photo of patient in previous figure. Result after vertical closure of incision, with restoration of perineal body.
Instead of primary closure of the skin, which is used in the techniques mentioned previously, perineal closure can be achieved by using island skin flaps.
Draganic and Solomon have reported their experience with island flap perineoplasty and concluded that wound dehiscence was significantly lower when skin flaps rather than simple wound approximation was used for perineal closure. [7] In their series, they performed a faecal diversion in 75% of their patients.
Despite the variety of techniques used in the correction of traumatic cloacal deformities, data on the outcome of these procedures is limited. Most reports are based on small series of patients; meta-analyses regarding results have not been performed.
The largest series of patients utilizing X-flaps for traumatic cloaca correction has been reported by Kaiser [4]. Twelve patients underwent this procedure in a 5-year period. No prophylactic diverting stomas were constructed. Three patients experienced postoperative rectovaginal fistula, and in one of these patients a stoma was necessary. Minor wound complications were observed in 8 patients. After surgical follow-up of 9.83 ± 2.8 months and long-term follow-up of 38.9 ± 6.9 months, all the patients were satisfied with regards to overall function, continence and cosmetic result.
In our own small series of 4 patients, mean hospital length stay was 5 ± 0.8 days (range 4-6). All patients were followed in the outpatient clinic weekly for the first month, then monthly for the next 3 months. Median follow-up was 27 months (range 10 – 52 months). One patient developed a small, superficial separation of the corner of the X-flap. This was resolved by placing sutures. One patient experienced significant constipation at 2 weeks postoperatively. This was treated with manual disimpaction and subsequent small-volume enemas. There were no wound infections, separation of the sphincter repair, or rectovaginal fistulas. At 1 month post-repair there was excellent healing of all wounds. (Fig 7)
At the latest follow-up visit, all patients reported significant improvement in fecal continence. The mean Cleveland Clinic Florida/Wexner score decreased from a mean preoperative value of 18.25 ± 0.95 to a postoperative value of 1.25 ±1.5. All patients required fibre supplement to maintain stool consistency. None of the patients required use of pads, and major incontinence episodes were ameliorated.
Regarding sexual activity, all patients were empirically advised to resume vaginal intercourse 3 months after repair. All patients resumed activity; one patient complained of dyspareunia at the end of the follow-up period.
Venkatesh and Ramanujam [3] reported surgical correction of traumatic cloaca in 44 patients over a 14-year period. The technique involved approximation of the internal and external sphincters together with the transverse perineal muscles in a vest-over- pants manner. The transverse perineal incision was closed vertically. The authors reported excellent functional results with minimal morbidity. A diverting stoma was not performed. Dyspareunia was reported in 27% of patients.
Excellent functional results have been also reported by Abcarian et al. [2] Their group performed surgical correction in 43 patients. Anal physiological testing was done preoperatively and postoperatively; most parameters (resting tone and voluntary squeeze pressures) return to normal after surgery.
Hollingshead et al reported on their experience in repairing traumatic cloacal deformities. [1] Twenty-nine patients underwent surgical repair over a 14-year period. The technique employed a curvilinear incision, division of the anovaginal septum, identification and mobilization of the cut sphincter edges. Then the anorectal defect was closed, a sphincteroplasty performed and the vagina repaired. In this series, the puborectalis or levator muscles were not plicated, and the initial incision was closed longitudinally. A stoma was constructed in 13 patients. Two out of 12 patients in whom a stoma was not fashioned at the time of repair developed a rectovaginal fistula and required subsequent bowel defunctioning. In most patients, improvement in incontinence scores, anophysiologic testing and sexual function.
Traumatic cloacal deformities are rare post-obstetric injuries characterized by diminution or disruption of the perineal body, anterior division and lateral retraction of the anal sphincters, and loss of the distal ano/rectovaginal septum of varying length. [1-6]
These deformities may be a result of a third or fourth degree perineal laceration, which may or may not have been repaired. Failure of repair may be a result of hematoma, infection or technical mishaps. Partial failure may result in a rectovaginal fistula; complete failure leads to a cloacal deformity. [8]
Most, if not all patients have complete loss of defecatory control. This leads to skin irritation, genitourinary infections and a significant degree of sexual dysfunction.
The effect on the quality of life of these patients is profound. Patients feel “deformed”, and may isolate themselves. Medical advice may be deferred for years. On the other hand, physicians may be unaware of the possibility of repair of such lesions and fail to refer to a specialist.
Conservative measures and/or pharmacologic manipulation of bowel movements do not improve quality of life and is not a sustainable treatment modality for this condition. Surgical treatment is required.
As in our small series, other authors have reported prior attempts to repair either a perineal laceration or a cloacal deformity before definitive correction. [2, 4] The timing of this definitive repair is extremely important. Early surgical repair may lead to failure as a result of active inflammation and absence of dissection planes. Therefore, most experts advise a waiting time of at least 3 months after index injury or previous repair before correction is undertaken. [2, 6]
Even though these lesions are rare, a variety of surgical techniques for correction have been described. [2-7] Most employ overlapping sphincteroplasty and division of the rectovaginal septum. Perineal body construction is performed in several ways. Closure at right angles to the original incision is one of the most common modes of perineal reconstruction. Others use island flaps or lotus petal flaps. [7, 10] Some authors advise partial closure of the wound, as to avoid wound infection, which is the most serious complication of cloacal repair. [2, 9]
Anovaginal and perineal reconstruction utilizing X flaps was initially described by Corman. [6] Kaiser reported a series of 12 patients in which improvement in fecal continence and sexual function was significant. [4] Possible downsides of this technique in comparison with others are the intrinsic complexity of the operation and potential problems with wound healing, specifically ischemia of the flaps. On the other hand, partial closure of a perineal wound may take time to heal, and may require multiple dressing changes and visits to clinic.
We believe the most important technical point in correction of cloacal deformities is the division of the anovaginal/rectovaginal septum, with mobilization of the anorectal and vaginal tubes. Herein lays the potential for failure of the repair. Several points regarding this division are important.
Firstly, meticulous hemostasis is mandatory. The vagina is an extremely vascular anatomical structure; large vessels (especially veins) surround it. A fair amount of bleeding may be encountered during anovaginal separation if an incorrect plane is entered. Hematoma formation with potential infection and abscess formation may lead to catastrophic results. Second, care should be taken not to “buttonhole” either the anorectum or vagina during this dissection. Unrepaired perforations in either structure may also lead to septic collections and failure of the operation. Third, interposition of tissue in the space created by the division must be utilized. This constructs a perineal body and supports the sphincteroplasty. It also fills dead space and eliminates the possibility of fluid collections, which, as mentioned before, may lead to failure. Plication of the puborectalis muscle seems to be the easiest choice of tissue for this interposition, since it is in immediate anatomical proximity. Other potential techniques should the puborectalis prove to be inadequate, are utilization of a bulbocarvernosus (Martius) flap, or a gracilis muscle flap. This could add significant complexity to the repair, however.
Some authors advocate construction of a temporary stoma. [1, 10] Proponents of fecal diversion maintain that this allows accelerated perineal healing and avoidance of infection and failure of the repair. The mechanics of defecation in the immediate perioperative period may also potentially lead to disruption of the puborectalis plication or other tissue interposition. Stomas are associated with significant morbidity (skin irritation, prolapse, hernia), and need a second operation for closure. [8, 9] In addition, they do not seem to improve outcomes. Some experts reserve diversion for major failure of the perineal repair. [3, 4]
The most common complications reported after correction of traumatic cloacal deformities are minor wound infection, rectovaginal fistula and breakdown of the perineal wound. These may occur in up to 40% of cases [1]. Expectant therapy and revision of repair are utilized in these instances [1, 4]
After definitive correction of traumatic cloacal deformities, sexual activity and subsequent pregnancies should be deferred; however, the exact “safe” time period remains to be determined. Some authors advise sexual abstinence for at least 6 weeks post repair. [8] Caesarean section after complex perineal repairs seems reasonable.
Finally, improvement in fecal incontinence is significant following cloacal repair in most series even after long-term follow-up. [1, 3, 4].
Traumatic cloacal deformities are the least frequent complications of obstetric injuries to the anorectum. Repair and reconstruction of all anatomical structures involved is feasible with satisfactory cosmetic and functional results. Correction of these deformities can be performed using a variety of techniques. Careful dissection and division of the anovaginal/rectovaginal septum is a crucial step in the creation of a new perineal body. Referral to a specialist colorectal surgeon familiar with repair of traumatic cloacal deformities may improve the quality of life of patients. Fecal diversion is not mandatory.
The Fourier transform has been the basis of digital signal processing since the development of the fast Fourier transform in 1965 by Cooley and Tukey in [1]. Its use for analysis goes back much farther with the development of the Fourier transform by Jean Baptiste Joseph Fourier in 1807 as a solution to thermodynamic equations. By using the Fourier transform, we can take any signal and obtain the amplitude of the sinusoids needed to recreate it. Then we can use this information to obtain the power spectrum of the signal, or we modify the amplitudes and take the inverse Fourier transform of the signal, which then filters the signal.
A fundamental limitation of the Fourier transform is that the all properties of a signal are global in scope. Information about local features of the signal, such as changes in frequency, becomes a global property of the signal in the frequency domain. There have been various methods proposed to address this limitation; the main two are the windowed Fourier transform and wavelets.
Gabor [2] created the windowed Fourier transform in 1946. It applies a window function of a short duration to the signal and the Fourier transform is applied to the resulting data. This method is frequently used; however, there are two limitations with this method. The first is that, since the filtering window is constant, it creates problems if the feature is larger or shorter than the window. The second is that the time resolution is the same for high frequencies as it is for low frequencies. Since as frequency increases, so does the rate of change of the signal, higher frequency signals can have more information in the same period of time as lower frequency signals, and so require a higher time resolution.
Wavelets overcome both these limitations in that the window is scaled in both time and frequency. The term wavelet was introduced by Ricker [3] in 1940 to describe the limited duration functions that he created to model seismic phenomena. The first wavelet was created earlier, in 1910, by Haar [4] as an alternative to the Fourier transform developed in 1807 by Fourier [5]. Work on the wavelet transform preceded slowly through the twentieth century until the 1980’s when work on them increased dramatically with the development of the continuous wavelet transform. In the 1990’s, the discrete wavelet transform and its inverse were developed, allowing filtering and compression of signals.
The wavelet transform has many more modes of operation and other options than the Fourier transform. This is one of the key problems with the use of wavelets; we can feel overwhelmed by all the options we have available with them. This chapter will go through some of these options and demonstrate their use.
The Fourier Transform was first published in 1822 by Joseph Fourier [6]. It converts a mathematical function from the time domain to the frequency domain. This enables us to find new properties of the function that would otherwise be hidden. There are several different variations of the Fourier transform equation. In this chapter, we are using the traditional electrical engineering equation
to convert f(t) to the frequency domain.
The Fourier transform itself is for continuous functions. The Discrete Fourier transform was developed for astronomical observations. The goal was to calculate a trigonometric equation for the orbit of an object in the sky based on observations of its ascensions and declinations at various points in time. Most datasets consist of discrete points sampled in time. These can be converted to the frequency domain as well with the discrete Fourier transform. The computational complexity of this is O(n2).
An interesting note about the Fast Fourier Transform is that it actually predates the Fourier Transform. While the Fast Fourier Transform that we now use was published in a paper by Cooley and Tukey [1] in 1967, a functionally equivalent algorithm was found in an unpublished work by Carl Friedrich Gauss [7] that is presumed to date to 1805. A fascinating history of the Fast Fourier Transform is in [8]. Gauss was computing the discrete Fourier transform of 12 points and noted that the problem could be broken down into subproblems that could simplify the number of steps used [5].
The Fast Fourier Transform reduces the computational complexity of the Discrete Fourier Transform from O(n2) to O(n log2 n). This enables efficient computation of time series. Table 1 shows how the computational complexity increases for an O(n2) process versus an O(n log2 n) process. The difference grows between the two processes until at 1 million data points, the discrete Fourier transform would require over 50,000 times the amount of time that the Fast Fourier transform would require.
n | O(n2) | O(n log2 n) | Ratio |
---|---|---|---|
10 | 100 | 34 | 2.94 |
100 | 10,000 | 665 | 15.04 |
1000 | 1,000,000 | 9966 | 100.34 |
10,000 | 100,000,000 | 132,878 | 752.57 |
100,000 | 10,000,000,000 | 1,660,965 | 6020.60 |
1,000,000 | 1E+12 | 1,9931,569 | 50171.66 |
Computational complexity of O(n2) versus O(n log2 n).
The drawback with the Fourier transform is that all signal information is across the entire range of the transform. As stated in [9], “A local characteristic of the signal becomes a global characteristic of the transform”. As illustrated by other authors [10], the best way that this can be explained is by a score of music as shown in Figure 1.
Opening of Beethoven’s 5th Symphony [11].
The score consists of many different notes, each with a finite duration, each happening at a precise time. A Fourier transform of this signal gives you the average amplitude of the individual frequencies over the entire piece, but obscures the duration and location of the notes. The Fourier power spectrum of music often approximates that of pink (1/f) noise [12]. That information is not lost, since the Fourier transform is reversible, but is encoded in the phase of the Fourier transform.
In 1946, Gabor [2] proposed the windowed Fourier transform as a way to deal with this problem. In it, a window function of a short duration is applied to the signal and the Fourier transform is taken. This is repeated at different locations in the signal. An example of the use of Hamming window function is shown in Figure 2.
Multiple Hamming window functions at successive locations in time.
One limitation of the windowed Fourier transform is that the window length is constant. When a signal feature is much shorter than the window, information about it can be difficult to extract, since the any local property within the time span of the window becomes a global property of the Fourier transform of the window, as noted previously. Conversely, when a signal feature is larger than the windowing function, information about it spans multiple windows, and can also be difficult to extract.
Another limitation is that the time resolution for the windowed Fourier transform is the same for high frequency signals as it is for low frequency signals. The Heisenberg uncertainty principle states that the time resolution of the window is inversely proportional to the frequency resolution. Since a high frequency signal changes much faster than a low frequency signal, it would be ideal to have a transform with better time resolution for high frequency portions of the signal, and better frequency resolution for lower frequency portions of the signal.
Going back to the music score, we can see this by looking at two different notes, Middle C, and one that is two octaves higher, called C6, as shown in Figure 3.
Musical notation for middle C and C6.
The frequency for middle C is 261.63 Hertz, and the frequency for C6 is 1046.50 [13]. The frequency for C6 is quadruple the one for Middle C, which means that for every complete cycle of the middle C note, four complete cycles of C6 have occurred, as shown in Figure 4. The windowed Fourier transform would have the limitation that both notes would be treated equally, when the time resolution for C6 needs to be 4 times that of middle C for analysis purposes.
Amplitude graph for Middle C and C6.
The wavelet transform overcomes the limitation of the windowed Fourier transform by scaling the bandwidth of the filter inversely to the frequency. According to [14], while each box of the windowed Fourier transform has the same bandwidth, each level of the wavelet transform has the same Q as defined as
This gives the transform the desired time resolution for the higher frequency portions of the signal and the desired frequency resolution for lower frequency portions.
The continuous wavelet has a long history spanning from the 1940’s to present. In 1940, Norman Ricker first proposed the term wavelet and various mathematical functions to model seismic waves as they traveled through the Earth’s crust in [3]. He further refined this in a series of papers [15, 16, 17]. This was the first continuous wavelet. The functions in the time domain are given by
called the three-loop equation, and
called the two-loop equation [18]. Graphs for both of these are in Figure 5.
Three loop (a) and two loop (b) Ricker wavelet equation with fm = 1.
The next development for continuous wavelets was in the 1980’s by Grossman and Morlet, and expanded on by Stephen Mallat and others [19]. The term continuous wavelet refers to the fact that it can be scaled to any time scale. Discrete wavelets can only use specific time scales, usually a power of 2.
Wavelet analysis centers around the use of a wavelet function, also called the mother function in literature, traditionally represented by the Greek letter upsilon (ψ). A key requirement is that it has finite energy, i.e.
The energy of the wavelet function is usually one. Functions such as sine and cosine cannot be used as analyzing functions, because they violate this condition by having infinite energy. There is an implicit requirement that, while it has finite energy, it must have some energy, so the integration of the function must be greater than zero.
The second requirement is known as the admissibility condition, which states that the Fourier transform of the wavelet function cannot have a zero-frequency component, i.e.
This can only be satisfied if
A third condition is usually that the wavelet function must have zero mean, which means that it must oscillate, hence be a wavelet. Mathematically this is [20]
Another condition is that the wavelet function has effective support. While the wavelet functions for the continuous wavelets are usually mathematical functions that extend to infinity, effective support means that the wavelet functions are effectively zero outside of a certain range. Since the continuous wavelet functions asymptotically approach 0 as x goes to either ∞ or -∞, the choice of the boundary of this range is a bit arbitrary and can vary from paper to paper.
Morlet and Grossman formalized the continuous wavelet transform in 1984 in [21]. For the continuous wavelet transform, the wavelet function itself is shifted in time and is scaled to do the wavelet transforms [22] as the following equation illustrates:
The continuous wavelet transform is defined as the integration of the function to be analyzed with the complex conjugate of the wavelet function:
In some papers such as [22], you will see the definition of the continuous wavelet transform without the complex conjugate definition. Since most wavelet functions are real valued and not complex, both definitions are equivalent, since the complex conjugate of a real number is equal to that number. The difference only comes up when the wavelet function is complex, such as the Gabor wavelet.
An alternate formula for the continuous wavelet transform is
where Wn(s) is the transformed sequence, xn′ is the original sequence, and ψ* is the complex conjugate of the analyzing wavelet function, n represents the time shift or dilation, and s represents the scale. Usually the time shift is calculated over the total number of data points of the function, and s goes over the scales that are being analyzed to give a two-dimensional picture of the data [23].
The first discrete wavelet was created in 1910 by Alfred Haar as an alternative to the Fourier transform. This consists of two functions as shown in Figure 6, one a scaling function and a wavelet function. The scaling function is the unit step function and the wavelet function consists of offsets from that.
Scaling (a) and wavelet (b) functions for Haar wavelet.
One of the drawbacks of the continuous wavelet transform is that it creates a lot of redundant data, since the coefficients between the scales are highly correlated. Ingrid Daubechies developed the theory of discrete wavelets in 1988, which generates compact data by eliminating the redundancy. Daubechies created an entire family of wavelet functions with the Haar wavelet forming the first level of the Daubechies wavelet.
The wavelet function for discrete wavelets is modified to
where s0 is the scale of the wavelet, usually 2 [20]. This condition as well as the condition that j and k are integers restricts the wavelet to only certain scales. The wavelet function has the properties of finite energy, oscillation, and the admissibility condition of the continuous wavelets, as well as the properties of compact support, vanishing moments, and orthogonality.
Compact support means that the wavelet function is defined by a series of coefficients over a finite region, and is zero at all other places. This contrasts with the continuous wavelets, which, as mentioned, are mathematical functions and have effective support in which the function continues to infinity, but is effectively zero outside of a finite range.
Vanishing moments are obtained when the following condition defined mathematically as
holds true for all integers 0 ≤ k < N, where N is the number of vanishing moments of the function [24]. This property is useful for analyzing functions that have an additive polynomial trend function given by
Here, g(x) is the function to be analyzed and N(x) is the polynomial trend function (also termed a nuisance function in Economics).
The orthogonality condition removes the redundancy of the continuous wavelet transform. As stated earlier, the discrete wavelet transform can only be used at certain scales, most often a power of 2. Mathematically it is stated as
An orthogonal basis ensures that the signal is represented in the most compact way possible. However, by removing all the redundant information, this also removes information to handle shift variance. The exact same function sampled at two different places can yield very different results. In order to deal with this, some discrete wavelet transforms retain some of this redundant information.
Each wavelet of the discrete wavelet family consists of two functions, a wavelet function (ψ), as in the continuous wavelet families, and also a new function called a scaling function (ϕ). In literature, these are termed the mother and father functions respectively. The scaling function has its own admissibility condition, which ensures that it has the zero-frequency component that the wavelet function does not:
This is necessary so that a discrete wavelet transform terminates in a finite number of steps and can completely regenerate the information in the signal [20]. Otherwise, the zero-frequency component could never be captured, since no amount of scaling value can cause the wavelet filter to have a zero-frequency component.
In addition, as specified in [25] the scaling equation is defined in terms of a finite set of coefficients pk that are defined by the following equation
that adheres to the following conditions as specified in [25] as well:
The wavelet function is defined by
where l is the length of the set of coefficients, so that the wavelet coefficients are basically the scaling coefficients in reverse order with alternating signs. These coefficients are used to implement the discrete wavelet transform as a filter bank of Finite Impulse Response (FIR) filters. Graph of the scaling and wavelet functions for Daubechies level 2 wavelet are shown in Figure 7 and the frequency response is shown in Figure 8. As with the Haar wavelet, the wavelet function is a high pass filter and the scaling function is a low pass filter. Both are symmetric around π/2.
Scaling (a) and Wavelet (b) functions for Daubechies level 2 wavelet.
Different papers and software implementations have different coefficients for the Haar and Daubechies wavelet, depending on how they are normalized and whether the scale parameter from Eq. (8) is included is included in the filter. The coefficients for the Haar and the Daubechies level 2 wavelet are in Tables 2 and 3 with b defined by the implementation. Mathematica uses 2 for b, which would normalize the sum of the coefficients to 1. PyWavelets uses
Scaling coefficients | Wavelet coefficients | ||
---|---|---|---|
c0 | 1/b | d0 | 1/b |
c1 | 1/b | d1 | −1/b |
Coefficients for Haar scaling and wavelet functions.
Scaling coefficients | Wavelet coefficients | ||
---|---|---|---|
c0 | d0 | ||
c1 | d1 | ||
c2 | d2 | ||
c3 | d3 |
Coefficients for Daubechies level 2 scaling and wavelet functions.
The class of discrete wavelet functions has many transforms available with the discrete wavelet transform in Figure 9 the most common. Since this was the transform introduced with the Haar wavelet, it is sometimes referred to as the Haar transform [26] as well as the decimated wavelet transform [10]. Essentially, it works as a pyramid algorithm, where the number of coefficients of each lower level is roughly twice that of the preceding level, but each coefficient is influenced by half as much of the data set as the preceding level. Each level has two sets of coefficients, one is called coarse and the other is called details.
Frequency response of scaling function (red) and wavelet function (blue) for Daubechies level 2 wavelet.
In Figure 9, g is the scaling filter defined by the set of scaling filter coefficients and h is the wavelet filter defined by the set of wavelet filter coefficients. At each level, the detail coefficients (W) are outputs, except for the final level, where the coarse coefficients (V) are given as outputs as well. Collectively, this set of coefficients contains enough information to reconstruct the signal perfectly.
One key part of the discrete wavelet transform is the down sampling operator, which is a function that removes every other position from a sequence. An example would be the sequence {a, b, c, d, e, f, g, h} would be {a, c, e, g} or {b, d, f, h}, after the down sampling operator is applied, depending on whether the even or the odd positions are eliminated. Both are valid, however, by convention with the discrete wavelet transform, the even positions are eliminated, leaving only the odd positions. The down sampling operator is what makes the discrete wavelet transform a pyramid function and also reduces the set of coefficients to the minimum amount necessary to reconstruct the signal.
A problem with the decimation operator is aliasing. This is when different sequences map to the same sequence after the application of the operator. An example would be that the sequences {a, b, c, d, e, f, g} and {a, h, c, i, e, j, g} would both map to the sequence {a, c, d, g}. Therefore, just given the sequence {a, c, d, g}, it would be impossible to reconstruct the original. The filters of the discrete wavelets are designed to compensate for this, ensuring that the original sequence can be recovered. The combination of these filters with the down sampling operator is referred to as decimation.
The discrete wavelet transform also has an inverse transform. This process combines as described in Figure 10 to form a perfect reconstruction of the signal, where
Diagram of a three-level discrete wavelet transform.
Implementing the discrete wavelet transform as a finite impulse response filter and using decimation gives it a computational complexity of O(n). As Table 4 shows, an O(n) process can be much faster than an O(n log2 n) process such as the fast Fourier Transform. At 1 million samples, an O(n) process requires almost 20 times less operations than an O(n log2 n) process (Table 4).
n | O(n log2 n) | O(n) | Ratio |
---|---|---|---|
10 | 34 | 10 | 3.40 |
100 | 665 | 100 | 6.65 |
1000 | 9966 | 1000 | 9.97 |
10,000 | 132,878 | 10,000 | 13.29 |
100,000 | 1,660,965 | 100,000 | 16.61 |
1,000,000 | 19,931,569 | 1,000,000 | 19.93 |
Computational complexity of O(n log2 n) versus O(n).
Another wavelet transform for discrete wavelet functions is the stationary wavelet transform, also known as the undecimated discrete wavelet transform. Essentially the stationary wavelet transform is the discrete wavelet transform without the decimation operation for the data. Whereas the number of coefficients for each level is half that of the preceding level in the discrete wavelet transform, the number of coefficients is the same for each level in the stationary wavelet transform.
The procedure is diagrammed in Figure 11, where gn is the set of the scaling filter coefficients and hn is the set of the wavelet filter coefficients. The reason that the scaling filter and wavelet filter coefficients are different for each level is that instead of the decimation operator being applied to the wavelet data coefficients after each level, the upsampling operator is applied to the wavelet and scaling filter coefficients. The wavelet and scaling coefficients for each level are upsampled from the previous level, as shown in Figure 12.
Diagram of a three-level inverse discrete wavelet transform.
Diagram of a three-level stationary wavelet transform.
Like the discrete wavelet transform, the stationary wavelet transform has an inverse transform, as shown in Figure 13. The difference between this and the inverse discrete wavelet transform is the absence of the upsampling operator. As with the stationary wavelet transform, the filter coefficients for the inverse stationary wavelet transform are changed instead of the data. In this case, the filters are down sampled. The retention of redundant data in the stationary wavelet transform helps to make it translation invariant, which is useful for filtering applications (Figure 13).
Diagram of a filter upsampling for the stationary wavelet transform.
Diagram of a three-level inverse stationary wavelet transform.
Since the decimation step is not used, the stationary wavelet transform has a computational complexity of O(n log2 n), the same as the Fast Fourier Transform. However, there is also memory complexity to consider. While the Fast Fourier Transform and the Discrete Wavelet Transform has an O(n) memory complexity, the stationary wavelet transform has an O(n log2 n) memory complexity. Therefore, the output will always be larger than the input.
The two previous transforms applied the detail and the coarse filters to the data at each level. The output of the coarse filter is given as the input to the next level and the output of the detail filter at that level is included in the set of the outputs of the transform. In the final level, the output of both the detail and the coarse filters were included in the set of outputs of the transform; however, that is not the only possibility. The packet transform creates a binary tree where the detail and coarse filters are applied to each node, diagrammed in Figure 14. The output of the detail filter becomes one child and the output of the coarse filter becomes the other. This process is repeated until the final level is reached, creating a set of output coefficients where each set is identified by the sequence of filters applied to it.
Diagram of a three-level discrete wavelet packet transform.
The stationary wavelet packet transform is yet another transform for discrete wavelet functions. Basically, it combines the stationary wavelet transform with the wavelet packet transform, as diagrammed in Figure 15. Instead of the decimation operator, the filters themselves are upsampled for each level. The transform creates a binary tree, as with the discrete wavelet packet transform, where both filters for each level are applied at each node. As with the wavelet packet transform, the output from the detail filter becomes one child and the output from the scaling filter becomes the other, and the process is repeated until the final level is reached. Each set of output coefficients are also identified by the sequence of filters applied to it, with the difference that since there is no decimation applied between levels, the number of each set of output coefficients is the same as the input data. This leads to the total number of output coefficients to be 2 times the number of levels multiplied by the length of the input data. Both the discrete wavelet packet transform and the stationary wavelet packet transform have inverse transforms.
Diagram of a three-level stationary wavelet packet transform.
The wavelet packet transform introduces many more possibilities for use, some of which are discussed here. Depending on the application, you can do different combinations of the scaling and wavelet filters. Computational complexity depends on the filter combinations selected. If it is taken to the maximum level with the maximum filter combinations, then the discrete wavelet packet transform has a complexity of O(n log2 n) and the stationary wavelet packet transform has a complexity of O(n2).
The Fast Fourier Transform has been listed as one of the top algorithms of the 20th century [27]. Its development has been instrumental to digital signal processing. However, recently a new algorithm, the wavelet transform, has started to have a significant impact on digital signal processing. The wavelet transform improves on the Fourier Transform in that it can analyze a signal by time and frequency simultaneously, thereby easily recovering localized signal information. This is key to many applications, including fractal and multifractal analysis, compression, and filtering.
The wavelet transform introduces many possibilities for use and this chapter has only touched the surface of it. Different wavelets can be used and the transform itself can be customized to fit the application as shown with the wavelet packet transform. Future research will be to determine the proper combination of features for various applications. In addition, there are other possibilities, such as the lifting wavelet transform, which wasn’t covered in this chapter. Only orthogonal wavelets that use the same set of wavelets for the forward and inverse transform were covered in this chapter. Biorthogonal wavelets that use different wavelets for the forward and inverse transforms are also available.
The key to wavelet compression and filtering is the sparse signal representation generated by the wavelet transform. The wavelet transform can reduce a signal to minimal set of coefficients. Coefficients that are near zero can be rounded to zero, reducing the size of the signal. In addition, fractional parts of the coefficients can be rounded, also reducing signal size. One of the first uses of this was to compress fingerprints for the FBI [28]. As stated in [29], in the 1990’s the FBI had 25 million cards, each containing 10 fingerprints. Digitized, each card contained 10 megabytes of information, for a total of 250 terabytes. Using the two-dimensional discrete wavelet packet transform gives a compression ratio of 20 to 1, enabling the archive to be stored on approximately 12.5 terabytes, while still being able to search and match unknown fingerprints against the ones in the archive. The recently developed JPEG format at the time was based on using the discrete cosine transform on blocks of the image, which left unacceptable artifacts in the image.
JPEG 2000 was developed using the two-dimensional wavelet transform to be the successor to JPEG, although it hasn’t caught on. JPEG 2000 allows both lossy and lossless compression. It also doesn’t have the lossy artifact generation that the JPEG format has as mentioned previously. Both lossy and lossless compression use the discrete wavelet transform, the difference is that the lossless one uses a wavelet transform that is reversible, while the lossy one uses a wavelet transform that introduces quantization noise that making it irreversible.
Compressed sensing deals with the fact that we that we can obtain a vast amount of information and a lot of it can be discarded and still retain what is relevant. As stated in [24], “singularities and irregular structures often carry the most important information in signals.” This is due to the fact that they represent changes to one or more of the properties of the signal. An example of this would be the edges in an image. Compressed sensing removes the redundant, unnecessary information from a signal and analyzes the remaining part of the signal. This is an ideal application for the wavelet transform.
The discrete wavelet transform has been used for Iris recognition for biometric identification in patent US 2002O150281A1 [30]. After taking a picture of the eye, the iris is extracted from the image and then converted to polar coordinates. Using the discrete wavelet transform, the high frequency components are extracted, which are the detail coefficients as referenced in this paper. These form the characteristic vector that is used to identify an iris from the previously recorded data.
The wavelet transform can provide an efficient way to filter white noise from a signal. The procedure consists of applying one of the discrete wavelet transforms to the data and then executing a threshold algorithm that modifies the detail coefficients. After the coefficients are modified, then the inverse transform is applied; the resulting output is a representation of the signal with the noise component significantly reduced.
There are numerous packages available for experimenting with the wavelet transform. The discrete and stationary wavelet transforms are available in Mathematica, Maple, Matlab, R, and PyWavelets to name a few, with the wavelet packet transform available in Mathematica, Matlab, and PyWavelets.
The wavelet transform provides many possibilities for signal analysis depending on the application. A few potential applications were touched on here. The reader is encouraged to develop their own uses and applications for the wavelet transform.
IntechOpen publishes different types of publications
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