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  1. Image Transforms Instructed by : J. Shanbezadeh Email : Shanbehzadeh@gmail.com Jamshid Shanbehzadeh

  2. Contents • Introduction • Applications of Image Transforms • Types of Image Transforms • 2-D Transform • Basis Image (m1,m2) • Reverse 2-D Transform • Basis Inverse Transform Image (m1,m2) • 2-D Unitary Transform • Separable Transform • Forward Transform • Reverse Transform • Unitary Matrix(Transform) • Orthogonal Matrix(Transform) • Separable Transform • Forward Separable Transform • Inverse Separable Transform • Fourier Transform • Forward Fourier Transform • Inverse Fourier Transform • Fourier Transform(Separable) • Fourier Transform Basis Functions • Fourier Transform Properties • Fourier Transform Phase Information • Translation Property • Rotation Property • Cosine transform • Basis functions • Basis Images • Cosine symmetry • Sine Transform • Basis functions • 2-D sine transform • Hartley Transform • Hadamard Transform • Basis Functions • Basis Images • Principle Components Analysis Jamshid Shanbehzadeh

  3. Applications of Image Transforms • Extracting Features from Images • In Fourier Transform, the average dc term is proportional to the average image amplitude • Image Compression • Dimensionality Reduction Jamshid Shanbehzadeh

  4. Contents • Introduction • Applications of Image Transforms • Types of Image Transforms • 2-D Transform • Basis Image (m1,m2) • Reverse 2-D Transform • Basis Inverse Transform Image (m1,m2) • 2-D Unitary Transform • Separable Transform • Forward Transform • Reverse Transform • Unitary Matrix(Transform) • Orthogonal Matrix(Transform) • Separable Transform • Forward Separable Transform • Inverse Separable Transform • Fourier Transform • Forward Fourier Transform • Inverse Fourier Transform • Fourier Transform(Separable) • Fourier Transform Basis Functions • Fourier Transform Properties • Fourier Transform Phase Information • Translation Property • Rotation Property • Cosine transform • Basis functions • Basis Images • Cosine symmetry • Sine Transform • Basis functions • 2-D sine transform • Hartley Transform • Hadamard Transform • Basis Functions • Basis Images • Principle Components Analysis Jamshid Shanbehzadeh

  5. Types of Image Transforms • Unitary Transforms • Fourier Transforms • Cosine, Sine, Hartley Transforms • Hadamard, Haar • Wavelet Transforms • Ridglet, Curvelet, Contourlet Jamshid Shanbehzadeh

  6. Contents • Introduction • Applications of Image Transforms • Types of Image Transforms • 2-D Transform • Basis Image (m1,m2) • Reverse 2-D Transform • Basis Inverse Transform Image (m1,m2) • 2-D Unitary Transform • Separable Transform • Forward Transform • Reverse Transform • Unitary Matrix(Transform) • Orthogonal Matrix(Transform) • Separable Transform • Forward Separable Transform • Inverse Separable Transform • Fourier Transform • Forward Fourier Transform • Inverse Fourier Transform • Fourier Transform(Separable) • Fourier Transform Basis Functions • Fourier Transform Properties • Fourier Transform Phase Information • Translation Property • Rotation Property • Cosine transform • Basis functions • Basis Images • Cosine symmetry • Sine Transform • Basis functions • 2-D sine transform • Hartley Transform • Hadamard Transform • Basis Functions • Basis Images • Principle Components Analysis Jamshid Shanbehzadeh

  7. 2-D Transforms Forward transform of the N1*N2 image array F(n1,n2) : F(0,0)=f(0,0).A(0,0,0,0)+f(0,1)A(0,1,0,0)+f(0,2)A(0,2,0,0)+….+f(0,N2-1)A(0,N2-1,0,0) : the Forward Transform Kernel • به ازای هر m1 و m2 یک تصویر پایه ساخته می شود. • n1 و n2 پیکسلهای تصویر در فضای جدید هستند. Jamshid Shanbehzadeh

  8. Basis Image (m1,m2) Jamshid Shanbehzadeh

  9. Basis Image (m1,m2) JamshidShanbehzadeh

  10. Jamshid Shanbehzadeh

  11. پیکسلهای تصویر اصلی را در پیکسلهای تصویر پایه، نظیر به نظیر در یکدیگر ضرب داخلی می نماییم. Jamshid Shanbehzadeh

  12. مقایسه تبدیل یک بعدی و دوبعدی یعنی کل تصویر را پیمایش می نمایند. Jamshid Shanbehzadeh

  13. Reverse 2-D Transforms A reverse or inverse transformation provides a mapping from the transform domain to the image space as given by : B(n1,n2; m1,m2) : the Inverse Transform Kernel کرنل مورد استفاده در تبدیل تصاویر بایستی معکوس پذیر باشد. Jamshid Shanbehzadeh

  14. Basis Inverse Transform Image (m1,m2) Jamshid Shanbehzadeh

  15. به ازای هر n1 و n2 یک تصویر پایه ساخته میشود، اگر تصاویر پایه بر هم عمود باشند. Jamshid Shanbehzadeh

  16. 2-D Unitary Transforms The transformation is unitary if the following orthonormality conditions are met: Jamshid Shanbehzadeh

  17. Inner Product Jamshid Shanbehzadeh

  18. Inner Product Jamshid Shanbehzadeh

  19. ضرب داخلی تصاویر Image Size(IS) =512 X 512 Number of Operations = IS X IS(Mul)+(IS X IS-1) (Addition) for one element =512 X 512(Mul) +(512 X 512 -1)(Addition) Number of operations for all = 512 X 512 (512 X 512(Mul) +(512 X 512 -1)(Addition) ) Jamshid Shanbehzadeh

  20. بلوک بندی تصاویر برای کاهش حجم محاسبات، تصاویر را به بلاکهایی تقسیم می نماییم: Image Size(IS) =512 X 512 Block Size(BS) =8 X 8 Number of Blocks(NB) =128 X 128 Size of Basis Image(SBI) =8 X 8 Number of Operations = NB X {BS X SBI(Mul)+(BS-1) (Addition)} =128 X 128{(64 X 64)Mult+63(Addition)} Number Operations = 67,108,864(Multiplications)+1,032,192(additions) حجم محاسبات علیرغم کاهش، زیاد است. Jamshid Shanbehzadeh

  21. Matrix multiplication • If we perform matrix multiplication, then we have for two N X N matrixes: • Number of operations (NO)= N X N {N(Mul) + (N-1)(addition)} • Number of Image Blocks (NIB) = Image Size/(NXN) • Total Number of Operations(TNO)=NIB X NO 16,384X(512(Mult)+448(additions))=8,388,608(Multi)+7,340,032(Additions) حجم محاسبات بسیار کاهش می باشد. Jamshid Shanbehzadeh

  22. Contents • Introduction • Applications of Image Transforms • Types of Image Transforms • 2-D Transform • Basis Image (m1,m2) • Reverse 2-D Transform • Basis Inverse Transform Image (m1,m2) • 2-D Unitary Transform • Separable Transform • Forward Transform • Reverse Transform • Unitary Matrix(Transform) • Orthogonal Matrix(Transform) • Separable Transform • Forward Separable Transform • Inverse Separable Transform • Fourier Transform • Forward Fourier Transform • Inverse Fourier Transform • Fourier Transform(Separable) • Fourier Transform Basis Functions • Fourier Transform Properties • Fourier Transform Phase Information • Translation Property • Rotation Property • Cosine transform • Basis functions • Basis Images • Cosine symmetry • Sine Transform • Basis functions • 2-D sine transform • Hartley Transform • Hadamard Transform • Basis Functions • Basis Images • Principle Components Analysis Jamshid Shanbehzadeh

  23. Separable Transforms The transformation is said to be separable if its kernels can be written in the form Where the kernel subscripts indicate row and column one-dimensional transform operations. Jamshid Shanbehzadeh

  24. Separable Transforms A separable two-dimensional unitary transform can be computed in two steps: First, a one-dimensional transform is taken along each column of the image, yielding Next, a second one-dimensional unitary transform is taken along each row of P(m1,m2), giving Jamshid Shanbehzadeh

  25. Separable Transforms Jamshid Shanbehzadeh

  26. Contents • Introduction • Applications of Image Transforms • Types of Image Transforms • 2-D Transform • Basis Image (m1,m2) • Reverse 2-D Transform • Basis Inverse Transform Image (m1,m2) • 2-D Unitary Transform • Separable Transform • Forward Transform • Reverse Transform • Unitary Matrix(Transform) • Orthogonal Matrix(Transform) • Separable Transform • Forward Separable Transform • Inverse Separable Transform • Fourier Transform • Forward Fourier Transform • Inverse Fourier Transform • Fourier Transform(Separable) • Fourier Transform Basis Functions • Fourier Transform Properties • Fourier Transform Phase Information • Translation Property • Rotation Property • Cosine transform • Basis functions • Basis Images • Cosine symmetry • Sine Transform • Basis functions • 2-D sine transform • Hartley Transform • Hadamard Transform • Basis Functions • Basis Images • Principle Components Analysis Jamshid Shanbehzadeh

  27. Forward Transform F and f denote the matrix and vector representations of a signal array. F and f be the matrix and vector forms of the transformed signal. The two-dimensional unitary transform is given by F=Af Where A is the forward transformation matrix. Jamshid Shanbehzadeh

  28. Contents • Introduction • Applications of Image Transforms • Types of Image Transforms • 2-D Transform • Basis Image (m1,m2) • Reverse 2-D Transform • Basis Inverse Transform Image (m1,m2) • 2-D Unitary Transform • Separable Transform • Forward Transform • Reverse Transform • Unitary Matrix(Transform) • Orthogonal Matrix(Transform) • Separable Transform • Forward Separable Transform • Inverse Separable Transform • Fourier Transform • Forward Fourier Transform • Inverse Fourier Transform • Fourier Transform(Separable) • Fourier Transform Basis Functions • Fourier Transform Properties • Fourier Transform Phase Information • Translation Property • Rotation Property • Cosine transform • Basis functions • Basis Images • Cosine symmetry • Sine Transform • Basis functions • 2-D sine transform • Hartley Transform • Hadamard Transform • Basis Functions • Basis Images • Principle Components Analysis Jamshid Shanbehzadeh

  29. Reverse Transform The inverse transform is f = Bf B represents the inverse transformation matrix B = A-1 Jamshid Shanbehzadeh

  30. Contents • Introduction • Applications of Image Transforms • Types of Image Transforms • 2-D Transform • Basis Image (m1,m2) • Reverse 2-D Transform • Basis Inverse Transform Image (m1,m2) • 2-D Unitary Transform • Separable Transform • Forward Transform • Reverse Transform • Unitary Matrix(Transform) • Orthogonal Matrix(Transform) • Separable Transform • Forward Separable Transform • Inverse Separable Transform • Fourier Transform • Forward Fourier Transform • Inverse Fourier Transform • Fourier Transform(Separable) • Fourier Transform Basis Functions • Fourier Transform Properties • Fourier Transform Phase Information • Translation Property • Rotation Property • Cosine transform • Basis functions • Basis Images • Cosine symmetry • Sine Transform • Basis functions • 2-D sine transform • Hartley Transform • Hadamard Transform • Basis Functions • Basis Images • Principle Components Analysis Jamshid Shanbehzadeh

  31. Unitary Matrix (Transform) For a unitary transformation, the matrix inverse is given by A-1 = A*T A is said to be a unitary matrix Jamshid Shanbehzadeh

  32. Contents • Introduction • Applications of Image Transforms • Types of Image Transforms • 2-D Transform • Basis Image (m1,m2) • Reverse 2-D Transform • Basis Inverse Transform Image (m1,m2) • 2-D Unitary Transform • Separable Transform • Forward Transform • Reverse Transform • Unitary Matrix(Transform) • Orthogonal Matrix(Transform) • Separable Transform • Forward Separable Transform • Inverse Separable Transform • Fourier Transform • Forward Fourier Transform • Inverse Fourier Transform • Fourier Transform(Separable) • Fourier Transform Basis Functions • Fourier Transform Properties • Fourier Transform Phase Information • Translation Property • Rotation Property • Cosine transform • Basis functions • Basis Images • Cosine symmetry • Sine Transform • Basis functions • 2-D sine transform • Hartley Transform • Hadamard Transform • Basis Functions • Basis Images • Principle Components Analysis Jamshid Shanbehzadeh

  33. Orthogonal Matrix (Transform) A real unitary matrix is called an orthogonal matrix. For such a matrix, A-1 = AT Jamshid Shanbehzadeh

  34. Contents • Introduction • Applications of Image Transforms • Types of Image Transforms • 2-D Transform • Basis Image (m1,m2) • Reverse 2-D Transform • Basis Inverse Transform Image (m1,m2) • 2-D Unitary Transform • Separable Transform • Forward Transform • Reverse Transform • Unitary Matrix(Transform) • Orthogonal Matrix(Transform) • Separable Transform • Forward Separable Transform • Inverse Separable Transform • Fourier Transform • Forward Fourier Transform • Inverse Fourier Transform • Fourier Transform(Separable) • Fourier Transform Basis Functions • Fourier Transform Properties • Fourier Transform Phase Information • Translation Property • Rotation Property • Cosine transform • Basis functions • Basis Images • Cosine symmetry • Sine Transform • Basis functions • 2-D sine transform • Hartley Transform • Hadamard Transform • Basis Functions • Basis Images • Principle Components Analysis Jamshid Shanbehzadeh

  35. Separable Transforms If the transform kernels are separable such that Where AR and AC are row and column unitary transform matrices. Jamshid Shanbehzadeh

  36. Forward Separable Transforms The transformedimagematrixcan be obtained from the image matrix by F Jamshid Shanbehzadeh

  37. Inverse Separable Transforms The inversetransformation is given by F = BC F BRT Where BC = AC-1 and BR = AR-1 Jamshid Shanbehzadeh

  38. Contents • Introduction • Applications of Image Transforms • Types of Image Transforms • 2-D Transform • Basis Image (m1,m2) • Reverse 2-D Transform • Basis Inverse Transform Image (m1,m2) • 2-D Unitary Transform • Separable Transform • Forward Transform • Reverse Transform • Unitary Matrix(Transform) • Orthogonal Matrix(Transform) • Separable Transform • Forward Separable Transform • Inverse Separable Transform • Fourier Transform • Forward Fourier Transform • Inverse Fourier Transform • Fourier Transform(Separable) • Fourier Transform Basis Functions • Fourier Transform Properties • Fourier Transform Phase Information • Translation Property • Rotation Property • Cosine transform • Basis functions • Basis Images • Cosine symmetry • Sine Transform • Basis functions • 2-D sine transform • Hartley Transform • Hadamard Transform • Basis Functions • Basis Images • Principle Components Analysis Jamshid Shanbehzadeh

  39. Forward Fourier Transform کرنل دوبعدی جدایی پذیر Jamshid Shanbehzadeh

  40. مقایسه تبدیل یک بعدی و دوبعدی Jamshid Shanbehzadeh

  41. Inverse Fourier Transform Fourier Transform : Inverse Fourier Transform : Jamshid Shanbehzadeh

  42. Fourier Transform (Separable) تبدیل دوبعدی را به صورت سینوسی و کسینوسی می نویسیم: Jamshid Shanbehzadeh

  43. Fourier Transform (Separable) Jamshid Shanbehzadeh

  44. Fourier transform basis functions , N=16 Jamshid Shanbehzadeh

  45. قسمت موهومی تصاویر پایه DFT قسمت حقیقی مقادیر تصاویر پایه DFT Jamshid Shanbehzadeh

  46. اندازه تبدیل فوریه تصویر اصلی (مبدا به وسط انتقال یافته است.) تصویر اصلی اندازه تبدیل فوریه تصویر اصلی با استفاده از لگاریتم اندازه ها فاز تبدیل فوریه Jamshid Shanbehzadeh

  47. تصویر اصلی تبدیل فوریه آن Jamshid Shanbehzadeh

  48. حساسیت تبدیل فوریه به چرخش دو نمونه تصویر اصلی تبدیل فوریه تصاویر چرخش یافته تصاویر تبدیل فوریه تصاویر Jamshid Shanbehzadeh

  49. Fourier Transform Properties The spectral component at the origin of the Fourier domain is equal to N times the spatial average of the image plane. Jamshid Shanbehzadeh

  50. Zero-frequency term at the center Multiplying the image function by factor (-1)j+k یعنی با ضرب f(x,y) در (-1)x+y مبدا تبدیل فوریه f(x,y) به مرکز مربع فرکانسی N X N متناظرش انتقال داده می شود. Jamshid Shanbehzadeh