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# Linear Algebra

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by

Tweet## Shravan Palyam

on 19 April 2013#### Transcript of Linear Algebra

Applied Mathematics Assignment By

Tanvi Ram Sunku

(1RV09TE057) Contents Image Processing

Representation of an image using matrices.

Operations on the images.

Image compression using SVD Image Processing What is a Digital Image? Image- collection of pixels Grey scale to Color Representation of an image in matrix form Example: 3×3 grayscale image Figure : Each layer is a grayscale image matrix, but when overlayed using

color, produce the full RGB spectrum OPERATIONS ON IMAGES USING MATRICES VERTICALLY FLIPPED IMAGE ORIGINAL IMAGE Flip Vertical

By multiplying the Original Matrix by the Inverted Identity Matrix, we can flip the image vertically. What this does is switch the bottom row with the top row and the second row with the second to the last row and so on. HORIZONTALLY FLIPPED IMAGE ORIGINAL IMAGE Flip Horizontal

By multiplying the Original Matrix to the Inverted Identity Matrix, we can flip the image horizontally. What this does is switch the first column with the last column and the second column with the second to the last column and so on. IMAGE ROTATED TO LEFT ORIGINAL IMAGE Rotate Left

By taking the Transpose of the Original Matrix, we can rotate the image to the left. What this does is take the first row of the matrix and make it the first column and then the second row becomes the second column and so on. IMAGE ROTATED TO RIGHT ORIGINAL IMAGE Rotate Right

By multiplying the Original Matrix to the Inverted Identity Matrix and then taking the Transpose of the Matrix, we can rotate the image to the right. What this does is first flipping the image vertically then transposing the matrix as in the example above. INVERTED IMAGE 0<--------------------------------------------------------------------------------------->255 Invert

There are 256 colors in the computers color spectrum, represented by the numbers 0 to 255. By subtracting the numbers in the Original Matrix by 255, we can get the color that is the opposite of the original color in the position of the original number. For example the number 1 (red) was in the top left hand corner of the Original Matrix, now 255 - 1 = 254 which is in the blue part of the color spectrum. LIGHTENED IMAGE ORIGINAL IMAGE Lighten

By multiplying the Original Matrix by 0.9 we are decreasing the value of the colors by 10%. The lower color values cause the image to appear lighter. DARKENED IMAGE ORIGINAL IMAGE Darken

By multiplying the Original Matrix by 1.1 we are increasing the value of the colors by 10%. The higher color values cause the image to appear darker. SMOOTH IMAGE ORIGINAL IMAGE Smooth

Blurs the active image or selection. This filter replaces each pixel with the average of its 3x3 neighbors. Introduction to SVD Data compression is an important application of linear algebra. The need to minimize the amount of digital information stored and transmitted is an ever growing concern in the modern world. Singular Value Decomposition is an effective tool for minimizing data storage and data transfer. SVD Overview Steps in SVD Therefore SVD helps in efficient utilization of bandwidth by condensing the image into smaller parts Conclusion Thank You

Full transcriptTanvi Ram Sunku

(1RV09TE057) Contents Image Processing

Representation of an image using matrices.

Operations on the images.

Image compression using SVD Image Processing What is a Digital Image? Image- collection of pixels Grey scale to Color Representation of an image in matrix form Example: 3×3 grayscale image Figure : Each layer is a grayscale image matrix, but when overlayed using

color, produce the full RGB spectrum OPERATIONS ON IMAGES USING MATRICES VERTICALLY FLIPPED IMAGE ORIGINAL IMAGE Flip Vertical

By multiplying the Original Matrix by the Inverted Identity Matrix, we can flip the image vertically. What this does is switch the bottom row with the top row and the second row with the second to the last row and so on. HORIZONTALLY FLIPPED IMAGE ORIGINAL IMAGE Flip Horizontal

By multiplying the Original Matrix to the Inverted Identity Matrix, we can flip the image horizontally. What this does is switch the first column with the last column and the second column with the second to the last column and so on. IMAGE ROTATED TO LEFT ORIGINAL IMAGE Rotate Left

By taking the Transpose of the Original Matrix, we can rotate the image to the left. What this does is take the first row of the matrix and make it the first column and then the second row becomes the second column and so on. IMAGE ROTATED TO RIGHT ORIGINAL IMAGE Rotate Right

By multiplying the Original Matrix to the Inverted Identity Matrix and then taking the Transpose of the Matrix, we can rotate the image to the right. What this does is first flipping the image vertically then transposing the matrix as in the example above. INVERTED IMAGE 0<--------------------------------------------------------------------------------------->255 Invert

There are 256 colors in the computers color spectrum, represented by the numbers 0 to 255. By subtracting the numbers in the Original Matrix by 255, we can get the color that is the opposite of the original color in the position of the original number. For example the number 1 (red) was in the top left hand corner of the Original Matrix, now 255 - 1 = 254 which is in the blue part of the color spectrum. LIGHTENED IMAGE ORIGINAL IMAGE Lighten

By multiplying the Original Matrix by 0.9 we are decreasing the value of the colors by 10%. The lower color values cause the image to appear lighter. DARKENED IMAGE ORIGINAL IMAGE Darken

By multiplying the Original Matrix by 1.1 we are increasing the value of the colors by 10%. The higher color values cause the image to appear darker. SMOOTH IMAGE ORIGINAL IMAGE Smooth

Blurs the active image or selection. This filter replaces each pixel with the average of its 3x3 neighbors. Introduction to SVD Data compression is an important application of linear algebra. The need to minimize the amount of digital information stored and transmitted is an ever growing concern in the modern world. Singular Value Decomposition is an effective tool for minimizing data storage and data transfer. SVD Overview Steps in SVD Therefore SVD helps in efficient utilization of bandwidth by condensing the image into smaller parts Conclusion Thank You