Convert image to pdf with Imagemagick keeping image resolution and placing it on top left corner

I found a solution of this problem. Here the bash code:

#!/bin/bash

if [ -z "$1" ] || [ -z "$2" ]
  then
    echo "Usage: $0 <input> <output>"
        exit 1;
fi


max () {
 echo $(( $1 > $2 ? $1 : $2 ))
}

IMAGE_X=$(identify -format "%w" "$1");
IMAGE_Y=$(identify -format "%h" "$1");

IMAGE_DENSITY_X=$(( $IMAGE_X * 100 / 827 ))
IMAGE_DENSITY_Y=$(( $IMAGE_Y * 100 / 1169 ))
IMAGE_DENSITY=$(max $IMAGE_DENSITY_X $IMAGE_DENSITY_Y)

echo "Image density: $IMAGE_DENSITY"

PAGE_SIZE_X=$(( $IMAGE_DENSITY * 827 / 100 ))
PAGE_SIZE_Y=$(( $IMAGE_DENSITY * 1169 / 100 )) 

echo "Page size X: $PAGE_SIZE_X"
echo "Page size Y: $PAGE_SIZE_Y"

OFFSET_X=$(( ($PAGE_SIZE_X - $IMAGE_X) / 2 * 72 / $IMAGE_DENSITY ))
OFFSET_Y=$(( ($PAGE_SIZE_Y - $IMAGE_Y) / 2 * 72 / $IMAGE_DENSITY )) 

echo "$IMAGE_DENSITY $PAGE_SIZE_X $OFFSET_X $OFFSET_Y"

convert "$1" \
-page ${PAGE_SIZE_X}x${PAGE_SIZE_Y}+${OFFSET_X}+${OFFSET_Y} \
-units PixelsPerInch \
-density $IMAGE_DENSITY \
-format pdf \
"$2"